{"id":1422,"date":"2018-02-06T16:19:17","date_gmt":"2018-02-06T16:19:17","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/?post_type=chapter&#038;p=1422"},"modified":"2018-02-06T16:19:17","modified_gmt":"2018-02-06T16:19:17","slug":"6-chapter-review","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/6-chapter-review\/","title":{"raw":"6 Chapter Review","rendered":"6 Chapter Review"},"content":{"raw":"<div class=\"os-glossary-container\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3><span class=\"os-text\">Key Terms<\/span><\/h3>\r\n<dl id=\"fs-id1165039331406\">\r\n \t<dt id=\"33239\"><strong>banked curve<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165039440814\">curve in a road that is sloping in a manner that helps a vehicle negotiate the curve<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165039353624\">\r\n \t<dt id=\"31765\"><strong>centripetal force<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165039106544\">any net force causing uniform circular motion<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165039329539\">\r\n \t<dt id=\"46238\"><strong>Coriolis force<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165039425267\">inertial force causing the apparent deflection of moving objects when viewed in a rotating frame of reference<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165039331836\">\r\n \t<dt id=\"47300\"><strong>drag force<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165039450582\">force that always opposes the motion of an object in a fluid; unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165038031500\">\r\n \t<dt id=\"72218\"><strong>friction<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165037166731\">force that opposes relative motion or attempts at motion between systems in contact<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165039255178\">\r\n \t<dt id=\"19445\"><strong>ideal banking<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165035708390\">sloping of a curve in a road, where the angle of the slope allows the vehicle to negotiate the curve at a certain speed without the aid of friction between the tires and the road; the net external force on the vehicle equals the horizontal centripetal force in the absence of friction<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165035708392\">\r\n \t<dt id=\"82109\"><strong>inertial force<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165039075055\">force that has no physical origin<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165038337854\">\r\n \t<dt id=\"77544\"><strong>kinetic friction<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165036846768\">force that opposes the motion of two systems that are in contact and moving relative to each other<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165039512549\">\r\n \t<dt id=\"89165\"><strong>noninertial frame of reference<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165039234658\">accelerated frame of reference<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165036846773\">\r\n \t<dt id=\"36669\"><strong>static friction<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165038327277\">force that opposes the motion of two systems that are in contact and are not moving relative to each other<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165039337073\">\r\n \t<dt id=\"65315\"><strong>terminal velocity<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165035673837\">constant velocity achieved by a falling object, which occurs when the weight of the object is balanced by the upward drag force<\/dd>\r\n<\/dl>\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"os-key-equations-container\">\r\n<div class=\"textbox shaded\">\r\n<h3><span class=\"os-text\">Key Equations<\/span><\/h3>\r\n<section id=\"fs-id1165035743931\" class=\"key-equations\">\r\n<table id=\"fs-id1170902765576\" class=\"unnumbered unstyled\" summary=\"This table gives the following formulae: Magnitude of static friction, f subscript s is less than or equal to mu subscript s N; Magnitude of kinetic friction, f subscript k equal to mu subscript k N; Centripetal force, F subscript C equal to m v squared by r or F subscript C equal to m r omega squared; Ideal angle of a banked curve, tan theta equal to v squared by rg; Drag force, F subscript D equal to half C rho A v squared; Stokes\u2019 law, F subscript S equal to 6 pi r eta v.\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td>Magnitude of static friction<\/td>\r\n<td><span id=\"MathJax-Element-1890-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40924\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40925\" class=\"mrow\"><span id=\"MathJax-Span-40926\" class=\"semantics\"><span id=\"MathJax-Span-40927\" class=\"mrow\"><span id=\"MathJax-Span-40928\" class=\"mrow\"><span id=\"MathJax-Span-40929\" class=\"msub\"><span id=\"MathJax-Span-40930\" class=\"mi\">f<\/span><span id=\"MathJax-Span-40931\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-40932\" class=\"mo\">\u2264<\/span><span id=\"MathJax-Span-40933\" class=\"msub\"><span id=\"MathJax-Span-40934\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-40935\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-40936\" class=\"mi\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">fs\u2264\u03bcsN<\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Magnitude of kinetic friction<\/td>\r\n<td><span id=\"MathJax-Element-1891-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40937\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40938\" class=\"mrow\"><span id=\"MathJax-Span-40939\" class=\"semantics\"><span id=\"MathJax-Span-40940\" class=\"mrow\"><span id=\"MathJax-Span-40941\" class=\"mrow\"><span id=\"MathJax-Span-40942\" class=\"msub\"><span id=\"MathJax-Span-40943\" class=\"mi\">f<\/span><span id=\"MathJax-Span-40944\" class=\"mi\">k<\/span><\/span><span id=\"MathJax-Span-40945\" class=\"mo\">=<\/span><span id=\"MathJax-Span-40946\" class=\"msub\"><span id=\"MathJax-Span-40947\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-40948\" class=\"mi\">k<\/span><\/span><span id=\"MathJax-Span-40949\" class=\"mi\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">fk=\u03bckN<\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Centripetal force<\/td>\r\n<td><span id=\"MathJax-Element-1892-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40950\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40951\" class=\"mrow\"><span id=\"MathJax-Span-40952\" class=\"semantics\"><span id=\"MathJax-Span-40953\" class=\"mrow\"><span id=\"MathJax-Span-40954\" class=\"mrow\"><span id=\"MathJax-Span-40955\" class=\"msub\"><span id=\"MathJax-Span-40956\" class=\"mi\">F<\/span><span id=\"MathJax-Span-40957\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-40958\" class=\"mo\">=<\/span><span id=\"MathJax-Span-40959\" class=\"mi\">m<\/span><span id=\"MathJax-Span-40960\" class=\"mfrac\"><span id=\"MathJax-Span-40961\" class=\"mrow\"><span id=\"MathJax-Span-40962\" class=\"msup\"><span id=\"MathJax-Span-40963\" class=\"mi\">v<\/span><span id=\"MathJax-Span-40964\" class=\"mn\">2<\/span><\/span><\/span><span id=\"MathJax-Span-40965\" class=\"mi\">r<\/span><\/span><span id=\"MathJax-Span-40966\" class=\"mspace\"><\/span><span id=\"MathJax-Span-40967\" class=\"mtext\">or<\/span><span id=\"MathJax-Span-40968\" class=\"mspace\"><\/span><span id=\"MathJax-Span-40969\" class=\"msub\"><span id=\"MathJax-Span-40970\" class=\"mi\">F<\/span><span id=\"MathJax-Span-40971\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-40972\" class=\"mo\">=<\/span><span id=\"MathJax-Span-40973\" class=\"mi\">m<\/span><span id=\"MathJax-Span-40974\" class=\"mi\">r<\/span><span id=\"MathJax-Span-40975\" class=\"msup\"><span id=\"MathJax-Span-40976\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-40977\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fc=mv2rorFc=mr\u03c92<\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Ideal angle of a banked curve<\/td>\r\n<td><span id=\"MathJax-Element-1893-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40978\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40979\" class=\"mrow\"><span id=\"MathJax-Span-40980\" class=\"semantics\"><span id=\"MathJax-Span-40981\" class=\"mrow\"><span id=\"MathJax-Span-40982\" class=\"mrow\"><span id=\"MathJax-Span-40983\" class=\"mtext\">tan<\/span><span id=\"MathJax-Span-40984\" class=\"mspace\"><\/span><span id=\"MathJax-Span-40985\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-40986\" class=\"mo\">=<\/span><span id=\"MathJax-Span-40987\" class=\"mfrac\"><span id=\"MathJax-Span-40988\" class=\"mrow\"><span id=\"MathJax-Span-40989\" class=\"msup\"><span id=\"MathJax-Span-40990\" class=\"mi\">v<\/span><span id=\"MathJax-Span-40991\" class=\"mn\">2<\/span><\/span><\/span><span id=\"MathJax-Span-40992\" class=\"mrow\"><span id=\"MathJax-Span-40993\" class=\"mi\">r<\/span><span id=\"MathJax-Span-40994\" class=\"mi\">g<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">tan\u03b8=v2rg<\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Drag force<\/td>\r\n<td><span id=\"MathJax-Element-1894-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40995\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40996\" class=\"mrow\"><span id=\"MathJax-Span-40997\" class=\"semantics\"><span id=\"MathJax-Span-40998\" class=\"mrow\"><span id=\"MathJax-Span-40999\" class=\"mrow\"><span id=\"MathJax-Span-41000\" class=\"msub\"><span id=\"MathJax-Span-41001\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41002\" class=\"mi\">D<\/span><\/span><span id=\"MathJax-Span-41003\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41004\" class=\"mfrac\"><span id=\"MathJax-Span-41005\" class=\"mn\">1<\/span><span id=\"MathJax-Span-41006\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-41007\" class=\"mi\">C<\/span><span id=\"MathJax-Span-41008\" class=\"mi\">\u03c1<\/span><span id=\"MathJax-Span-41009\" class=\"mi\">A<\/span><span id=\"MathJax-Span-41010\" class=\"msup\"><span id=\"MathJax-Span-41011\" class=\"mi\">v<\/span><span id=\"MathJax-Span-41012\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">FD=12C\u03c1Av2<\/span><\/span><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td>Stokes\u2019 law<\/td>\r\n<td><span id=\"MathJax-Element-1895-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41013\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41014\" class=\"mrow\"><span id=\"MathJax-Span-41015\" class=\"semantics\"><span id=\"MathJax-Span-41016\" class=\"mrow\"><span id=\"MathJax-Span-41017\" class=\"mrow\"><span id=\"MathJax-Span-41018\" class=\"msub\"><span id=\"MathJax-Span-41019\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41020\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41021\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41022\" class=\"mn\">6<\/span><span id=\"MathJax-Span-41023\" class=\"mi\">\u03c0<\/span><span id=\"MathJax-Span-41024\" class=\"mi\">r<\/span><span id=\"MathJax-Span-41025\" class=\"mi\">\u03b7<\/span><span id=\"MathJax-Span-41026\" class=\"mi\">v<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fs=6\u03c0r\u03b7v<\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><\/div>\r\n<\/div>\r\n<div class=\"os-key-concepts-container\">\r\n<div class=\"textbox\">\r\n<h3><span class=\"os-text\">Summary<\/span><\/h3>\r\n<div class=\"os-key-concepts\">\r\n<div class=\"os-section-area\"><section id=\"fs-id1165036775374\" class=\"key-concepts\">\r\n<h4 id=\"59779_copy_1\"><span class=\"os-number\">6.1<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Solving Problems with Newton\u2019s Laws<\/span><\/h4>\r\n<ul id=\"fs-id1165036852024\">\r\n \t<li>Newton\u2019s laws of motion can be applied in numerous situations to solve motion problems.<\/li>\r\n \t<li>Some problems contain multiple force vectors acting in different directions on an object. Be sure to draw diagrams, resolve all force vectors into horizontal and vertical components, and draw a free-body diagram. Always analyze the direction in which an object accelerates so that you can determine whether\u00a0<span id=\"MathJax-Element-1896-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41027\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41028\" class=\"mrow\"><span id=\"MathJax-Span-41029\" class=\"semantics\"><span id=\"MathJax-Span-41030\" class=\"mrow\"><span id=\"MathJax-Span-41031\" class=\"mrow\"><span id=\"MathJax-Span-41032\" class=\"msub\"><span id=\"MathJax-Span-41033\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41034\" class=\"mrow\"><span id=\"MathJax-Span-41035\" class=\"mtext\">net<\/span><\/span><\/span><span id=\"MathJax-Span-41036\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41037\" class=\"mi\">m<\/span><span id=\"MathJax-Span-41038\" class=\"mi\">a<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fnet=ma<\/span><\/span>\u00a0or\u00a0<span id=\"MathJax-Element-1897-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41039\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41040\" class=\"mrow\"><span id=\"MathJax-Span-41041\" class=\"semantics\"><span id=\"MathJax-Span-41042\" class=\"mrow\"><span id=\"MathJax-Span-41043\" class=\"mrow\"><span id=\"MathJax-Span-41044\" class=\"msub\"><span id=\"MathJax-Span-41045\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41046\" class=\"mrow\"><span id=\"MathJax-Span-41047\" class=\"mtext\">net<\/span><\/span><\/span><span id=\"MathJax-Span-41048\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41049\" class=\"mn\">0<\/span><span id=\"MathJax-Span-41050\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fnet=0.<\/span><\/span><\/li>\r\n \t<li>The normal force on an object is not always equal in magnitude to the weight of the object. If an object is accelerating vertically, the normal force is less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force is always less than the full weight of the object.<\/li>\r\n \t<li>Some problems contain several physical quantities, such as forces, acceleration, velocity, or position. You can apply concepts from kinematics and dynamics to solve these problems.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<div class=\"os-section-area\"><section id=\"fs-id1165037218490\" class=\"key-concepts\">\r\n<h4 id=\"11340_copy_1\"><span class=\"os-number\">6.2<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Friction<\/span><\/h4>\r\n<ul id=\"fs-id1165038192411\">\r\n \t<li>Friction is a contact force that opposes the motion or attempted motion between two systems. Simple friction is proportional to the normal force\u00a0<em>N<\/em>\u00a0supporting the two systems.<\/li>\r\n \t<li>The magnitude of static friction force between two materials stationary relative to each other is determined using the coefficient of static friction, which depends on both materials.<\/li>\r\n \t<li>The kinetic friction force between two materials moving relative to each other is determined using the coefficient of kinetic friction, which also depends on both materials and is always less than the coefficient of static friction.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<div class=\"os-section-area\"><section id=\"fs-id1165039291121\" class=\"key-concepts\">\r\n<h4 id=\"71765_copy_1\"><span class=\"os-number\">6.3<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Centripetal Force<\/span><\/h4>\r\n<ul id=\"fs-id1165038989422\">\r\n \t<li>Centripetal force\u00a0<span id=\"MathJax-Element-1898-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41051\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41052\" class=\"mrow\"><span id=\"MathJax-Span-41053\" class=\"semantics\"><span id=\"MathJax-Span-41054\" class=\"mrow\"><span id=\"MathJax-Span-41055\" class=\"mrow\"><span id=\"MathJax-Span-41056\" class=\"msub\"><span id=\"MathJax-Span-41057\" class=\"mstyle\"><span id=\"MathJax-Span-41058\" class=\"mrow\"><span id=\"MathJax-Span-41059\" class=\"mover\"><span id=\"MathJax-Span-41060\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41061\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41062\" class=\"mtext\">c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192c<\/span><\/span>\u00a0is a \u201ccenter-seeking\u201d force that always points toward the center of rotation. It is perpendicular to linear velocity and has the magnitude\r\n<div id=\"40902\"><\/div>\r\n<div id=\"fs-id1165035627257\" class=\"unnumbered\">\r\n<div class=\"MathJax_Display\"><span id=\"MathJax-Element-1899-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41063\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41064\" class=\"mrow\"><span id=\"MathJax-Span-41065\" class=\"semantics\"><span id=\"MathJax-Span-41066\" class=\"mrow\"><span id=\"MathJax-Span-41067\" class=\"mrow\"><span id=\"MathJax-Span-41068\" class=\"msub\"><span id=\"MathJax-Span-41069\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41070\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-41071\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41072\" class=\"mi\">m<\/span><span id=\"MathJax-Span-41073\" class=\"msub\"><span id=\"MathJax-Span-41074\" class=\"mi\">a<\/span><span id=\"MathJax-Span-41075\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-41076\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\">Fc=mac.<\/span><\/span><\/div>\r\n<\/div><\/li>\r\n \t<li>Rotating and accelerated frames of reference are noninertial. Inertial forces, such as the Coriolis force, are needed to explain motion in such frames.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<div class=\"os-section-area\"><section id=\"fs-id1165039277022\" class=\"key-concepts\">\r\n<h4 id=\"47362_copy_1\"><span class=\"os-number\">6.4<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Drag Force and Terminal Speed<\/span><\/h4>\r\n<ul id=\"fs-id1165035867590\">\r\n \t<li>Drag forces acting on an object moving in a fluid oppose the motion. For larger objects (such as a baseball) moving at a velocity in air, the drag force is determined using the drag coefficient (typical values are given in\u00a0<a class=\"autogenerated-content\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:36a2674d-4cdb-4dfd-a2b0-5c63cc5a969b@5#fs-id1165035723394\">Table 6.2<\/a>), the area of the object facing the fluid, and the fluid density.<\/li>\r\n \t<li>For small objects (such as a bacterium) moving in a denser medium (such as water), the drag force is given by Stokes\u2019 law.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-review-conceptual-questions-container\">\r\n<div class=\"textbox learning-objectives\">\r\n<h3><span class=\"os-text\">Conceptual Questions<\/span><\/h3>\r\n<div class=\"os-review-conceptual-questions\">\r\n<div class=\"os-section-area\"><section id=\"fs-id1165036735565\" class=\"review-conceptual-questions\">\r\n<h4 id=\"59779_copy_2\"><span class=\"os-number\">6.1<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Solving Problems with Newton\u2019s Laws<\/span><\/h4>\r\n<div id=\"fs-id1165036983215\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165036983217\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036983215-solution\">1<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036983219\">To simulate the apparent weightlessness of space orbit, astronauts are trained in the hold of a cargo aircraft that is accelerating downward at\u00a0<em>g<\/em>. Why do they appear to be weightless, as measured by standing on a bathroom scale, in this accelerated frame of reference? Is there any difference between their apparent weightlessness in orbit and in the aircraft?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"os-section-area\"><section id=\"fs-id1165037183984\" class=\"review-conceptual-questions\">\r\n<h4 id=\"11340_copy_2\"><span class=\"os-number\">6.2<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Friction<\/span><\/h4>\r\n<div id=\"fs-id1165037088661\" class=\"\"><section>\r\n<div id=\"fs-id1165038383771\"><span class=\"os-number\">2<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038383773\">The glue on a piece of tape can exert forces. Can these forces be a type of simple friction? Explain, considering especially that tape can stick to vertical walls and even to ceilings.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038006273\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037982195\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038006273-solution\">3<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037982197\">When you learn to drive, you discover that you need to let up slightly on the brake pedal as you come to a stop or the car will stop with a jerk. Explain this in terms of the relationship between static and kinetic friction.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038331619\" class=\"\"><section>\r\n<div id=\"fs-id1165038331621\"><span class=\"os-number\">4<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036884400\">When you push a piece of chalk across a chalkboard, it sometimes screeches because it rapidly alternates between slipping and sticking to the board. Describe this process in more detail, in particular, explaining how it is related to the fact that kinetic friction is less than static friction. (The same slip-grab process occurs when tires screech on pavement.)<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038360288\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165038360290\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038360288-solution\">5<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037062457\">A physics major is cooking breakfast when she notices that the frictional force between her steel spatula and Teflon frying pan is only 0.200 N. Knowing the coefficient of kinetic friction between the two materials, she quickly calculates the normal force. What is it?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"os-section-area\"><section id=\"fs-id1165039026731\" class=\"review-conceptual-questions\">\r\n<h4 id=\"71765_copy_2\"><span class=\"os-number\">6.3<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Centripetal Force<\/span><\/h4>\r\n<div id=\"fs-id1165039075190\" class=\"\"><section>\r\n<div id=\"fs-id1165035760150\"><span class=\"os-number\">6<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035718736\">If you wish to reduce the stress (which is related to centripetal force) on high-speed tires, would you use large- or small-diameter tires? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039326296\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035715382\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039326296-solution\">7<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038959750\">Define centripetal force. Can any type of force (for example, tension, gravitational force, friction, and so on) be a centripetal force? Can any combination of forces be a centripetal force?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039510688\" class=\"\"><section>\r\n<div id=\"fs-id1165038990778\"><span class=\"os-number\">8<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039496835\">If centripetal force is directed toward the center, why do you feel that you are \u2018thrown\u2019 away from the center as a car goes around a curve? Explain.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039003212\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039384949\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039003212-solution\">9<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039113906\">Race car drivers routinely cut corners, as shown below (Path 2). Explain how this allows the curve to be taken at the greatest speed.<\/p>\r\n\r\n<span id=\"fs-id1165039497191\"><img id=\"69252\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/41b71f50ff2da85e0b90b8a9acf6f61e6febe734\" alt=\"Two paths are shown inside a race track through a ninety degree curve. Two cars, a red and a blue one,  and their paths of travel are shown. The blue car is making a tight turn on path one, which is the inside path along the track. The red car is shown overtaking the first car, while taking a wider turn and crossing in front of the blue car into the inside path and then back out of it.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039393190\" class=\"\"><section>\r\n<div id=\"fs-id1165039073338\"><span class=\"os-number\">10<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035726450\">Many amusement parks have rides that make vertical loops like the one shown below. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if:<\/p>\r\n<p id=\"fs-id1165039284952\">(a) The car goes over the top at faster than this speed?<\/p>\r\n<p id=\"fs-id1165039000642\">(b) The car goes over the top at slower than this speed?<\/p>\r\n\r\n<span id=\"fs-id1165039287645\"><img id=\"79571\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/2cd9784e5ca0ebf1365ff2205c09c89539af5f76\" alt=\"A photo of a roller coaster with a vertical loop. The loop has a tighter curvature at the top than at the bottom, making an inverted teardrop shape.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039264324\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039066680\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039264324-solution\">11<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035685329\">What causes water to be removed from clothes in a spin-dryer?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039302968\" class=\"\"><section>\r\n<div id=\"fs-id1165039458614\"><span class=\"os-number\">12<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035703891\">As a skater forms a circle, what force is responsible for making his turn? Use a free-body diagram in your answer.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039099187\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039108353\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039099187-solution\">13<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039483406\">Suppose a child is riding on a merry-go-round at a distance about halfway between its center and edge. She has a lunch box resting on wax paper, so that there is very little friction between it and the merry-go-round. Which path shown below will the lunch box take when she lets go? The lunch box leaves a trail in the dust on the merry-go-round. Is that trail straight, curved to the left, or curved to the right? Explain your answer.<\/p>\r\n\r\n<span id=\"fs-id1165039286742\"><img id=\"98583\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/12198ae05fdad562467c8f2e7576986bd851dc72\" alt=\"An illustration of the circular base of a merry-go-round with a single horse and child on it. The angular velocity, omega, is clockwise, shown here with an arrow. A point P is shown near the horse, on a circle concentric with the merry-go-round. Three arrows are shown coming out of point P, depicting the three possible path of the lunch box. Path A curves into the circle, to the right from the perspective of the box. Path B is straight, tangent to the circle. Path C curves to the left from the perspective of the box, out of the circle.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039122580\" class=\"\"><section>\r\n<div id=\"fs-id1165035708917\"><span class=\"os-number\">14<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039376236\">Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your car\u2019s speed? What is the direction of the force exerted on you by the car seat?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035731478\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039297560\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035731478-solution\">15<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039098694\">Suppose a mass is moving in a circular path on a frictionless table as shown below. In Earth\u2019s frame of reference, there is no centrifugal force pulling the mass away from the center of rotation, yet there is a force stretching the string attaching the mass to the nail. Using concepts related to centripetal force and Newton\u2019s third law, explain what force stretches the string, identifying its physical origin.<\/p>\r\n\r\n<span id=\"fs-id1165039121255\"><img id=\"27241\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/5965080d38ed2d7862b7dcc73e44dc80aa0950df\" alt=\"An illustration of a mass moving in a circular path on a table. The mass is attached to a string that is pinned at the center of the circle to the table at the other end.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039064683\" class=\"\"><section>\r\n<div id=\"fs-id1165039419790\"><span class=\"os-number\">16<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036155941\">When a toilet is flushed or a sink is drained, the water (and other material) begins to rotate about the drain on the way down. Assuming no initial rotation and a flow initially directly straight toward the drain, explain what causes the rotation and which direction it has in the Northern Hemisphere. (Note that this is a small effect and in most toilets the rotation is caused by directional water jets.) Would the direction of rotation reverse if water were forced up the drain?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039347286\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039021031\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039347286-solution\">17<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039264802\">A car rounds a curve and encounters a patch of ice with a very low coefficient of kinetic fiction. The car slides off the road. Describe the path of the car as it leaves the road.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039315339\" class=\"\"><section>\r\n<div id=\"fs-id1165039293336\"><span class=\"os-number\">18<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039376622\">In one amusement park ride, riders enter a large vertical barrel and stand against the wall on its horizontal floor. The barrel is spun up and the floor drops away. Riders feel as if they are pinned to the wall by a force something like the gravitational force. This is an inertial force sensed and used by the riders to explain events in the rotating frame of reference of the barrel. Explain in an inertial frame of reference (Earth is nearly one) what pins the riders to the wall, and identify all forces acting on them.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039098388\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039087924\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039098388-solution\">19<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038974400\">Two friends are having a conversation. Anna says a satellite in orbit is in free fall because the satellite keeps falling toward Earth. Tom says a satellite in orbit is not in free fall because the acceleration due to gravity is not\u00a0<span id=\"MathJax-Element-1900-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41077\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41078\" class=\"mrow\"><span id=\"MathJax-Span-41079\" class=\"semantics\"><span id=\"MathJax-Span-41080\" class=\"mrow\"><span id=\"MathJax-Span-41081\" class=\"mrow\"><span id=\"MathJax-Span-41082\" class=\"mn\">9.80<\/span><span id=\"MathJax-Span-41083\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41084\" class=\"msup\"><span id=\"MathJax-Span-41085\" class=\"mrow\"><span id=\"MathJax-Span-41086\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41087\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">9.80m\/s2<\/span><\/span>. Who do you agree with and why?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039224165\" class=\"\"><section>\r\n<div id=\"fs-id1165035756498\"><span class=\"os-number\">20<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039027867\">A nonrotating frame of reference placed at the center of the Sun is very nearly an inertial one. Why is it not exactly an inertial frame?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"os-section-area\"><section id=\"fs-id1165035772600\" class=\"review-conceptual-questions\">\r\n<h4 id=\"47362_copy_2\"><span class=\"os-number\">6.4<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Drag Force and Terminal Speed<\/span><\/h4>\r\n<div id=\"fs-id1165036157277\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165038999340\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036157277-solution\">21<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036144143\">Athletes such as swimmers and bicyclists wear body suits in competition. Formulate a list of pros and cons of such suits.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039276994\" class=\"\"><section>\r\n<div id=\"fs-id1165039512659\"><span class=\"os-number\">22<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039079000\">Two expressions were used for the drag force experienced by a moving object in a liquid. One depended upon the speed, while the other was proportional to the square of the speed. In which types of motion would each of these expressions be more applicable than the other one?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036143556\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165036148102\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036143556-solution\">23<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035639634\">As cars travel, oil and gasoline leaks onto the road surface. If a light rain falls, what does this do to the control of the car? Does a heavy rain make any difference?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035977406\" class=\"\"><section>\r\n<div id=\"fs-id1165039391020\"><span class=\"os-number\">24<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039291834\">Why can a squirrel jump from a tree branch to the ground and run away undamaged, while a human could break a bone in such a fall?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-review-problems-container\">\r\n<div class=\"textbox exercises\">\r\n<div class=\"os-review-problems-container\">\r\n<h3><span class=\"os-text\">Problems<\/span><\/h3>\r\n<div class=\"os-review-problems\">\r\n<div class=\"os-section-area\"><section id=\"fs-id1165036776336\" class=\"review-problems\">\r\n<h4 id=\"59779_copy_3\"><span class=\"os-number\">6.1<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Solving Problems with Newton\u2019s Laws<\/span><\/h4>\r\n<div id=\"fs-id1165037186198\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037186200\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037186198-solution\">25<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037186202\">A 30.0-kg girl in a swing is pushed to one side and held at rest by a horizontal force\u00a0<span id=\"MathJax-Element-1901-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41088\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41089\" class=\"mrow\"><span id=\"MathJax-Span-41090\" class=\"semantics\"><span id=\"MathJax-Span-41091\" class=\"mrow\"><span id=\"MathJax-Span-41092\" class=\"mstyle\"><span id=\"MathJax-Span-41093\" class=\"mrow\"><span id=\"MathJax-Span-41094\" class=\"mover\"><span id=\"MathJax-Span-41095\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41096\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192<\/span><\/span>\u00a0so that the swing ropes are\u00a0<span id=\"MathJax-Element-1902-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41097\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41098\" class=\"mrow\"><span id=\"MathJax-Span-41099\" class=\"semantics\"><span id=\"MathJax-Span-41100\" class=\"mrow\"><span id=\"MathJax-Span-41101\" class=\"mrow\"><span id=\"MathJax-Span-41102\" class=\"mn\">30.0<\/span><span id=\"MathJax-Span-41103\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">30.0\u00b0<\/span><\/span>with respect to the vertical. (a) Calculate the tension in each of the two ropes supporting the swing under these conditions. (b) Calculate the magnitude of\u00a0<span id=\"MathJax-Element-1903-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41104\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41105\" class=\"mrow\"><span id=\"MathJax-Span-41106\" class=\"semantics\"><span id=\"MathJax-Span-41107\" class=\"mrow\"><span id=\"MathJax-Span-41108\" class=\"mrow\"><span id=\"MathJax-Span-41109\" class=\"mstyle\"><span id=\"MathJax-Span-41110\" class=\"mrow\"><span id=\"MathJax-Span-41111\" class=\"mover\"><span id=\"MathJax-Span-41112\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41113\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41114\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192.<\/span><\/span><\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037155160\" class=\"\"><section>\r\n<div id=\"fs-id1165037155163\"><span class=\"os-number\">26<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037155165\">Find the tension in each of the three cables supporting the traffic light if it weighs 2.00 \u00d7 10<sup>2<\/sup>\u00a0N.<\/p>\r\n\r\n<span id=\"fs-id1165037155169\"><img id=\"7843\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/6712294d8942dbcf6876aea4500ec11115ca2cc9\" alt=\"A sketch of a traffic light suspended by a cable that is in turn suspended from two other cables is shown. Tension T sub 3 is the tension in the cable connecting the traffic light to the upper cables. Tension T sub one is the tension in the upper cable pulling up and to the left, making a 41 degree angle with the horizontal. Tension T sub two is the tension pulling up and to the right, making a 63 degree angle with the horizontal. Force vector w equal to 200 Newtons pulls vertically downward on the traffic light.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037151556\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037151559\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037151556-solution\">27<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037151561\">Three forces act on an object, considered to be a particle, which moves with constant velocity\u00a0<span id=\"MathJax-Element-1904-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41115\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41116\" class=\"mrow\"><span id=\"MathJax-Span-41117\" class=\"semantics\"><span id=\"MathJax-Span-41118\" class=\"mrow\"><span id=\"MathJax-Span-41119\" class=\"mrow\"><span id=\"MathJax-Span-41120\" class=\"mi\">v<\/span><span id=\"MathJax-Span-41121\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41122\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41123\" class=\"mn\">3<\/span><span id=\"MathJax-Span-41124\" class=\"mstyle\"><span id=\"MathJax-Span-41125\" class=\"mrow\"><span id=\"MathJax-Span-41126\" class=\"mover\"><span id=\"MathJax-Span-41127\" class=\"mi\">i<\/span><span id=\"MathJax-Span-41128\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41129\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-41130\" class=\"mn\">2<\/span><span id=\"MathJax-Span-41131\" class=\"mstyle\"><span id=\"MathJax-Span-41132\" class=\"mrow\"><span id=\"MathJax-Span-41133\" class=\"mover\"><span id=\"MathJax-Span-41134\" class=\"mi\">j<\/span><span id=\"MathJax-Span-41135\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41136\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41137\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41138\" class=\"mtext\">m\/s<\/span><span id=\"MathJax-Span-41139\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v=(3i^\u22122j^)m\/s.<\/span><\/span>\u00a0Two of the forces are\u00a0<span id=\"MathJax-Element-1905-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41140\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41141\" class=\"mrow\"><span id=\"MathJax-Span-41142\" class=\"semantics\"><span id=\"MathJax-Span-41143\" class=\"mrow\"><span id=\"MathJax-Span-41144\" class=\"mrow\"><span id=\"MathJax-Span-41145\" class=\"msub\"><span id=\"MathJax-Span-41146\" class=\"mstyle\"><span id=\"MathJax-Span-41147\" class=\"mrow\"><span id=\"MathJax-Span-41148\" class=\"mover\"><span id=\"MathJax-Span-41149\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41150\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41151\" class=\"mn\">1<\/span><\/span><span id=\"MathJax-Span-41152\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41153\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41154\" class=\"mn\">3<\/span><span id=\"MathJax-Span-41155\" class=\"mstyle\"><span id=\"MathJax-Span-41156\" class=\"mrow\"><span id=\"MathJax-Span-41157\" class=\"mover\"><span id=\"MathJax-Span-41158\" class=\"mi\">i<\/span><span id=\"MathJax-Span-41159\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41160\" class=\"mo\">+<\/span><span id=\"MathJax-Span-41161\" class=\"mn\">5<\/span><span id=\"MathJax-Span-41162\" class=\"mstyle\"><span id=\"MathJax-Span-41163\" class=\"mrow\"><span id=\"MathJax-Span-41164\" class=\"mover\"><span id=\"MathJax-Span-41165\" class=\"mi\">j<\/span><span id=\"MathJax-Span-41166\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41167\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-41168\" class=\"mn\">6<\/span><span id=\"MathJax-Span-41169\" class=\"mstyle\"><span id=\"MathJax-Span-41170\" class=\"mrow\"><span id=\"MathJax-Span-41171\" class=\"mover\"><span id=\"MathJax-Span-41172\" class=\"mi\">k<\/span><span id=\"MathJax-Span-41173\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41174\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41175\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41176\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u21921=(3i^+5j^\u22126k^)N<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1906-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41177\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41178\" class=\"mrow\"><span id=\"MathJax-Span-41179\" class=\"semantics\"><span id=\"MathJax-Span-41180\" class=\"mrow\"><span id=\"MathJax-Span-41181\" class=\"mrow\"><span id=\"MathJax-Span-41182\" class=\"msub\"><span id=\"MathJax-Span-41183\" class=\"mstyle\"><span id=\"MathJax-Span-41184\" class=\"mrow\"><span id=\"MathJax-Span-41185\" class=\"mover\"><span id=\"MathJax-Span-41186\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41187\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41188\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-41189\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41190\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41191\" class=\"mn\">4<\/span><span id=\"MathJax-Span-41192\" class=\"mstyle\"><span id=\"MathJax-Span-41193\" class=\"mrow\"><span id=\"MathJax-Span-41194\" class=\"mover\"><span id=\"MathJax-Span-41195\" class=\"mi\">i<\/span><span id=\"MathJax-Span-41196\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41197\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-41198\" class=\"mn\">7<\/span><span id=\"MathJax-Span-41199\" class=\"mstyle\"><span id=\"MathJax-Span-41200\" class=\"mrow\"><span id=\"MathJax-Span-41201\" class=\"mover\"><span id=\"MathJax-Span-41202\" class=\"mi\">j<\/span><span id=\"MathJax-Span-41203\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41204\" class=\"mo\">+<\/span><span id=\"MathJax-Span-41205\" class=\"mn\">2<\/span><span id=\"MathJax-Span-41206\" class=\"mstyle\"><span id=\"MathJax-Span-41207\" class=\"mrow\"><span id=\"MathJax-Span-41208\" class=\"mover\"><span id=\"MathJax-Span-41209\" class=\"mi\">k<\/span><span id=\"MathJax-Span-41210\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41211\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41212\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41213\" class=\"mtext\">N<\/span><span id=\"MathJax-Span-41214\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u21922=(4i^\u22127j^+2k^)N.<\/span><\/span>\u00a0Find the third force.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037154683\" class=\"\"><section>\r\n<div id=\"fs-id1165037154685\"><span class=\"os-number\">28<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037154687\">A flea jumps by exerting a force of\u00a0<span id=\"MathJax-Element-1907-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41215\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41216\" class=\"mrow\"><span id=\"MathJax-Span-41217\" class=\"semantics\"><span id=\"MathJax-Span-41218\" class=\"mrow\"><span id=\"MathJax-Span-41219\" class=\"mrow\"><span id=\"MathJax-Span-41220\" class=\"mn\">1.20<\/span><span id=\"MathJax-Span-41221\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41222\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41223\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41224\" class=\"msup\"><span id=\"MathJax-Span-41225\" class=\"mrow\"><span id=\"MathJax-Span-41226\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41227\" class=\"mrow\"><span id=\"MathJax-Span-41228\" class=\"mn\">\u22125<\/span><\/span><\/span><span id=\"MathJax-Span-41229\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41230\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.20\u00d710\u22125N<\/span><\/span>\u00a0straight down on the ground. A breeze blowing on the flea parallel to the ground exerts a force of\u00a0<span id=\"MathJax-Element-1908-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41231\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41232\" class=\"mrow\"><span id=\"MathJax-Span-41233\" class=\"semantics\"><span id=\"MathJax-Span-41234\" class=\"mrow\"><span id=\"MathJax-Span-41235\" class=\"mrow\"><span id=\"MathJax-Span-41236\" class=\"mn\">0.500<\/span><span id=\"MathJax-Span-41237\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41238\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41239\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41240\" class=\"msup\"><span id=\"MathJax-Span-41241\" class=\"mrow\"><span id=\"MathJax-Span-41242\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41243\" class=\"mrow\"><span id=\"MathJax-Span-41244\" class=\"mn\">\u22126<\/span><\/span><\/span><span id=\"MathJax-Span-41245\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41246\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.500\u00d710\u22126N<\/span><\/span>\u00a0on the flea while the flea is still in contact with the ground. Find the direction and magnitude of the acceleration of the flea if its mass is\u00a0<span id=\"MathJax-Element-1909-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41247\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41248\" class=\"mrow\"><span id=\"MathJax-Span-41249\" class=\"semantics\"><span id=\"MathJax-Span-41250\" class=\"mrow\"><span id=\"MathJax-Span-41251\" class=\"mrow\"><span id=\"MathJax-Span-41252\" class=\"mn\">6.00<\/span><span id=\"MathJax-Span-41253\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41254\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41255\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41256\" class=\"msup\"><span id=\"MathJax-Span-41257\" class=\"mrow\"><span id=\"MathJax-Span-41258\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41259\" class=\"mrow\"><span id=\"MathJax-Span-41260\" class=\"mn\">\u22127<\/span><\/span><\/span><span id=\"MathJax-Span-41261\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41262\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">6.00\u00d710\u22127kg<\/span><\/span>. Do not neglect the gravitational force.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036901084\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165036901086\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036901084-solution\">29<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036901088\">Two muscles in the back of the leg pull upward on the Achilles tendon, as shown below. (These muscles are called the medial and lateral heads of the gastrocnemius muscle.) Find the magnitude and direction of the total force on the Achilles tendon. What type of movement could be caused by this force?<\/p>\r\n\r\n<span id=\"fs-id1165037262483\"><img id=\"64788\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/9c6d40ff32e9ba34cbb98d3100d0d990d42af261\" alt=\"An Achilles tendon is shown in the figure with two forces exerted on it by the lateral and medial heads of the gastrocnemius muscle. F sub one, equal to two hundred Newtons, is shown as a vector making an angle twenty degrees to the right of vertical, and F sub two, equal to two hundred Newtons, is shown making an angle of twenty degrees left of vertical.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038018567\" class=\"\"><section>\r\n<div id=\"fs-id1165036834090\"><span class=\"os-number\">30<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036834092\">After a mishap, a 76.0-kg circus performer clings to a trapeze, which is being pulled to the side by another circus artist, as shown here. Calculate the tension in the two ropes if the person is momentarily motionless. Include a free-body diagram in your solution.<\/p>\r\n\r\n<span id=\"fs-id1165036834098\"><img id=\"52754\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/feaad2318d1eed2fdee8dae38210d0b706a006a1\" alt=\"A circus performer hanging from a trapeze is being pulled to the right by another performer using a rope. Her weight is shown by a vector w acting vertically downward. The trapeze rope exerts a tension, T sub one, up and to the left, making an angle of fifteen degrees with the vertical. The second performer pulls with tension T sub two, making an angle of ten degrees above the positive x direction.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037271925\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037271927\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037271925-solution\">31<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037271929\">A 35.0-kg dolphin decelerates from 12.0 to 7.50 m\/s in 2.30 s to join another dolphin in play. What average force was exerted to slow the first dolphin if it was moving horizontally? (The gravitational force is balanced by the buoyant force of the water.)<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037017931\" class=\"\"><section>\r\n<div id=\"fs-id1165037017933\"><span class=\"os-number\">32<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037017935\">When starting a foot race, a 70.0-kg sprinter exerts an average force of 650 N backward on the ground for 0.800 s. (a) What is his final speed? (b) How far does he travel?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037985556\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037985558\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037985556-solution\">33<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037985560\">A large rocket has a mass of\u00a0<span id=\"MathJax-Element-1910-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41263\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41264\" class=\"mrow\"><span id=\"MathJax-Span-41265\" class=\"semantics\"><span id=\"MathJax-Span-41266\" class=\"mrow\"><span id=\"MathJax-Span-41267\" class=\"mrow\"><span id=\"MathJax-Span-41268\" class=\"mn\">2.00<\/span><span id=\"MathJax-Span-41269\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41270\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41271\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41272\" class=\"msup\"><span id=\"MathJax-Span-41273\" class=\"mrow\"><span id=\"MathJax-Span-41274\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41275\" class=\"mn\">6<\/span><\/span><span id=\"MathJax-Span-41276\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41277\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.00\u00d7106kg<\/span><\/span>\u00a0at takeoff, and its engines produce a thrust of\u00a0<span id=\"MathJax-Element-1911-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41278\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41279\" class=\"mrow\"><span id=\"MathJax-Span-41280\" class=\"semantics\"><span id=\"MathJax-Span-41281\" class=\"mrow\"><span id=\"MathJax-Span-41282\" class=\"mrow\"><span id=\"MathJax-Span-41283\" class=\"mn\">3.50<\/span><span id=\"MathJax-Span-41284\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41285\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41286\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41287\" class=\"msup\"><span id=\"MathJax-Span-41288\" class=\"mrow\"><span id=\"MathJax-Span-41289\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41290\" class=\"mn\">7<\/span><\/span><span id=\"MathJax-Span-41291\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41292\" class=\"mtext\">N<\/span><span id=\"MathJax-Span-41293\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.50\u00d7107N.<\/span><\/span>\u00a0(a) Find its initial acceleration if it takes off vertically. (b) How long does it take to reach a velocity of 120 km\/h straight up, assuming constant mass and thrust?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037221234\" class=\"\"><section>\r\n<div id=\"fs-id1165037221236\"><span class=\"os-number\">34<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037221238\">A basketball player jumps straight up for a ball. To do this, he lowers his body 0.300 m and then accelerates through this distance by forcefully straightening his legs. This player leaves the floor with a vertical velocity sufficient to carry him 0.900 m above the floor. (a) Calculate his velocity when he leaves the floor. (b) Calculate his acceleration while he is straightening his legs. He goes from zero to the velocity found in (a) in a distance of 0.300 m. (c) Calculate the force he exerts on the floor to do this, given that his mass is 110.0 kg.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037223036\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037223038\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037223036-solution\">35<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037223040\">A 2.50-kg fireworks shell is fired straight up from a mortar and reaches a height of 110.0 m. (a) Neglecting air resistance (a poor assumption, but we will make it for this example), calculate the shell\u2019s velocity when it leaves the mortar. (b) The mortar itself is a tube 0.450 m long. Calculate the average acceleration of the shell in the tube as it goes from zero to the velocity found in (a). (c) What is the average force on the shell in the mortar? Express your answer in newtons and as a ratio to the weight of the shell.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037183422\" class=\"\"><section>\r\n<div id=\"fs-id1165037183424\"><span class=\"os-number\">36<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037183426\">A 0.500-kg potato is fired at an angle of\u00a0<span id=\"MathJax-Element-1912-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41294\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41295\" class=\"mrow\"><span id=\"MathJax-Span-41296\" class=\"semantics\"><span id=\"MathJax-Span-41297\" class=\"mrow\"><span id=\"MathJax-Span-41298\" class=\"mrow\"><span id=\"MathJax-Span-41299\" class=\"mn\">80.0<\/span><span id=\"MathJax-Span-41300\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">80.0\u00b0<\/span><\/span>\u00a0above the horizontal from a PVC pipe used as a \u201cpotato gun\u201d and reaches a height of 110.0 m. (a) Neglecting air resistance, calculate the potato\u2019s velocity when it leaves the gun. (b) The gun itself is a tube 0.450 m long. Calculate the average acceleration of the potato in the tube as it goes from zero to the velocity found in (a). (c) What is the average force on the potato in the gun? Express your answer in newtons and as a ratio to the weight of the potato.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036784614\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165036784616\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036784614-solution\">37<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036784618\">An elevator filled with passengers has a mass of\u00a0<span id=\"MathJax-Element-1913-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41301\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41302\" class=\"mrow\"><span id=\"MathJax-Span-41303\" class=\"semantics\"><span id=\"MathJax-Span-41304\" class=\"mrow\"><span id=\"MathJax-Span-41305\" class=\"mrow\"><span id=\"MathJax-Span-41306\" class=\"mn\">1.70<\/span><span id=\"MathJax-Span-41307\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41308\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41309\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41310\" class=\"msup\"><span id=\"MathJax-Span-41311\" class=\"mrow\"><span id=\"MathJax-Span-41312\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41313\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41314\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41315\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.70\u00d7103kg<\/span><\/span>. (a) The elevator accelerates upward from rest at a rate of\u00a0<span id=\"MathJax-Element-1914-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41316\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41317\" class=\"mrow\"><span id=\"MathJax-Span-41318\" class=\"semantics\"><span id=\"MathJax-Span-41319\" class=\"mrow\"><span id=\"MathJax-Span-41320\" class=\"mrow\"><span id=\"MathJax-Span-41321\" class=\"mn\">1.20<\/span><span id=\"MathJax-Span-41322\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41323\" class=\"msup\"><span id=\"MathJax-Span-41324\" class=\"mrow\"><span id=\"MathJax-Span-41325\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41326\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.20m\/s2<\/span><\/span>\u00a0for 1.50 s. Calculate the tension in the cable supporting the elevator. (b) The elevator continues upward at constant velocity for 8.50 s. What is the tension in the cable during this time? (c) The elevator decelerates at a rate of\u00a0<span id=\"MathJax-Element-1915-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41327\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41328\" class=\"mrow\"><span id=\"MathJax-Span-41329\" class=\"semantics\"><span id=\"MathJax-Span-41330\" class=\"mrow\"><span id=\"MathJax-Span-41331\" class=\"mrow\"><span id=\"MathJax-Span-41332\" class=\"mn\">0.600<\/span><span id=\"MathJax-Span-41333\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41334\" class=\"msup\"><span id=\"MathJax-Span-41335\" class=\"mrow\"><span id=\"MathJax-Span-41336\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41337\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.600m\/s2<\/span><\/span>\u00a0for 3.00 s. What is the tension in the cable during deceleration? (d) How high has the elevator moved above its original starting point, and what is its final velocity?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038203868\" class=\"\"><section>\r\n<div id=\"fs-id1165038203870\"><span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038203872\">A 20.0-g ball hangs from the roof of a freight car by a string. When the freight car begins to move, the string makes an angle of\u00a0<span id=\"MathJax-Element-1916-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41338\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41339\" class=\"mrow\"><span id=\"MathJax-Span-41340\" class=\"semantics\"><span id=\"MathJax-Span-41341\" class=\"mrow\"><span id=\"MathJax-Span-41342\" class=\"mrow\"><span id=\"MathJax-Span-41343\" class=\"mn\">35.0<\/span><span id=\"MathJax-Span-41344\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">35.0\u00b0<\/span><\/span>\u00a0with the vertical. (a) What is the acceleration of the freight car? (b) What is the tension in the string?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036887553\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165036887555\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036887553-solution\">39<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036887557\">A student\u2019s backpack, full of textbooks, is hung from a spring scale attached to the ceiling of an elevator. When the elevator is accelerating downward at\u00a0<span id=\"MathJax-Element-1917-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41345\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41346\" class=\"mrow\"><span id=\"MathJax-Span-41347\" class=\"semantics\"><span id=\"MathJax-Span-41348\" class=\"mrow\"><span id=\"MathJax-Span-41349\" class=\"mrow\"><span id=\"MathJax-Span-41350\" class=\"mn\">3.8<\/span><span id=\"MathJax-Span-41351\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41352\" class=\"msup\"><span id=\"MathJax-Span-41353\" class=\"mrow\"><span id=\"MathJax-Span-41354\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41355\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.8m\/s2<\/span><\/span>, the scale reads 60 N. (a) What is the mass of the backpack? (b) What does the scale read if the elevator moves upward while slowing down at a rate\u00a0<span id=\"MathJax-Element-1918-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41356\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41357\" class=\"mrow\"><span id=\"MathJax-Span-41358\" class=\"semantics\"><span id=\"MathJax-Span-41359\" class=\"mrow\"><span id=\"MathJax-Span-41360\" class=\"mrow\"><span id=\"MathJax-Span-41361\" class=\"mn\">3.8<\/span><span id=\"MathJax-Span-41362\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41363\" class=\"msup\"><span id=\"MathJax-Span-41364\" class=\"mrow\"><span id=\"MathJax-Span-41365\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41366\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.8m\/s2<\/span><\/span>? (c) What does the scale read if the elevator moves upward at constant velocity? (d) If the elevator had no brakes and the cable supporting it were to break loose so that the elevator could fall freely, what would the spring scale read?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038183857\" class=\"\"><section>\r\n<div id=\"fs-id1165038183859\"><span class=\"os-number\">40<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038183861\">A service elevator takes a load of garbage, mass 10.0 kg, from a floor of a skyscraper under construction, down to ground level, accelerating downward at a rate of\u00a0<span id=\"MathJax-Element-1919-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41367\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41368\" class=\"mrow\"><span id=\"MathJax-Span-41369\" class=\"semantics\"><span id=\"MathJax-Span-41370\" class=\"mrow\"><span id=\"MathJax-Span-41371\" class=\"mrow\"><span id=\"MathJax-Span-41372\" class=\"mn\">1.2<\/span><span id=\"MathJax-Span-41373\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41374\" class=\"msup\"><span id=\"MathJax-Span-41375\" class=\"mrow\"><span id=\"MathJax-Span-41376\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41377\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.2m\/s2<\/span><\/span>. Find the magnitude of the force the garbage exerts on the floor of the service elevator?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037265785\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037265787\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037265785-solution\">41<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037265789\">A roller coaster car starts from rest at the top of a track 30.0 m long and inclined at\u00a0<span id=\"MathJax-Element-1920-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41378\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41379\" class=\"mrow\"><span id=\"MathJax-Span-41380\" class=\"semantics\"><span id=\"MathJax-Span-41381\" class=\"mrow\"><span id=\"MathJax-Span-41382\" class=\"mrow\"><span id=\"MathJax-Span-41383\" class=\"mn\">20.0<\/span><span id=\"MathJax-Span-41384\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">20.0\u00b0<\/span><\/span>\u00a0to the horizontal. Assume that friction can be ignored. (a) What is the acceleration of the car? (b) How much time elapses before it reaches the bottom of the track?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036842411\" class=\"\"><section>\r\n<div id=\"fs-id1165036842413\"><span class=\"os-number\">42<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036842415\">The device shown below is the Atwood\u2019s machine considered in\u00a0<a class=\"autogenerated-content\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e811c980-cf62-4f66-8fef-c399eb88dd2a@5#fs-id1165037923338\">Example 6.5<\/a>. Assuming that the masses of the string and the frictionless pulley are negligible, (a) find an equation for the acceleration of the two blocks; (b) find an equation for the tension in the string; and (c) find both the acceleration and tension when block 1 has mass 2.00 kg and block 2 has mass 4.00 kg.<\/p>\r\n\r\n<span id=\"fs-id1165037265194\"><img id=\"30115\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/6ae358105f6272d7737ebb4e55bf3014cc04b8bd\" alt=\"An Atwood machine consisting of masses suspended on either side of a pulley by a string passing over the pulley is shown. Mass m sub 1 is on the left and mass m sub 2 is on the right.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036779432\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165036779434\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036779432-solution\">43<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036779436\">Two blocks are connected by a massless rope as shown below. The mass of the block on the table is 4.0 kg and the hanging mass is 1.0 kg. The table and the pulley are frictionless. (a) Find the acceleration of the system. (b) Find the tension in the rope. (c) Find the speed with which the hanging mass hits the floor if it starts from rest and is initially located 1.0 m from the floor.<\/p>\r\n\r\n<span id=\"fs-id1165036963759\"><img id=\"11684\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/282733f27565e7ecb908869cf2f96f5759d35fe0\" alt=\"Block m sub 1 is on a horizontal table. It is connected to a string that passes over a pulley at the edge of the table. The string then hangs straight down and connects to block m sub 2, which is not in contact with the table. Block m sub 1 has acceleration a sub 1 directed to the right. Block m sub 2 has acceleration a sub 2 directed downward.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038010136\" class=\"\"><section>\r\n<div id=\"fs-id1165038010139\"><span class=\"os-number\">44<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038010141\">Shown below are two carts connected by a cord that passes over a small frictionless pulley. Each cart rolls freely with negligible friction. Calculate the acceleration of the carts and the tension in the cord.<\/p>\r\n\r\n<span id=\"fs-id1165038010146\"><img id=\"57774\" src=\"https:\/\/cnx.org\/resources\/0e76b273a6a0729619717c847a9f7a6f50bebfdf\" alt=\"Two carts connected by a string passing over a pulley are on either side of a double inclined plane. The string passes over a pulley attached to the top of the double incline. On the left, the incline makes an angle of 37 degrees with the horizontal and the cart on that side has mass 10 kilograms. On the right, the incline makes an angle of 53 degrees with the horizontal and the cart on that side has mass 15 kilograms.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037268006\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037063523\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037268006-solution\">45<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037063525\">A 2.00 kg block (mass 1) and a 4.00 kg block (mass 2) are connected by a light string as shown; the inclination of the ramp is\u00a0<span id=\"MathJax-Element-1921-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41385\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41386\" class=\"mrow\"><span id=\"MathJax-Span-41387\" class=\"semantics\"><span id=\"MathJax-Span-41388\" class=\"mrow\"><span id=\"MathJax-Span-41389\" class=\"mrow\"><span id=\"MathJax-Span-41390\" class=\"mn\">40.0<\/span><span id=\"MathJax-Span-41391\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">40.0\u00b0<\/span><\/span>. Friction is negligible. What is (a) the acceleration of each block and (b) the tension in the string?<\/p>\r\n\r\n<span id=\"fs-id1165037063537\"><img id=\"59880\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/62dac4eaa2d3fa779fad11d99af3ebfe423000ca\" alt=\"Block 1 is on a ramp inclined up and to the right at an angle of 40 degrees above the horizontal. It is connected to a string that passes over a pulley at the top of the ramp, then hangs straight down and connects to block 2. Block 2 is not in contact with the ramp.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"os-section-area\"><section id=\"fs-id1165036748145\" class=\"review-problems\">\r\n<h4 id=\"11340_copy_3\"><span class=\"os-number\">6.2<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Friction<\/span><\/h4>\r\n<div id=\"fs-id1165037064164\" class=\"\"><section>\r\n<div id=\"fs-id1165037064166\"><span class=\"os-number\">46<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036783994\">(a) When rebuilding his car\u2019s engine, a physics major must exert\u00a0<span id=\"MathJax-Element-1922-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41392\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41393\" class=\"mrow\"><span id=\"MathJax-Span-41394\" class=\"semantics\"><span id=\"MathJax-Span-41395\" class=\"mrow\"><span id=\"MathJax-Span-41396\" class=\"mrow\"><span id=\"MathJax-Span-41397\" class=\"mn\">3.00<\/span><span id=\"MathJax-Span-41398\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41399\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41400\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41401\" class=\"msup\"><span id=\"MathJax-Span-41402\" class=\"mrow\"><span id=\"MathJax-Span-41403\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41404\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.00\u00d7102<\/span><\/span>\u00a0N of force to insert a dry steel piston into a steel cylinder. What is the normal force between the piston and cylinder? (b) What force would he have to exert if the steel parts were oiled?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037233270\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037233273\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037233270-solution\">47<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037262674\">(a) What is the maximum frictional force in the knee joint of a person who supports 66.0 kg of her mass on that knee? (b) During strenuous exercise, it is possible to exert forces to the joints that are easily 10 times greater than the weight being supported. What is the maximum force of friction under such conditions? The frictional forces in joints are relatively small in all circumstances except when the joints deteriorate, such as from injury or arthritis. Increased frictional forces can cause further damage and pain.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037046771\" class=\"\"><section>\r\n<div id=\"fs-id1165037046773\"><span class=\"os-number\">48<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036846790\">Suppose you have a 120-kg wooden crate resting on a wood floor, with coefficient of static friction 0.500 between these wood surfaces. (a) What maximum force can you exert horizontally on the crate without moving it? (b) If you continue to exert this force once the crate starts to slip, what will its acceleration then be? The coefficient of sliding friction is known to be 0.300 for this situation.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037027752\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165038360212\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037027752-solution\">49<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038360214\">(a) If half of the weight of a small\u00a0<span id=\"MathJax-Element-1923-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41405\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41406\" class=\"mrow\"><span id=\"MathJax-Span-41407\" class=\"semantics\"><span id=\"MathJax-Span-41408\" class=\"mrow\"><span id=\"MathJax-Span-41409\" class=\"mrow\"><span id=\"MathJax-Span-41410\" class=\"mn\">1.00<\/span><span id=\"MathJax-Span-41411\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41412\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41413\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41414\" class=\"msup\"><span id=\"MathJax-Span-41415\" class=\"mrow\"><span id=\"MathJax-Span-41416\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41417\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41418\" class=\"mtext\">-kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.00\u00d7103-kg<\/span><\/span>\u00a0utility truck is supported by its two drive wheels, what is the maximum acceleration it can achieve on dry concrete? (b) Will a metal cabinet lying on the wooden bed of the truck slip if it accelerates at this rate? (c) Solve both problems assuming the truck has four-wheel drive.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037263604\" class=\"\"><section>\r\n<div id=\"fs-id1165038155161\"><span class=\"os-number\">50<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038155163\">A team of eight dogs pulls a sled with waxed wood runners on wet snow (mush!). The dogs have average masses of 19.0 kg, and the loaded sled with its rider has a mass of 210 kg. (a) Calculate the acceleration of the dogs starting from rest if each dog exerts an average force of 185 N backward on the snow. (b) Calculate the force in the coupling between the dogs and the sled.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038313967\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165038313969\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038313967-solution\">51<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036785328\">Consider the 65.0-kg ice skater being pushed by two others shown below. (a) Find the direction and magnitude of\u00a0<span id=\"MathJax-Element-1924-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41419\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41420\" class=\"mrow\"><span id=\"MathJax-Span-41421\" class=\"semantics\"><span id=\"MathJax-Span-41422\" class=\"mrow\"><span id=\"MathJax-Span-41423\" class=\"mrow\"><span id=\"MathJax-Span-41424\" class=\"msub\"><span id=\"MathJax-Span-41425\" class=\"mstyle\"><span id=\"MathJax-Span-41426\" class=\"mrow\"><span id=\"MathJax-Span-41427\" class=\"mi\">F<\/span><\/span><\/span><span id=\"MathJax-Span-41428\" class=\"mrow\"><span id=\"MathJax-Span-41429\" class=\"mtext\">tot<\/span><\/span><\/span><span id=\"MathJax-Span-41430\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Ftot,<\/span><\/span>\u00a0the total force exerted on her by the others, given that the magnitudes\u00a0<span id=\"MathJax-Element-1925-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41431\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41432\" class=\"mrow\"><span id=\"MathJax-Span-41433\" class=\"semantics\"><span id=\"MathJax-Span-41434\" class=\"mrow\"><span id=\"MathJax-Span-41435\" class=\"mrow\"><span id=\"MathJax-Span-41436\" class=\"msub\"><span id=\"MathJax-Span-41437\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41438\" class=\"mn\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F1<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1926-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41439\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41440\" class=\"mrow\"><span id=\"MathJax-Span-41441\" class=\"semantics\"><span id=\"MathJax-Span-41442\" class=\"mrow\"><span id=\"MathJax-Span-41443\" class=\"mrow\"><span id=\"MathJax-Span-41444\" class=\"msub\"><span id=\"MathJax-Span-41445\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41446\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F2<\/span><\/span>\u00a0are 26.4 N and 18.6 N, respectively. (b) What is her initial acceleration if she is initially stationary and wearing steel-bladed skates that point in the direction of\u00a0<span id=\"MathJax-Element-1927-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41447\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41448\" class=\"mrow\"><span id=\"MathJax-Span-41449\" class=\"semantics\"><span id=\"MathJax-Span-41450\" class=\"mrow\"><span id=\"MathJax-Span-41451\" class=\"mrow\"><span id=\"MathJax-Span-41452\" class=\"msub\"><span id=\"MathJax-Span-41453\" class=\"mstyle\"><span id=\"MathJax-Span-41454\" class=\"mrow\"><span id=\"MathJax-Span-41455\" class=\"mi\">F<\/span><\/span><\/span><span id=\"MathJax-Span-41456\" class=\"mrow\"><span id=\"MathJax-Span-41457\" class=\"mtext\">tot<\/span><\/span><\/span><span id=\"MathJax-Span-41458\" class=\"mo\">?<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Ftot?<\/span><\/span>\u00a0(c) What is her acceleration assuming she is already moving in the direction of\u00a0<span id=\"MathJax-Element-1928-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41459\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41460\" class=\"mrow\"><span id=\"MathJax-Span-41461\" class=\"semantics\"><span id=\"MathJax-Span-41462\" class=\"mrow\"><span id=\"MathJax-Span-41463\" class=\"mrow\"><span id=\"MathJax-Span-41464\" class=\"msub\"><span id=\"MathJax-Span-41465\" class=\"mstyle\"><span id=\"MathJax-Span-41466\" class=\"mrow\"><span id=\"MathJax-Span-41467\" class=\"mi\">F<\/span><\/span><\/span><span id=\"MathJax-Span-41468\" class=\"mrow\"><span id=\"MathJax-Span-41469\" class=\"mtext\">tot<\/span><\/span><\/span><span id=\"MathJax-Span-41470\" class=\"mo\">?<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Ftot?<\/span><\/span>\u00a0(Remember that friction always acts in the direction opposite that of motion or attempted motion between surfaces in contact.)<\/p>\r\n\r\n<span id=\"fs-id1165037171810\"><img id=\"52665\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/00c2f606486ab5e6defc00a56a9f5c43a94541b4\" alt=\"(a) Overhead view of two ice skaters pushing on a third. One skater pushes with a force F one, represented by an arrow pointing to the right, and a second skater pushes with a force F two, represented by an arrow pointing up. Vector F one and vector F two are along the arms of the two skaters acting on the third skater. A vector diagram is shown in the form of a right triangle in which the base is vector F one pointing to the right, and perpendicular to F one is vector F two pointing up. The resultant vector is shown by the hypotenuse pointing up and to the right and is labeled as vector F sub tot. (b) Free body diagram showing only the forces F sub one and F sub 2 acting on the skater.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037014648\" class=\"\"><section>\r\n<div id=\"fs-id1165036965255\"><span class=\"os-number\">52<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036965257\">Show that the acceleration of any object down a frictionless incline that makes an angle\u00a0<span id=\"MathJax-Element-1929-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41471\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41472\" class=\"mrow\"><span id=\"MathJax-Span-41473\" class=\"semantics\"><span id=\"MathJax-Span-41474\" class=\"mrow\"><span id=\"MathJax-Span-41475\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0with the horizontal is\u00a0<span id=\"MathJax-Element-1930-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41476\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41477\" class=\"mrow\"><span id=\"MathJax-Span-41478\" class=\"semantics\"><span id=\"MathJax-Span-41479\" class=\"mrow\"><span id=\"MathJax-Span-41480\" class=\"mrow\"><span id=\"MathJax-Span-41481\" class=\"mi\">a<\/span><span id=\"MathJax-Span-41482\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41483\" class=\"mi\">g<\/span><span id=\"MathJax-Span-41484\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41485\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-41486\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41487\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">a=gsin\u03b8<\/span><\/span>. (Note that this acceleration is independent of mass.)<\/p>\r\n\r\n<span id=\"fs-id1165038017726\"><img id=\"98748\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/76d33d47d237582d2d0c36930d628dab7a69e64d\" alt=\"An illustration of  block on  a slope. The slope angles down and to the right at an angle of theta degrees to the horizontal. The block has an acceleration a parallel to the slope, toward its bottom. The following forces are shown: N perpendicular to the slope and pointing out of it, and w which equals m times g vertically down. An x y coordinate system is shown tilted so that positive x is downslope, parallel to the surface, and positive y is perpendicular to the slope, pointing out of the surface.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038010694\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165038244292\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038010694-solution\">53<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038244294\">Show that the acceleration of any object down an incline where friction behaves simply (that is, where\u00a0<span id=\"MathJax-Element-1931-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41488\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41489\" class=\"mrow\"><span id=\"MathJax-Span-41490\" class=\"semantics\"><span id=\"MathJax-Span-41491\" class=\"mrow\"><span id=\"MathJax-Span-41492\" class=\"mrow\"><span id=\"MathJax-Span-41493\" class=\"msub\"><span id=\"MathJax-Span-41494\" class=\"mi\">f<\/span><span id=\"MathJax-Span-41495\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-41496\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41497\" class=\"msub\"><span id=\"MathJax-Span-41498\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41499\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-41500\" class=\"mi\">N<\/span><span id=\"MathJax-Span-41501\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">fk=\u03bckN)<\/span><\/span>\u00a0is\u00a0<span id=\"MathJax-Element-1932-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41502\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41503\" class=\"mrow\"><span id=\"MathJax-Span-41504\" class=\"semantics\"><span id=\"MathJax-Span-41505\" class=\"mrow\"><span id=\"MathJax-Span-41506\" class=\"mrow\"><span id=\"MathJax-Span-41507\" class=\"mi\">a<\/span><span id=\"MathJax-Span-41508\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41509\" class=\"mi\">g<\/span><span id=\"MathJax-Span-41510\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41511\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-41512\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41513\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-41514\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-41515\" class=\"msub\"><span id=\"MathJax-Span-41516\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41517\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-41518\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41519\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-41520\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41521\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-41522\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41523\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">a=g(sin\u03b8\u2212\u03bckcos\u03b8).<\/span><\/span>\u00a0Note that the acceleration is independent of mass and reduces to the expression found in the previous problem when friction becomes negligibly small\u00a0<span id=\"MathJax-Element-1933-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41524\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41525\" class=\"mrow\"><span id=\"MathJax-Span-41526\" class=\"semantics\"><span id=\"MathJax-Span-41527\" class=\"mrow\"><span id=\"MathJax-Span-41528\" class=\"mrow\"><span id=\"MathJax-Span-41529\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41530\" class=\"msub\"><span id=\"MathJax-Span-41531\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41532\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-41533\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41534\" class=\"mn\">0<\/span><span id=\"MathJax-Span-41535\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41536\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(\u03bck=0).<\/span><\/span><\/p>\r\n\r\n<span id=\"fs-id1165036982731\"><img id=\"58987\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/617457a351deabedf57f3137a869928efd7ad8e0\" alt=\"An illustration of  block on  a slope. The slope angles down and to the right at an angle of theta degrees to the horizontal. The block has an acceleration, a, parallel to the slope, toward its bottom. The following forces are shown:  f in a direction parallel to the slope toward its top, N perpendicular to the slope and pointing out of it, w sub x in a direction parallel to the slope toward its bottom, and w sub y perpendicular to the slope and pointing into it. An x y coordinate system is shown tilted so that positive x is downslope, parallel to the surface, and positive y is perpendicular to the slope, pointing out of the surface.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037218975\" class=\"\"><section>\r\n<div id=\"fs-id1165037218978\"><span class=\"os-number\">54<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036859341\">Calculate the deceleration of a snow boarder going up a\u00a0<span id=\"MathJax-Element-1934-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41537\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41538\" class=\"mrow\"><span id=\"MathJax-Span-41539\" class=\"semantics\"><span id=\"MathJax-Span-41540\" class=\"mrow\"><span id=\"MathJax-Span-41541\" class=\"mrow\"><span id=\"MathJax-Span-41542\" class=\"mn\">5.00<\/span><span id=\"MathJax-Span-41543\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.00\u00b0<\/span><\/span>\u00a0slope, assuming the coefficient of friction for waxed wood on wet snow. The result of the preceding problem may be useful, but be careful to consider the fact that the snow boarder is going uphill.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037987755\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037987757\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037987755-solution\">55<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037159582\">A machine at a post office sends packages out a chute and down a ramp to be loaded into delivery vehicles. (a) Calculate the acceleration of a box heading down a\u00a0<span id=\"MathJax-Element-1935-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41544\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41545\" class=\"mrow\"><span id=\"MathJax-Span-41546\" class=\"semantics\"><span id=\"MathJax-Span-41547\" class=\"mrow\"><span id=\"MathJax-Span-41548\" class=\"mrow\"><span id=\"MathJax-Span-41549\" class=\"mn\">10.0<\/span><span id=\"MathJax-Span-41550\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">10.0\u00b0<\/span><\/span>\u00a0slope, assuming the coefficient of friction for a parcel on waxed wood is 0.100. (b) Find the angle of the slope down which this box could move at a constant velocity. You can neglect air resistance in both parts.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037229369\" class=\"\"><section>\r\n<div id=\"fs-id1165037229371\"><span class=\"os-number\">56<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038250675\">If an object is to rest on an incline without slipping, then friction must equal the component of the weight of the object parallel to the incline. This requires greater and greater friction for steeper slopes. Show that the maximum angle of an incline above the horizontal for which an object will not slide down is\u00a0<span id=\"MathJax-Element-1936-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41551\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41552\" class=\"mrow\"><span id=\"MathJax-Span-41553\" class=\"semantics\"><span id=\"MathJax-Span-41554\" class=\"mrow\"><span id=\"MathJax-Span-41555\" class=\"mrow\"><span id=\"MathJax-Span-41556\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-41557\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41558\" class=\"msup\"><span id=\"MathJax-Span-41559\" class=\"mrow\"><span id=\"MathJax-Span-41560\" class=\"mtext\">tan<\/span><\/span><span id=\"MathJax-Span-41561\" class=\"mrow\"><span id=\"MathJax-Span-41562\" class=\"mn\">\u22121<\/span><\/span><\/span><span id=\"MathJax-Span-41563\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41564\" class=\"msub\"><span id=\"MathJax-Span-41565\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41566\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41567\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8=tan\u22121\u03bcs.<\/span><\/span>\u00a0You may use the result of the previous problem. Assume that\u00a0<span id=\"MathJax-Element-1937-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41568\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41569\" class=\"mrow\"><span id=\"MathJax-Span-41570\" class=\"semantics\"><span id=\"MathJax-Span-41571\" class=\"mrow\"><span id=\"MathJax-Span-41572\" class=\"mrow\"><span id=\"MathJax-Span-41573\" class=\"mi\">a<\/span><span id=\"MathJax-Span-41574\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41575\" class=\"mn\">0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">a=0<\/span><\/span>\u00a0and that static friction has reached its maximum value.<\/p>\r\n\r\n<span id=\"fs-id1165038163415\"><img id=\"3620\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/d38c3860133768d93151529b68829c3123e41e0c\" alt=\"An illustration of  a block mass m on  a slope. The slope angles up and to the right at an angle of theta degrees to the horizontal. The mass feels force w sub parallel in a direction parallel to the slope toward its bottom, and f in a direction parallel to the slope toward its top.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037112519\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037112521\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037112519-solution\">57<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037112524\">Calculate the maximum acceleration of a car that is heading down a\u00a0<span id=\"MathJax-Element-1938-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41576\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41577\" class=\"mrow\"><span id=\"MathJax-Span-41578\" class=\"semantics\"><span id=\"MathJax-Span-41579\" class=\"mrow\"><span id=\"MathJax-Span-41580\" class=\"mrow\"><span id=\"MathJax-Span-41581\" class=\"mn\">6.00<\/span><span id=\"MathJax-Span-41582\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">6.00\u00b0<\/span><\/span>\u00a0slope (one that makes an angle of\u00a0<span id=\"MathJax-Element-1939-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41583\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41584\" class=\"mrow\"><span id=\"MathJax-Span-41585\" class=\"semantics\"><span id=\"MathJax-Span-41586\" class=\"mrow\"><span id=\"MathJax-Span-41587\" class=\"mrow\"><span id=\"MathJax-Span-41588\" class=\"mn\">6.00<\/span><span id=\"MathJax-Span-41589\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">6.00\u00b0<\/span><\/span>\u00a0with the horizontal) under the following road conditions. You may assume that the weight of the car is evenly distributed on all four tires and that the coefficient of static friction is involved\u2014that is, the tires are not allowed to slip during the deceleration. (Ignore rolling.) Calculate for a car: (a) On dry concrete. (b) On wet concrete. (c) On ice, assuming that\u00a0<span id=\"MathJax-Element-1940-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41590\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41591\" class=\"mrow\"><span id=\"MathJax-Span-41592\" class=\"semantics\"><span id=\"MathJax-Span-41593\" class=\"mrow\"><span id=\"MathJax-Span-41594\" class=\"mrow\"><span id=\"MathJax-Span-41595\" class=\"msub\"><span id=\"MathJax-Span-41596\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41597\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41598\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41599\" class=\"mn\">0.100<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bcs=0.100<\/span><\/span>, the same as for shoes on ice.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038247675\" class=\"\"><section>\r\n<div id=\"fs-id1165038247677\"><span class=\"os-number\">58<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038247679\">Calculate the maximum acceleration of a car that is heading up a\u00a0<span id=\"MathJax-Element-1941-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41600\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41601\" class=\"mrow\"><span id=\"MathJax-Span-41602\" class=\"semantics\"><span id=\"MathJax-Span-41603\" class=\"mrow\"><span id=\"MathJax-Span-41604\" class=\"mrow\"><span id=\"MathJax-Span-41605\" class=\"mn\">4.00<\/span><span id=\"MathJax-Span-41606\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">4.00\u00b0<\/span><\/span>\u00a0slope (one that makes an angle of\u00a0<span id=\"MathJax-Element-1942-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41607\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41608\" class=\"mrow\"><span id=\"MathJax-Span-41609\" class=\"semantics\"><span id=\"MathJax-Span-41610\" class=\"mrow\"><span id=\"MathJax-Span-41611\" class=\"mrow\"><span id=\"MathJax-Span-41612\" class=\"mn\">4.00<\/span><span id=\"MathJax-Span-41613\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">4.00\u00b0<\/span><\/span>\u00a0with the horizontal) under the following road conditions. Assume that only half the weight of the car is supported by the two drive wheels and that the coefficient of static friction is involved\u2014that is, the tires are not allowed to slip during the acceleration. (Ignore rolling.) (a) On dry concrete. (b) On wet concrete. (c) On ice, assuming that\u00a0<span id=\"MathJax-Element-1943-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41614\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41615\" class=\"mrow\"><span id=\"MathJax-Span-41616\" class=\"semantics\"><span id=\"MathJax-Span-41617\" class=\"mrow\"><span id=\"MathJax-Span-41618\" class=\"mrow\"><span id=\"MathJax-Span-41619\" class=\"msub\"><span id=\"MathJax-Span-41620\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41621\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41622\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41623\" class=\"mn\">0.100<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bcs=0.100<\/span><\/span>, the same as for shoes on ice.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038248967\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037089573\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038248967-solution\">59<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037089575\">Repeat the preceding problem for a car with four-wheel drive.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036967840\" class=\"\"><section>\r\n<div id=\"fs-id1165038009903\"><span class=\"os-number\">60<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038009905\">A freight train consists of two\u00a0<span id=\"MathJax-Element-1944-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41624\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41625\" class=\"mrow\"><span id=\"MathJax-Span-41626\" class=\"semantics\"><span id=\"MathJax-Span-41627\" class=\"mrow\"><span id=\"MathJax-Span-41628\" class=\"mrow\"><span id=\"MathJax-Span-41629\" class=\"mn\">8.00<\/span><span id=\"MathJax-Span-41630\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41631\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41632\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41633\" class=\"msup\"><span id=\"MathJax-Span-41634\" class=\"mrow\"><span id=\"MathJax-Span-41635\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41636\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-41637\" class=\"mtext\">-kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">8.00\u00d7105-kg<\/span><\/span>\u00a0engines and 45 cars with average masses of\u00a0<span id=\"MathJax-Element-1945-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41638\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41639\" class=\"mrow\"><span id=\"MathJax-Span-41640\" class=\"semantics\"><span id=\"MathJax-Span-41641\" class=\"mrow\"><span id=\"MathJax-Span-41642\" class=\"mrow\"><span id=\"MathJax-Span-41643\" class=\"mn\">5.50<\/span><span id=\"MathJax-Span-41644\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41645\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41646\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41647\" class=\"msup\"><span id=\"MathJax-Span-41648\" class=\"mrow\"><span id=\"MathJax-Span-41649\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41650\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-41651\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41652\" class=\"mtext\">kg<\/span><span id=\"MathJax-Span-41653\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.50\u00d7105kg.<\/span><\/span>\u00a0(a) What force must each engine exert backward on the track to accelerate the train at a rate of\u00a0<span id=\"MathJax-Element-1946-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41654\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41655\" class=\"mrow\"><span id=\"MathJax-Span-41656\" class=\"semantics\"><span id=\"MathJax-Span-41657\" class=\"mrow\"><span id=\"MathJax-Span-41658\" class=\"mrow\"><span id=\"MathJax-Span-41659\" class=\"mn\">5.00<\/span><span id=\"MathJax-Span-41660\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41661\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41662\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41663\" class=\"msup\"><span id=\"MathJax-Span-41664\" class=\"mrow\"><span id=\"MathJax-Span-41665\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41666\" class=\"mrow\"><span id=\"MathJax-Span-41667\" class=\"mn\">\u22122<\/span><\/span><\/span><span id=\"MathJax-Span-41668\" class=\"mtext\">m<\/span><span id=\"MathJax-Span-41669\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-41670\" class=\"msup\"><span id=\"MathJax-Span-41671\" class=\"mtext\">s<\/span><span id=\"MathJax-Span-41672\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.00\u00d710\u22122m\/s2<\/span><\/span>\u00a0if the force of friction is\u00a0<span id=\"MathJax-Element-1947-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41673\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41674\" class=\"mrow\"><span id=\"MathJax-Span-41675\" class=\"semantics\"><span id=\"MathJax-Span-41676\" class=\"mrow\"><span id=\"MathJax-Span-41677\" class=\"mrow\"><span id=\"MathJax-Span-41678\" class=\"mn\">7.50<\/span><span id=\"MathJax-Span-41679\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41680\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41681\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41682\" class=\"msup\"><span id=\"MathJax-Span-41683\" class=\"mrow\"><span id=\"MathJax-Span-41684\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41685\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-41686\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">7.50\u00d7105N<\/span><\/span>, assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently, trains are very energy-efficient transportation systems. (b) What is the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038191800\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165038191802\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038191800-solution\">61<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038191805\">Consider the 52.0-kg mountain climber shown below. (a) Find the tension in the rope and the force that the mountain climber must exert with her feet on the vertical rock face to remain stationary. Assume that the force is exerted parallel to her legs. Also, assume negligible force exerted by her arms. (b) What is the minimum coefficient of friction between her shoes and the cliff?<\/p>\r\n\r\n<span id=\"fs-id1165037089825\"><img id=\"80081\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/9d7620350be4ecabd3c0406466d44890a59c9ec6\" alt=\"A mountain climber is drawn leaning away from the rock face with her feet against the rock face. The rope extends up from the climber  at an angle of 31 degrees to the vertical. The climbers legs are straight and make an angle of fifteen degrees with the rock face. The force vector F sub T starts at the harness and points away from the climber, along the rope. The force vector F sub legs starts at climber\u2019s feet and points away from the rock, parallel to her legs.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036765363\" class=\"\"><section>\r\n<div id=\"fs-id1165038132570\"><span class=\"os-number\">62<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038132572\">A contestant in a winter sporting event pushes a 45.0-kg block of ice across a frozen lake as shown below. (a) Calculate the minimum force\u00a0<em>F<\/em>\u00a0he must exert to get the block moving. (b) What is its acceleration once it starts to move, if that force is maintained?<\/p>\r\n\r\n<span id=\"fs-id1165037166007\"><img id=\"53067\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/9e698d2bb8ad6d6b3aa5ea3f56df6b354680c701\" alt=\"A block of ice is being pushed with a force F that is directed at an angle of twenty five degrees below the horizontal.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165037150306\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165037150308\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037150306-solution\">63<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165037843834\">The contestant now pulls the block of ice with a rope over his shoulder at the same angle above the horizontal as shown below. Calculate the minimum force\u00a0<em>F<\/em>\u00a0he must exert to get the block moving. (b) What is its acceleration once it starts to move, if that force is maintained?<\/p>\r\n\r\n<span id=\"fs-id1165037063269\"><img id=\"64358\" src=\"https:\/\/cnx.org\/resources\/1f0c563cc53fe02196a93993dfc155d7532aa7de\" alt=\"A block of ice is being pulled with a force F that is directed at an angle of twenty five degrees above the horizontal.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165034581253\" class=\"\"><section>\r\n<div id=\"fs-id1165034581255\"><span class=\"os-number\">64<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038000706\">At a post office, a parcel that is a 20.0-kg box slides down a ramp inclined at\u00a0<span id=\"MathJax-Element-1948-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41687\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41688\" class=\"mrow\"><span id=\"MathJax-Span-41689\" class=\"semantics\"><span id=\"MathJax-Span-41690\" class=\"mrow\"><span id=\"MathJax-Span-41691\" class=\"mrow\"><span id=\"MathJax-Span-41692\" class=\"mn\">30.0<\/span><span id=\"MathJax-Span-41693\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">30.0\u00b0<\/span><\/span>\u00a0with the horizontal. The coefficient of kinetic friction between the box and plane is 0.0300. (a) Find the acceleration of the box. (b) Find the velocity of the box as it reaches the end of the plane, if the length of the plane is 2 m and the box starts at rest.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"os-section-area\"><section id=\"fs-id1165035734743\" class=\"review-problems\">\r\n<h4 id=\"71765_copy_3\"><span class=\"os-number\">6.3<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Centripetal Force<\/span><\/h4>\r\n<div id=\"fs-id1165039087650\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039479306\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039087650-solution\">65<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035682097\">(a) A 22.0-kg child is riding a playground merry-go-round that is rotating at 40.0 rev\/min. What centripetal force is exerted if he is 1.25 m from its center? (b) What centripetal force is exerted if the merry-go-round rotates at 3.00 rev\/min and he is 8.00 m from its center? (c) Compare each force with his weight.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039100614\" class=\"\"><section>\r\n<div id=\"fs-id1165039242547\"><span class=\"os-number\">66<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039209500\">Calculate the centripetal force on the end of a 100-m (radius) wind turbine blade that is rotating at 0.5 rev\/s. Assume the mass is 4 kg.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039098944\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035636247\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039098944-solution\">67<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039438998\">What is the ideal banking angle for a gentle turn of 1.20-km radius on a highway with a 105 km\/h speed limit (about 65 mi\/h), assuming everyone travels at the limit?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036159224\" class=\"\"><section>\r\n<div id=\"fs-id1165039085659\"><span class=\"os-number\">68<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165038974988\">What is the ideal speed to take a 100.0-m-radius curve banked at a\u00a0<span id=\"MathJax-Element-1949-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41694\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41695\" class=\"mrow\"><span id=\"MathJax-Span-41696\" class=\"semantics\"><span id=\"MathJax-Span-41697\" class=\"mrow\"><span id=\"MathJax-Span-41698\" class=\"mrow\"><span id=\"MathJax-Span-41699\" class=\"mn\">20.0<\/span><span id=\"MathJax-Span-41700\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">20.0\u00b0<\/span><\/span>\u00a0angle?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039399485\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035619493\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039399485-solution\">69<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039104015\">(a) What is the radius of a bobsled turn banked at\u00a0<span id=\"MathJax-Element-1950-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41701\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41702\" class=\"mrow\"><span id=\"MathJax-Span-41703\" class=\"semantics\"><span id=\"MathJax-Span-41704\" class=\"mrow\"><span id=\"MathJax-Span-41705\" class=\"mrow\"><span id=\"MathJax-Span-41706\" class=\"mn\">75.0<\/span><span id=\"MathJax-Span-41707\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">75.0\u00b0<\/span><\/span>\u00a0and taken at 30.0 m\/s, assuming it is ideally banked? (b) Calculate the centripetal acceleration. (c) Does this acceleration seem large to you?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039426342\" class=\"\"><section>\r\n<div id=\"fs-id1165038981681\"><span class=\"os-number\">70<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039276298\">Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen below. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components\u2014friction parallel to the road (this must supply the centripetal force) and the vertical normal force (which must equal the system\u2019s weight). (a) Show that\u00a0<span id=\"MathJax-Element-1951-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41708\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41709\" class=\"mrow\"><span id=\"MathJax-Span-41710\" class=\"semantics\"><span id=\"MathJax-Span-41711\" class=\"mrow\"><span id=\"MathJax-Span-41712\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0(as defined as shown) is related to the speed\u00a0<em>v<\/em>\u00a0and radius of curvature\u00a0<em>r<\/em>\u00a0of the turn in the same way as for an ideally banked roadway\u2014that is,\u00a0<span id=\"MathJax-Element-1952-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41713\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41714\" class=\"mrow\"><span id=\"MathJax-Span-41715\" class=\"semantics\"><span id=\"MathJax-Span-41716\" class=\"mrow\"><span id=\"MathJax-Span-41717\" class=\"mrow\"><span id=\"MathJax-Span-41718\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-41719\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41720\" class=\"msup\"><span id=\"MathJax-Span-41721\" class=\"mrow\"><span id=\"MathJax-Span-41722\" class=\"mtext\">tan<\/span><\/span><span id=\"MathJax-Span-41723\" class=\"mrow\"><span id=\"MathJax-Span-41724\" class=\"mn\">\u22121<\/span><\/span><\/span><span id=\"MathJax-Span-41725\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41726\" class=\"msup\"><span id=\"MathJax-Span-41727\" class=\"mi\">v<\/span><span id=\"MathJax-Span-41728\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-41729\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-41730\" class=\"mi\">r<\/span><span id=\"MathJax-Span-41731\" class=\"mi\">g<\/span><span id=\"MathJax-Span-41732\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41733\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8=tan\u22121(v2\/rg).<\/span><\/span>\u00a0(b) Calculate\u00a0<span id=\"MathJax-Element-1953-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41734\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41735\" class=\"mrow\"><span id=\"MathJax-Span-41736\" class=\"semantics\"><span id=\"MathJax-Span-41737\" class=\"mrow\"><span id=\"MathJax-Span-41738\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0for a 12.0-m\/s turn of radius 30.0 m (as in a race).<\/p>\r\n\r\n<span id=\"fs-id1165039285032\"><img id=\"37715\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/adc368237be4c9d5ccf02856a9df08c8af510b82\" alt=\"The figure is an illustration of a man riding a bicycle, viewed from the front. The rider and bike are tilted to the right at an angle theta to the vertical. Three force vectors are shown as solid line arrows. One is from the bottom of the front wheel to the right showing the centripetal force F sub c. A second is from the same point vertically upward showing the force N. The third is from the chest of the rider vertically downward showing his weight, w. An additional broken line arrow from the bottom of the wheel to the chest point, at an angle theta to the right of vertical, is also shown and labeled with force F exerting on it.  The vectors F sub c, w and F form a right triangle whose hypotenuse is F. A free-body diagram is also given above the figure showing vectors w and F. The vector relations F equals the sum of N and F sub c, and N equals w are also given alongside the figure.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039512072\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039115696\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039512072-solution\">71<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039115698\">If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a problem on icy mountain roads). (a) Calculate the ideal speed to take a 100.0 m radius curve banked at\u00a0<span id=\"MathJax-Element-1954-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41739\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41740\" class=\"mrow\"><span id=\"MathJax-Span-41741\" class=\"semantics\"><span id=\"MathJax-Span-41742\" class=\"mrow\"><span id=\"MathJax-Span-41743\" class=\"mrow\"><span id=\"MathJax-Span-41744\" class=\"mn\">15.0<\/span><span id=\"MathJax-Span-41745\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">15.0\u00b0<\/span><\/span>. (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km\/h?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039045013\" class=\"\"><section>\r\n<div id=\"fs-id1165039045015\"><span class=\"os-number\">72<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039125365\">Modern roller coasters have vertical loops like the one shown here. The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into their seats. (a) What is the speed of the roller coaster at the top of the loop if the radius of curvature there is 15.0 m and the downward acceleration of the car is 1.50\u00a0<em>g<\/em>? (b) How high above the top of the loop must the roller coaster start from rest, assuming negligible friction? (c) If it actually starts 5.00 m higher than your answer to (b), how much energy did it lose to friction? Its mass is\u00a0<span id=\"MathJax-Element-1955-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41746\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41747\" class=\"mrow\"><span id=\"MathJax-Span-41748\" class=\"semantics\"><span id=\"MathJax-Span-41749\" class=\"mrow\"><span id=\"MathJax-Span-41750\" class=\"mrow\"><span id=\"MathJax-Span-41751\" class=\"mn\">1.50<\/span><span id=\"MathJax-Span-41752\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41753\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41754\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41755\" class=\"msup\"><span id=\"MathJax-Span-41756\" class=\"mrow\"><span id=\"MathJax-Span-41757\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41758\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41759\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41760\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.50\u00d7103kg<\/span><\/span>.<\/p>\r\n\r\n<span id=\"fs-id1165035733795\"><img id=\"48524\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/0cd33aa8c952fc54cd87c3388a9be18abd783fe7\" alt=\"An illustration of a loop of a roller. The radius of curvature is smaller at the top than on the sides and bottom. The radius of the loop at the tom is shown and labeled as r sub minimum. The radius at the lowest part of the loop is labeled as r sub maximum.  The track is on the inside surface of the loop. The motion is indicated by arrows, starting at ground level to the right of the loop, going up inside the loop on the left, then down the inside right of the loop, and out again at ground level on the left. Four location on the track, A, B, C, and D and B, are labeled. Point A is at ground level, to the right of the loop, where the track is straight and horizontal. Point B is part way up the left side of the loop. Point C is part way up the right side of the loop, at the same level as point B. Point D is at ground level, to the left of the loop, where the track is straight and horizontal.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039091562\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039234647\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039091562-solution\">73<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039234649\">A child of mass 40.0 kg is in a roller coaster car that travels in a loop of radius 7.00 m. At point A the speed of the car is 10.0 m\/s, and at point B, the speed is 10.5 m\/s. Assume the child is not holding on and does not wear a seat belt. (a) What is the force of the car seat on the child at point A? (b) What is the force of the car seat on the child at point B? (c) What minimum speed is required to keep the child in his seat at point A?<\/p>\r\n\r\n<span id=\"fs-id1165039439426\"><img id=\"12769\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/f93261c5ebe5098833fc3e47126687e2fc34168f\" alt=\"An illustration of a loop of a roller coaster with a child seated in a car approaching the loop. The track is on the inside surface of the loop. Two location on the loop, A and B, are labeled. Point A is at the top of the loop. Point B is down and to the left of A. The angle between the radii to points A and B is thirty degrees.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039445011\" class=\"\"><section>\r\n<div id=\"fs-id1165039445013\"><span class=\"os-number\">74<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039516436\">In the simple Bohr model of the ground state of the hydrogen atom, the electron travels in a circular orbit around a fixed proton. The radius of the orbit is\u00a0<span id=\"MathJax-Element-1956-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41761\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41762\" class=\"mrow\"><span id=\"MathJax-Span-41763\" class=\"semantics\"><span id=\"MathJax-Span-41764\" class=\"mrow\"><span id=\"MathJax-Span-41765\" class=\"mrow\"><span id=\"MathJax-Span-41766\" class=\"mn\">5.28<\/span><span id=\"MathJax-Span-41767\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41768\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41769\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41770\" class=\"msup\"><span id=\"MathJax-Span-41771\" class=\"mrow\"><span id=\"MathJax-Span-41772\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41773\" class=\"mrow\"><span id=\"MathJax-Span-41774\" class=\"mn\">\u221211<\/span><\/span><\/span><span id=\"MathJax-Span-41775\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41776\" class=\"mtext\">m,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.28\u00d710\u221211m,<\/span><\/span>\u00a0and the speed of the electron is\u00a0<span id=\"MathJax-Element-1957-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41777\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41778\" class=\"mrow\"><span id=\"MathJax-Span-41779\" class=\"semantics\"><span id=\"MathJax-Span-41780\" class=\"mrow\"><span id=\"MathJax-Span-41781\" class=\"mrow\"><span id=\"MathJax-Span-41782\" class=\"mn\">2.18<\/span><span id=\"MathJax-Span-41783\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41784\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41785\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41786\" class=\"msup\"><span id=\"MathJax-Span-41787\" class=\"mrow\"><span id=\"MathJax-Span-41788\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41789\" class=\"mn\">6<\/span><\/span><span id=\"MathJax-Span-41790\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41791\" class=\"mrow\"><span id=\"MathJax-Span-41792\" class=\"mtext\">m<\/span><span id=\"MathJax-Span-41793\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-41794\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41795\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.18\u00d7106m\/s.<\/span><\/span>\u00a0The mass of an electron is\u00a0<span id=\"MathJax-Element-1958-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41796\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41797\" class=\"mrow\"><span id=\"MathJax-Span-41798\" class=\"semantics\"><span id=\"MathJax-Span-41799\" class=\"mrow\"><span id=\"MathJax-Span-41800\" class=\"mrow\"><span id=\"MathJax-Span-41801\" class=\"mn\">9.11<\/span><span id=\"MathJax-Span-41802\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41803\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41804\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41805\" class=\"msup\"><span id=\"MathJax-Span-41806\" class=\"mrow\"><span id=\"MathJax-Span-41807\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41808\" class=\"mrow\"><span id=\"MathJax-Span-41809\" class=\"mn\">\u221231<\/span><\/span><\/span><span id=\"MathJax-Span-41810\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41811\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">9.11\u00d710\u221231kg<\/span><\/span>. What is the force on the electron?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035610496\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035610498\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035610496-solution\">75<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039421074\">Railroad tracks follow a circular curve of radius 500.0 m and are banked at an angle of\u00a0<span id=\"MathJax-Element-1959-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41812\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41813\" class=\"mrow\"><span id=\"MathJax-Span-41814\" class=\"semantics\"><span id=\"MathJax-Span-41815\" class=\"mrow\"><span id=\"MathJax-Span-41816\" class=\"mrow\"><span id=\"MathJax-Span-41817\" class=\"mn\">5.0<\/span><span id=\"MathJax-Span-41818\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.0\u00b0<\/span><\/span>. For trains of what speed are these tracks designed?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039295768\" class=\"\"><section>\r\n<div id=\"fs-id1165039440552\"><span class=\"os-number\">76<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039440554\">The CERN particle accelerator is circular with a circumference of 7.0 km. (a) What is the acceleration of the protons\u00a0<span id=\"MathJax-Element-1960-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41819\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41820\" class=\"mrow\"><span id=\"MathJax-Span-41821\" class=\"semantics\"><span id=\"MathJax-Span-41822\" class=\"mrow\"><span id=\"MathJax-Span-41823\" class=\"mrow\"><span id=\"MathJax-Span-41824\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41825\" class=\"mi\">m<\/span><span id=\"MathJax-Span-41826\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41827\" class=\"mn\">1.67<\/span><span id=\"MathJax-Span-41828\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41829\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41830\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41831\" class=\"msup\"><span id=\"MathJax-Span-41832\" class=\"mrow\"><span id=\"MathJax-Span-41833\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41834\" class=\"mrow\"><span id=\"MathJax-Span-41835\" class=\"mn\">\u221227<\/span><\/span><\/span><span id=\"MathJax-Span-41836\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41837\" class=\"mtext\">kg<\/span><span id=\"MathJax-Span-41838\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(m=1.67\u00d710\u221227kg)<\/span><\/span>\u00a0that move around the accelerator at\u00a0<span id=\"MathJax-Element-1961-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41839\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41840\" class=\"mrow\"><span id=\"MathJax-Span-41841\" class=\"semantics\"><span id=\"MathJax-Span-41842\" class=\"mrow\"><span id=\"MathJax-Span-41843\" class=\"mrow\"><span id=\"MathJax-Span-41844\" class=\"mn\">5<\/span><span id=\"MathJax-Span-41845\" class=\"mi\">%<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5%<\/span><\/span>\u00a0of the speed of light? (The speed of light is\u00a0<span id=\"MathJax-Element-1962-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41846\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41847\" class=\"mrow\"><span id=\"MathJax-Span-41848\" class=\"semantics\"><span id=\"MathJax-Span-41849\" class=\"mrow\"><span id=\"MathJax-Span-41850\" class=\"mrow\"><span id=\"MathJax-Span-41851\" class=\"mi\">v<\/span><span id=\"MathJax-Span-41852\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41853\" class=\"mn\">3.00<\/span><span id=\"MathJax-Span-41854\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41855\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41856\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41857\" class=\"msup\"><span id=\"MathJax-Span-41858\" class=\"mrow\"><span id=\"MathJax-Span-41859\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41860\" class=\"mn\">8<\/span><\/span><span id=\"MathJax-Span-41861\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41862\" class=\"mtext\">m\/s<\/span><span id=\"MathJax-Span-41863\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v=3.00\u00d7108m\/s.<\/span><\/span>) (b) What is the force on the protons?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039071024\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039237696\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039071024-solution\">77<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039237698\">A car rounds an unbanked curve of radius 65 m. If the coefficient of static friction between the road and car is 0.70, what is the maximum speed at which the car traverse the curve without slipping?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039315651\" class=\"\"><section>\r\n<div id=\"fs-id1165039513150\"><span class=\"os-number\">78<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039513152\">A banked highway is designed for traffic moving at 90.0 km\/h. The radius of the curve is 310 m. What is the angle of banking of the highway?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"os-section-area\"><section id=\"fs-id1165039390902\" class=\"review-problems\">\r\n<h4 id=\"47362_copy_3\"><span class=\"os-number\">6.4<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Drag Force and Terminal Speed<\/span><\/h4>\r\n<div id=\"fs-id1165039251723\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039092545\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039251723-solution\">79<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035682260\">The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Find the terminal velocity (in meters per second and kilometers per hour) of an 80.0-kg skydiver falling in a pike (headfirst) position with a surface area of\u00a0<span id=\"MathJax-Element-1963-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41864\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41865\" class=\"mrow\"><span id=\"MathJax-Span-41866\" class=\"semantics\"><span id=\"MathJax-Span-41867\" class=\"mrow\"><span id=\"MathJax-Span-41868\" class=\"mrow\"><span id=\"MathJax-Span-41869\" class=\"mn\">0.140<\/span><span id=\"MathJax-Span-41870\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41871\" class=\"msup\"><span id=\"MathJax-Span-41872\" class=\"mrow\"><span id=\"MathJax-Span-41873\" class=\"mtext\">m<\/span><\/span><span id=\"MathJax-Span-41874\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.140m2<\/span><\/span>.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035906453\" class=\"\"><section>\r\n<div id=\"fs-id1165036153774\"><span class=\"os-number\">80<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035682242\">A 60.0-kg and a 90.0-kg skydiver jump from an airplane at an altitude of\u00a0<span id=\"MathJax-Element-1964-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41875\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41876\" class=\"mrow\"><span id=\"MathJax-Span-41877\" class=\"semantics\"><span id=\"MathJax-Span-41878\" class=\"mrow\"><span id=\"MathJax-Span-41879\" class=\"mrow\"><span id=\"MathJax-Span-41880\" class=\"mn\">6.00<\/span><span id=\"MathJax-Span-41881\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41882\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41883\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41884\" class=\"msup\"><span id=\"MathJax-Span-41885\" class=\"mrow\"><span id=\"MathJax-Span-41886\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41887\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41888\" class=\"mtext\">m<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">6.00\u00d7103m<\/span><\/span>, both falling in the pike position. Make some assumption on their frontal areas and calculate their terminal velocities. How long will it take for each skydiver to reach the ground (assuming the time to reach terminal velocity is small)? Assume all values are accurate to three significant digits.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036158571\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id11650394586140\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036158571-solution\">81<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035663631\">A 560-g squirrel with a surface area of\u00a0<span id=\"MathJax-Element-1965-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41889\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41890\" class=\"mrow\"><span id=\"MathJax-Span-41891\" class=\"semantics\"><span id=\"MathJax-Span-41892\" class=\"mrow\"><span id=\"MathJax-Span-41893\" class=\"mrow\"><span id=\"MathJax-Span-41894\" class=\"mn\">930<\/span><span id=\"MathJax-Span-41895\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41896\" class=\"msup\"><span id=\"MathJax-Span-41897\" class=\"mrow\"><span id=\"MathJax-Span-41898\" class=\"mtext\">cm<\/span><\/span><span id=\"MathJax-Span-41899\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">930cm2<\/span><\/span>\u00a0falls from a 5.0-m tree to the ground. Estimate its terminal velocity. (Use a drag coefficient for a horizontal skydiver.) What will be the velocity of a 56-kg person hitting the ground, assuming no drag contribution in such a short distance?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039111535\" class=\"\"><section>\r\n<div id=\"fs-id1165036148072\"><span class=\"os-number\">82<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036141290\">To maintain a constant speed, the force provided by a car\u2019s engine must equal the drag force plus the force of friction of the road (the rolling resistance). (a) What are the drag forces at 70 km\/h and 100 km\/h for a Toyota Camry? (Drag area is\u00a0<span id=\"MathJax-Element-1966-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41900\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41901\" class=\"mrow\"><span id=\"MathJax-Span-41902\" class=\"semantics\"><span id=\"MathJax-Span-41903\" class=\"mrow\"><span id=\"MathJax-Span-41904\" class=\"mrow\"><span id=\"MathJax-Span-41905\" class=\"mn\">0.70<\/span><span id=\"MathJax-Span-41906\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41907\" class=\"msup\"><span id=\"MathJax-Span-41908\" class=\"mrow\"><span id=\"MathJax-Span-41909\" class=\"mtext\">m<\/span><\/span><span id=\"MathJax-Span-41910\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.70m2<\/span><\/span>) (b) What is the drag force at 70 km\/h and 100 km\/h for a Hummer H2? (Drag area is\u00a0<span id=\"MathJax-Element-1967-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41911\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41912\" class=\"mrow\"><span id=\"MathJax-Span-41913\" class=\"semantics\"><span id=\"MathJax-Span-41914\" class=\"mrow\"><span id=\"MathJax-Span-41915\" class=\"mrow\"><span id=\"MathJax-Span-41916\" class=\"mn\">2.44<\/span><span id=\"MathJax-Span-41917\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41918\" class=\"msup\"><span id=\"MathJax-Span-41919\" class=\"mrow\"><span id=\"MathJax-Span-41920\" class=\"mtext\">m<\/span><\/span><span id=\"MathJax-Span-41921\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-41922\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.44m2)<\/span><\/span>\u00a0Assume all values are accurate to three significant digits.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039477110\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039103252\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039477110-solution\">83<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039312187\">By what factor does the drag force on a car increase as it goes from 65 to 110 km\/h?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035834936\" class=\"\"><section>\r\n<div id=\"fs-id1165036080908\"><span class=\"os-number\">84<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035939537\">Calculate the velocity a spherical rain drop would achieve falling from 5.00 km (a) in the absence of air drag (b) with air drag. Take the size across of the drop to be 4 mm, the density to be\u00a0<span id=\"MathJax-Element-1968-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41923\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41924\" class=\"mrow\"><span id=\"MathJax-Span-41925\" class=\"semantics\"><span id=\"MathJax-Span-41926\" class=\"mrow\"><span id=\"MathJax-Span-41927\" class=\"mrow\"><span id=\"MathJax-Span-41928\" class=\"mn\">1.00<\/span><span id=\"MathJax-Span-41929\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41930\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41931\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41932\" class=\"msup\"><span id=\"MathJax-Span-41933\" class=\"mrow\"><span id=\"MathJax-Span-41934\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41935\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41936\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41937\" class=\"msup\"><span id=\"MathJax-Span-41938\" class=\"mrow\"><span id=\"MathJax-Span-41939\" class=\"mtext\">kg\/m<\/span><\/span><span id=\"MathJax-Span-41940\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.00\u00d7103kg\/m3<\/span><\/span>, and the surface area to be\u00a0<span id=\"MathJax-Element-1969-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41941\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41942\" class=\"mrow\"><span id=\"MathJax-Span-41943\" class=\"semantics\"><span id=\"MathJax-Span-41944\" class=\"mrow\"><span id=\"MathJax-Span-41945\" class=\"mrow\"><span id=\"MathJax-Span-41946\" class=\"mi\">\u03c0<\/span><span id=\"MathJax-Span-41947\" class=\"msup\"><span id=\"MathJax-Span-41948\" class=\"mi\">r<\/span><span id=\"MathJax-Span-41949\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03c0r2<\/span><\/span>.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039434228\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165036152519\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039434228-solution\">85<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035698600\">Using Stokes\u2019 law, verify that the units for viscosity are kilograms per meter per second.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039415482\" class=\"\"><section>\r\n<div id=\"fs-id1165039292226\"><span class=\"os-number\">86<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035730440\">Find the terminal velocity of a spherical bacterium (diameter\u00a0<span id=\"MathJax-Element-1970-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41950\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41951\" class=\"mrow\"><span id=\"MathJax-Span-41952\" class=\"semantics\"><span id=\"MathJax-Span-41953\" class=\"mrow\"><span id=\"MathJax-Span-41954\" class=\"mrow\"><span id=\"MathJax-Span-41955\" class=\"mn\">2.00<\/span><span id=\"MathJax-Span-41956\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41957\" class=\"mtext\">\u03bcm<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.00\u03bcm<\/span><\/span>) falling in water. You will first need to note that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be\u00a0<span id=\"MathJax-Element-1971-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41958\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41959\" class=\"mrow\"><span id=\"MathJax-Span-41960\" class=\"semantics\"><span id=\"MathJax-Span-41961\" class=\"mrow\"><span id=\"MathJax-Span-41962\" class=\"mrow\"><span id=\"MathJax-Span-41963\" class=\"mn\">1.10<\/span><span id=\"MathJax-Span-41964\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41965\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41966\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41967\" class=\"msup\"><span id=\"MathJax-Span-41968\" class=\"mrow\"><span id=\"MathJax-Span-41969\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41970\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41971\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41972\" class=\"msup\"><span id=\"MathJax-Span-41973\" class=\"mrow\"><span id=\"MathJax-Span-41974\" class=\"mtext\">kg\/m<\/span><\/span><span id=\"MathJax-Span-41975\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.10\u00d7103kg\/m3<\/span><\/span>.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039396162\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039423411\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039396162-solution\">87<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039028296\">Stokes\u2019 law describes sedimentation of particles in liquids and can be used to measure viscosity. Particles in liquids achieve terminal velocity quickly. One can measure the time it takes for a particle to fall a certain distance and then use Stokes\u2019 law to calculate the viscosity of the liquid. Suppose a steel ball bearing (density\u00a0<span id=\"MathJax-Element-1972-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41976\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41977\" class=\"mrow\"><span id=\"MathJax-Span-41978\" class=\"semantics\"><span id=\"MathJax-Span-41979\" class=\"mrow\"><span id=\"MathJax-Span-41980\" class=\"mrow\"><span id=\"MathJax-Span-41981\" class=\"mn\">7.8<\/span><span id=\"MathJax-Span-41982\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41983\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41984\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41985\" class=\"msup\"><span id=\"MathJax-Span-41986\" class=\"mrow\"><span id=\"MathJax-Span-41987\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41988\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41989\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41990\" class=\"msup\"><span id=\"MathJax-Span-41991\" class=\"mrow\"><span id=\"MathJax-Span-41992\" class=\"mtext\">kg\/m<\/span><\/span><span id=\"MathJax-Span-41993\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">7.8\u00d7103kg\/m3<\/span><\/span>, diameter 3.0 mm) is dropped in a container of motor oil. It takes 12 s to fall a distance of 0.60 m. Calculate the viscosity of the oil.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165033377748\" class=\"\"><section>\r\n<div id=\"fs-id1165039192558\"><span class=\"os-number\">88<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036153540\">Suppose that the resistive force of the air on a skydiver can be approximated by\u00a0<span id=\"MathJax-Element-1973-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41994\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41995\" class=\"mrow\"><span id=\"MathJax-Span-41996\" class=\"semantics\"><span id=\"MathJax-Span-41997\" class=\"mrow\"><span id=\"MathJax-Span-41998\" class=\"mrow\"><span id=\"MathJax-Span-41999\" class=\"mi\">f<\/span><span id=\"MathJax-Span-42000\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42001\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-42002\" class=\"mi\">b<\/span><span id=\"MathJax-Span-42003\" class=\"msup\"><span id=\"MathJax-Span-42004\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42005\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42006\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">f=\u2212bv2.<\/span><\/span>\u00a0If the terminal velocity of a 50.0-kg skydiver is 60.0 m\/s, what is the value of\u00a0<em>b<\/em>?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035732089\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035938648\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035732089-solution\">89<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036151480\">A small diamond of mass 10.0 g drops from a swimmer\u2019s earring and falls through the water, reaching a terminal velocity of 2.0 m\/s. (a) Assuming the frictional force on the diamond obeys\u00a0<span id=\"MathJax-Element-1974-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42007\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42008\" class=\"mrow\"><span id=\"MathJax-Span-42009\" class=\"semantics\"><span id=\"MathJax-Span-42010\" class=\"mrow\"><span id=\"MathJax-Span-42011\" class=\"mrow\"><span id=\"MathJax-Span-42012\" class=\"mi\">f<\/span><span id=\"MathJax-Span-42013\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42014\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-42015\" class=\"mi\">b<\/span><span id=\"MathJax-Span-42016\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42017\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">f=\u2212bv,<\/span><\/span>\u00a0what is\u00a0<em>b<\/em>? (b) How far does the diamond fall before it reaches 90 percent of its terminal speed?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039345130\" class=\"\"><section>\r\n<div id=\"fs-id1165038989787\"><span class=\"os-number\">90<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165033370856\">(a) What is the final velocity of a car originally traveling at 50.0 km\/h that decelerates at a rate of\u00a0<span id=\"MathJax-Element-1975-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42018\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42019\" class=\"mrow\"><span id=\"MathJax-Span-42020\" class=\"semantics\"><span id=\"MathJax-Span-42021\" class=\"mrow\"><span id=\"MathJax-Span-42022\" class=\"mrow\"><span id=\"MathJax-Span-42023\" class=\"mn\">0.400<\/span><span id=\"MathJax-Span-42024\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42025\" class=\"msup\"><span id=\"MathJax-Span-42026\" class=\"mrow\"><span id=\"MathJax-Span-42027\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-42028\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.400m\/s2<\/span><\/span>\u00a0for 50.0 s? Assume a coefficient of friction of 1.0. (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035761606\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035729743\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035761606-solution\">91<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039464292\">A 75.0-kg woman stands on a bathroom scale in an elevator that accelerates from rest to 30.0 m\/s in 2.00 s. (a) Calculate the scale reading in newtons and compare it with her weight. (The scale exerts an upward force on her equal to its reading.) (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039351979\" class=\"\"><section>\r\n<div id=\"fs-id1165039241179\"><span class=\"os-number\">92<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039396054\">(a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked 50.0 m radius curve at 30.0 m\/s. (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036006942\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035863813\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036006942-solution\">93<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039411492\">As shown below, if\u00a0<span id=\"MathJax-Element-1976-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42029\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42030\" class=\"mrow\"><span id=\"MathJax-Span-42031\" class=\"semantics\"><span id=\"MathJax-Span-42032\" class=\"mrow\"><span id=\"MathJax-Span-42033\" class=\"mrow\"><span id=\"MathJax-Span-42034\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42035\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42036\" class=\"mn\">5.50<\/span><span id=\"MathJax-Span-42037\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42038\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=5.50kg,<\/span><\/span>\u00a0what is the tension in string 1?<\/p>\r\n\r\n<span id=\"fs-id1165035723285\"><img id=\"16243\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/86a15128bbd2a06be7a45fe7e705ed0472536502\" alt=\"Mass M is suspended from strings 1 and 2. String 1 connects to a wall at a point below and to the left of the mass. String 1 makes an angle of 40 degrees below the horizontal. String 2 connects to a ceiling at a point above and to the right of the mass. String 2 makes an angle of 40 degrees to the right of vertical.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165038979966\" class=\"\"><section>\r\n<div id=\"fs-id1165035641506\"><span class=\"os-number\">94<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035721482\">As shown below, if\u00a0<span id=\"MathJax-Element-1977-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42039\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42040\" class=\"mrow\"><span id=\"MathJax-Span-42041\" class=\"semantics\"><span id=\"MathJax-Span-42042\" class=\"mrow\"><span id=\"MathJax-Span-42043\" class=\"mrow\"><span id=\"MathJax-Span-42044\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42045\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42046\" class=\"mn\">60.0<\/span><span id=\"MathJax-Span-42047\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42048\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F=60.0N<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1978-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42049\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42050\" class=\"mrow\"><span id=\"MathJax-Span-42051\" class=\"semantics\"><span id=\"MathJax-Span-42052\" class=\"mrow\"><span id=\"MathJax-Span-42053\" class=\"mrow\"><span id=\"MathJax-Span-42054\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42055\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42056\" class=\"mn\">4.00<\/span><span id=\"MathJax-Span-42057\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42058\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=4.00kg,<\/span><\/span>\u00a0what is the magnitude of the acceleration of the suspended object? All surfaces are frictionless.<\/p>\r\n\r\n<span id=\"fs-id1165039112562\"><img id=\"6391\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/11b895b8d74a85765b176ee0f621a79ec3d459b3\" alt=\"Two blocks are shown. One block, labeled 2 M is on a horizontal table. A force F pulls on the 2 M block up and to the left at an angle of 30 degrees above the horizontal. On the opposite side, the block is connected to a string that pulls it to the right. The string passes over a pulley at edge of the table, then hangs straight down and connects to  the second block, labeled M. Block 2 is not in contact with the ramp.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035615426\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035868420\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035615426-solution\">95<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035939480\">As shown below, if\u00a0<span id=\"MathJax-Element-1979-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42059\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42060\" class=\"mrow\"><span id=\"MathJax-Span-42061\" class=\"semantics\"><span id=\"MathJax-Span-42062\" class=\"mrow\"><span id=\"MathJax-Span-42063\" class=\"mrow\"><span id=\"MathJax-Span-42064\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42065\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42066\" class=\"mn\">6.0<\/span><span id=\"MathJax-Span-42067\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42068\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=6.0kg,<\/span><\/span>\u00a0what is the tension in the connecting string? The pulley and all surfaces are frictionless.<\/p>\r\n\r\n<span id=\"fs-id1165039485639\"><img id=\"26047\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/bf04ac78a612bd5be9c5903ce270c7784153c097\" alt=\"Two blocks, both mass M are connected by a string that passes over a pulley between the blocks. The upper block is on a surface that slopes down and to the right at an angle of 30 degrees to the horizontal. The pulley is attached to the corner at the bottom of the slope, where the surface then bends and goes vertically down. The lower mass hangs straight down. It is not in contact with the surface.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039103753\" class=\"\"><section>\r\n<div id=\"fs-id1165035676708\"><span class=\"os-number\">96<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039337931\">A small space probe is released from a spaceship. The space probe has mass 20.0 kg and contains 90.0 kg of fuel. It starts from rest in deep space, from the origin of a coordinate system based on the spaceship, and burns fuel at the rate of 3.00 kg\/s. The engine provides a constant thrust of 120.0 N. (a) Write an expression for the mass of the space probe as a function of time, between 0 and 30 seconds, assuming that the engine ignites fuel beginning at\u00a0<span id=\"MathJax-Element-1980-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42069\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42070\" class=\"mrow\"><span id=\"MathJax-Span-42071\" class=\"semantics\"><span id=\"MathJax-Span-42072\" class=\"mrow\"><span id=\"MathJax-Span-42073\" class=\"mrow\"><span id=\"MathJax-Span-42074\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42075\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42076\" class=\"mn\">0<\/span><span id=\"MathJax-Span-42077\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0.<\/span><\/span>\u00a0(b) What is the velocity after 15.0 s? (c) What is the position of the space probe after 15.0 s, with initial position at the origin? (d) Write an expression for the position as a function of time, for\u00a0<span id=\"MathJax-Element-1981-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42078\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42079\" class=\"mrow\"><span id=\"MathJax-Span-42080\" class=\"semantics\"><span id=\"MathJax-Span-42081\" class=\"mrow\"><span id=\"MathJax-Span-42082\" class=\"mrow\"><span id=\"MathJax-Span-42083\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42084\" class=\"mo\">&gt;<\/span><span id=\"MathJax-Span-42085\" class=\"mn\">30.0<\/span><span id=\"MathJax-Span-42086\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42087\" class=\"mtext\">s<\/span><span id=\"MathJax-Span-42088\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t&gt;30.0s.<\/span><\/span><\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039109864\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039367916\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039109864-solution\">97<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035868296\">A half-full recycling bin has mass 3.0 kg and is pushed up a\u00a0<span id=\"MathJax-Element-1982-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42089\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42090\" class=\"mrow\"><span id=\"MathJax-Span-42091\" class=\"semantics\"><span id=\"MathJax-Span-42092\" class=\"mrow\"><span id=\"MathJax-Span-42093\" class=\"mrow\"><span id=\"MathJax-Span-42094\" class=\"mn\">40.0<\/span><span id=\"MathJax-Span-42095\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">40.0\u00b0<\/span><\/span>\u00a0incline with constant speed under the action of a 26-N force acting up and parallel to the incline. The incline has friction. What magnitude force must act up and parallel to the incline for the bin to move down the incline at constant velocity?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039268406\" class=\"\"><section>\r\n<div id=\"fs-id1165035866354\"><span class=\"os-number\">98<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039353641\">A child has mass 6.0 kg and slides down a\u00a0<span id=\"MathJax-Element-1983-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42096\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42097\" class=\"mrow\"><span id=\"MathJax-Span-42098\" class=\"semantics\"><span id=\"MathJax-Span-42099\" class=\"mrow\"><span id=\"MathJax-Span-42100\" class=\"mrow\"><span id=\"MathJax-Span-42101\" class=\"mn\">35<\/span><span id=\"MathJax-Span-42102\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">35\u00b0<\/span><\/span>\u00a0incline with constant speed under the action of a 34-N force acting up and parallel to the incline. What is the coefficient of kinetic friction between the child and the surface of the incline?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-review-additional-problems-container\">\r\n<h3><span class=\"os-text\">Additional Problems<\/span><\/h3>\r\n<section id=\"fs-id1165035682534\" class=\"review-additional-problems\">\r\n<div id=\"fs-id1165039234785\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039219140\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039234785-solution\">99<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036160772\">The two barges shown here are coupled by a cable of negligible mass. The mass of the front barge is\u00a0<span id=\"MathJax-Element-1984-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42103\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42104\" class=\"mrow\"><span id=\"MathJax-Span-42105\" class=\"semantics\"><span id=\"MathJax-Span-42106\" class=\"mrow\"><span id=\"MathJax-Span-42107\" class=\"mrow\"><span id=\"MathJax-Span-42108\" class=\"mn\">2.00<\/span><span id=\"MathJax-Span-42109\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42110\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42111\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42112\" class=\"msup\"><span id=\"MathJax-Span-42113\" class=\"mrow\"><span id=\"MathJax-Span-42114\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42115\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42116\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42117\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.00\u00d7103kg<\/span><\/span>\u00a0and the mass of the rear barge is\u00a0<span id=\"MathJax-Element-1985-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42118\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42119\" class=\"mrow\"><span id=\"MathJax-Span-42120\" class=\"semantics\"><span id=\"MathJax-Span-42121\" class=\"mrow\"><span id=\"MathJax-Span-42122\" class=\"mrow\"><span id=\"MathJax-Span-42123\" class=\"mn\">3.00<\/span><span id=\"MathJax-Span-42124\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42125\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42126\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42127\" class=\"msup\"><span id=\"MathJax-Span-42128\" class=\"mrow\"><span id=\"MathJax-Span-42129\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42130\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42131\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42132\" class=\"mtext\">kg<\/span><span id=\"MathJax-Span-42133\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.00\u00d7103kg.<\/span><\/span>\u00a0A tugboat pulls the front barge with a horizontal force of magnitude\u00a0<span id=\"MathJax-Element-1986-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42134\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42135\" class=\"mrow\"><span id=\"MathJax-Span-42136\" class=\"semantics\"><span id=\"MathJax-Span-42137\" class=\"mrow\"><span id=\"MathJax-Span-42138\" class=\"mrow\"><span id=\"MathJax-Span-42139\" class=\"mn\">20.0<\/span><span id=\"MathJax-Span-42140\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42141\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42142\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42143\" class=\"msup\"><span id=\"MathJax-Span-42144\" class=\"mrow\"><span id=\"MathJax-Span-42145\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42146\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42147\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42148\" class=\"mtext\">N,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">20.0\u00d7103N,<\/span><\/span>\u00a0and the frictional forces of the water on the front and rear barges are\u00a0<span id=\"MathJax-Element-1987-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42149\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42150\" class=\"mrow\"><span id=\"MathJax-Span-42151\" class=\"semantics\"><span id=\"MathJax-Span-42152\" class=\"mrow\"><span id=\"MathJax-Span-42153\" class=\"mrow\"><span id=\"MathJax-Span-42154\" class=\"mn\">8.00<\/span><span id=\"MathJax-Span-42155\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42156\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42157\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42158\" class=\"msup\"><span id=\"MathJax-Span-42159\" class=\"mrow\"><span id=\"MathJax-Span-42160\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42161\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42162\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42163\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">8.00\u00d7103N<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1988-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42164\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42165\" class=\"mrow\"><span id=\"MathJax-Span-42166\" class=\"semantics\"><span id=\"MathJax-Span-42167\" class=\"mrow\"><span id=\"MathJax-Span-42168\" class=\"mrow\"><span id=\"MathJax-Span-42169\" class=\"mn\">10.0<\/span><span id=\"MathJax-Span-42170\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42171\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42172\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42173\" class=\"msup\"><span id=\"MathJax-Span-42174\" class=\"mrow\"><span id=\"MathJax-Span-42175\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42176\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42177\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42178\" class=\"mtext\">N,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">10.0\u00d7103N,<\/span><\/span>respectively. Find the horizontal acceleration of the barges and the tension in the connecting cable.<\/p>\r\n\r\n<span id=\"fs-id1165033379354\"><img id=\"58353\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/a5325babf6dbc08087b6e367b47c75ad4bd381bc\" alt=\"An illustration showing a tug boat pulling two barges. The barge directly attached to the tug boat has mass 2.00 times 10 to the third kilograms. The barge at the end,  behind the first barge, has mass 3.00 times 10 to the third kilograms.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036035448\" class=\"\"><section>\r\n<div id=\"fs-id1165038990654\"><span class=\"os-number\">100<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035669137\">If the order of the barges of the preceding exercise is reversed so that the tugboat pulls the\u00a0<span id=\"MathJax-Element-1989-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42179\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42180\" class=\"mrow\"><span id=\"MathJax-Span-42181\" class=\"semantics\"><span id=\"MathJax-Span-42182\" class=\"mrow\"><span id=\"MathJax-Span-42183\" class=\"mrow\"><span id=\"MathJax-Span-42184\" class=\"mn\">3.00<\/span><span id=\"MathJax-Span-42185\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42186\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42187\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42188\" class=\"msup\"><span id=\"MathJax-Span-42189\" class=\"mrow\"><span id=\"MathJax-Span-42190\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42191\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42192\" class=\"mtext\">-kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.00\u00d7103-kg<\/span><\/span>\u00a0barge with a force of\u00a0<span id=\"MathJax-Element-1990-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42193\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42194\" class=\"mrow\"><span id=\"MathJax-Span-42195\" class=\"semantics\"><span id=\"MathJax-Span-42196\" class=\"mrow\"><span id=\"MathJax-Span-42197\" class=\"mrow\"><span id=\"MathJax-Span-42198\" class=\"mn\">20.0<\/span><span id=\"MathJax-Span-42199\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42200\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42201\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42202\" class=\"msup\"><span id=\"MathJax-Span-42203\" class=\"mrow\"><span id=\"MathJax-Span-42204\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42205\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42206\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42207\" class=\"mtext\">N<\/span><span id=\"MathJax-Span-42208\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">20.0\u00d7103N,<\/span><\/span>\u00a0what are the acceleration of the barges and the tension in the coupling cable?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039027173\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039293063\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039027173-solution\">101<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039296223\">An object with mass\u00a0<em>m<\/em>\u00a0moves along the\u00a0<em>x<\/em>-axis. Its position at any time is given by\u00a0<span id=\"MathJax-Element-1991-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42209\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42210\" class=\"mrow\"><span id=\"MathJax-Span-42211\" class=\"semantics\"><span id=\"MathJax-Span-42212\" class=\"mrow\"><span id=\"MathJax-Span-42213\" class=\"mrow\"><span id=\"MathJax-Span-42214\" class=\"mi\">x<\/span><span id=\"MathJax-Span-42215\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42216\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42217\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42218\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42219\" class=\"mi\">p<\/span><span id=\"MathJax-Span-42220\" class=\"msup\"><span id=\"MathJax-Span-42221\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42222\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42223\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42224\" class=\"mi\">q<\/span><span id=\"MathJax-Span-42225\" class=\"msup\"><span id=\"MathJax-Span-42226\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42227\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">x(t)=pt3+qt2<\/span><\/span>\u00a0where\u00a0<em>p<\/em>\u00a0and\u00a0<em>q<\/em>\u00a0are constants. Find the net force on this object for any time\u00a0<em>t<\/em>.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039335542\" class=\"\"><section>\r\n<div id=\"fs-id1165035792005\"><span class=\"os-number\">102<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039352836\">A helicopter with mass\u00a0<span id=\"MathJax-Element-1992-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42228\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42229\" class=\"mrow\"><span id=\"MathJax-Span-42230\" class=\"semantics\"><span id=\"MathJax-Span-42231\" class=\"mrow\"><span id=\"MathJax-Span-42232\" class=\"mrow\"><span id=\"MathJax-Span-42233\" class=\"mn\">2.35<\/span><span id=\"MathJax-Span-42234\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42235\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42236\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42237\" class=\"msup\"><span id=\"MathJax-Span-42238\" class=\"mrow\"><span id=\"MathJax-Span-42239\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42240\" class=\"mn\">4<\/span><\/span><span id=\"MathJax-Span-42241\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42242\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.35\u00d7104kg<\/span><\/span>\u00a0has a position given by\u00a0<span id=\"MathJax-Element-1993-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42243\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42244\" class=\"mrow\"><span id=\"MathJax-Span-42245\" class=\"semantics\"><span id=\"MathJax-Span-42246\" class=\"mrow\"><span id=\"MathJax-Span-42247\" class=\"mrow\"><span id=\"MathJax-Span-42248\" class=\"mstyle\"><span id=\"MathJax-Span-42249\" class=\"mrow\"><span id=\"MathJax-Span-42250\" class=\"mover\"><span id=\"MathJax-Span-42251\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42252\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42253\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42254\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42255\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42256\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42257\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42258\" class=\"mn\">0.020<\/span><span id=\"MathJax-Span-42259\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42260\" class=\"msup\"><span id=\"MathJax-Span-42261\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42262\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42263\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42264\" class=\"mstyle\"><span id=\"MathJax-Span-42265\" class=\"mrow\"><span id=\"MathJax-Span-42266\" class=\"mover\"><span id=\"MathJax-Span-42267\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42268\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42269\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42270\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42271\" class=\"mn\">2.2<\/span><span id=\"MathJax-Span-42272\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42273\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42274\" class=\"mstyle\"><span id=\"MathJax-Span-42275\" class=\"mrow\"><span id=\"MathJax-Span-42276\" class=\"mover\"><span id=\"MathJax-Span-42277\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42278\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42279\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-42280\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42281\" class=\"mn\">0.060<\/span><span id=\"MathJax-Span-42282\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42283\" class=\"msup\"><span id=\"MathJax-Span-42284\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42285\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42286\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42287\" class=\"mstyle\"><span id=\"MathJax-Span-42288\" class=\"mrow\"><span id=\"MathJax-Span-42289\" class=\"mover\"><span id=\"MathJax-Span-42290\" class=\"mi\">k<\/span><span id=\"MathJax-Span-42291\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42292\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)=(0.020t3)i^+(2.2t)j^\u2212(0.060t2)k^.<\/span><\/span>\u00a0Find the net force on the helicopter at\u00a0<span id=\"MathJax-Element-1994-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42293\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42294\" class=\"mrow\"><span id=\"MathJax-Span-42295\" class=\"semantics\"><span id=\"MathJax-Span-42296\" class=\"mrow\"><span id=\"MathJax-Span-42297\" class=\"mrow\"><span id=\"MathJax-Span-42298\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42299\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42300\" class=\"mn\">3.0<\/span><span id=\"MathJax-Span-42301\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42302\" class=\"mtext\">s<\/span><span id=\"MathJax-Span-42303\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=3.0s.<\/span><\/span><\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039453675\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035980452\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039453675-solution\">103<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035688942\">Located at the origin, an electric car of mass\u00a0<em>m<\/em>\u00a0is at rest and in equilibrium. A time dependent force of\u00a0<span id=\"MathJax-Element-1995-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42304\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42305\" class=\"mrow\"><span id=\"MathJax-Span-42306\" class=\"semantics\"><span id=\"MathJax-Span-42307\" class=\"mrow\"><span id=\"MathJax-Span-42308\" class=\"mrow\"><span id=\"MathJax-Span-42309\" class=\"mstyle\"><span id=\"MathJax-Span-42310\" class=\"mrow\"><span id=\"MathJax-Span-42311\" class=\"mover\"><span id=\"MathJax-Span-42312\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42313\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42314\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42315\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42316\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192(t)<\/span><\/span>\u00a0is applied at time\u00a0<span id=\"MathJax-Element-1996-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42317\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42318\" class=\"mrow\"><span id=\"MathJax-Span-42319\" class=\"semantics\"><span id=\"MathJax-Span-42320\" class=\"mrow\"><span id=\"MathJax-Span-42321\" class=\"mrow\"><span id=\"MathJax-Span-42322\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42323\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42324\" class=\"mn\">0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0<\/span><\/span>, and its components are\u00a0<span id=\"MathJax-Element-1997-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42325\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42326\" class=\"mrow\"><span id=\"MathJax-Span-42327\" class=\"semantics\"><span id=\"MathJax-Span-42328\" class=\"mrow\"><span id=\"MathJax-Span-42329\" class=\"mrow\"><span id=\"MathJax-Span-42330\" class=\"msub\"><span id=\"MathJax-Span-42331\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42332\" class=\"mi\">x<\/span><\/span><span id=\"MathJax-Span-42333\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42334\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42335\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42336\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42337\" class=\"mi\">p<\/span><span id=\"MathJax-Span-42338\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42339\" class=\"mi\">n<\/span><span id=\"MathJax-Span-42340\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fx(t)=p+nt<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1998-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42341\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42342\" class=\"mrow\"><span id=\"MathJax-Span-42343\" class=\"semantics\"><span id=\"MathJax-Span-42344\" class=\"mrow\"><span id=\"MathJax-Span-42345\" class=\"mrow\"><span id=\"MathJax-Span-42346\" class=\"msub\"><span id=\"MathJax-Span-42347\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42348\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-42349\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42350\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42351\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42352\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42353\" class=\"mi\">q<\/span><span id=\"MathJax-Span-42354\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fy(t)=qt<\/span><\/span>\u00a0where\u00a0<em>p<\/em>,\u00a0<em>q<\/em>, and\u00a0<em>n<\/em>\u00a0are constants. Find the position\u00a0<span id=\"MathJax-Element-1999-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42355\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42356\" class=\"mrow\"><span id=\"MathJax-Span-42357\" class=\"semantics\"><span id=\"MathJax-Span-42358\" class=\"mrow\"><span id=\"MathJax-Span-42359\" class=\"mrow\"><span id=\"MathJax-Span-42360\" class=\"mstyle\"><span id=\"MathJax-Span-42361\" class=\"mrow\"><span id=\"MathJax-Span-42362\" class=\"mover\"><span id=\"MathJax-Span-42363\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42364\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42365\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42366\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42367\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)<\/span><\/span>\u00a0and velocity\u00a0<span id=\"MathJax-Element-2000-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42368\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42369\" class=\"mrow\"><span id=\"MathJax-Span-42370\" class=\"semantics\"><span id=\"MathJax-Span-42371\" class=\"mrow\"><span id=\"MathJax-Span-42372\" class=\"mrow\"><span id=\"MathJax-Span-42373\" class=\"mstyle\"><span id=\"MathJax-Span-42374\" class=\"mrow\"><span id=\"MathJax-Span-42375\" class=\"mover\"><span id=\"MathJax-Span-42376\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42377\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42378\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42379\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42380\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v\u2192(t)<\/span><\/span>\u00a0as functions of time\u00a0<em>t<\/em>.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165033378093\" class=\"\"><section>\r\n<div id=\"fs-id1165039402799\"><span class=\"os-number\">104<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035867149\">A particle of mass\u00a0<em>m<\/em>\u00a0is located at the origin. It is at rest and in equilibrium. A time-dependent force of\u00a0<span id=\"MathJax-Element-2001-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42381\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42382\" class=\"mrow\"><span id=\"MathJax-Span-42383\" class=\"semantics\"><span id=\"MathJax-Span-42384\" class=\"mrow\"><span id=\"MathJax-Span-42385\" class=\"mrow\"><span id=\"MathJax-Span-42386\" class=\"mstyle\"><span id=\"MathJax-Span-42387\" class=\"mrow\"><span id=\"MathJax-Span-42388\" class=\"mover\"><span id=\"MathJax-Span-42389\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42390\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42391\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42392\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42393\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192(t)<\/span><\/span>\u00a0is applied at time\u00a0<span id=\"MathJax-Element-2002-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42394\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42395\" class=\"mrow\"><span id=\"MathJax-Span-42396\" class=\"semantics\"><span id=\"MathJax-Span-42397\" class=\"mrow\"><span id=\"MathJax-Span-42398\" class=\"mrow\"><span id=\"MathJax-Span-42399\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42400\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42401\" class=\"mn\">0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0<\/span><\/span>, and its components are\u00a0<span id=\"MathJax-Element-2003-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42402\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42403\" class=\"mrow\"><span id=\"MathJax-Span-42404\" class=\"semantics\"><span id=\"MathJax-Span-42405\" class=\"mrow\"><span id=\"MathJax-Span-42406\" class=\"mrow\"><span id=\"MathJax-Span-42407\" class=\"msub\"><span id=\"MathJax-Span-42408\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42409\" class=\"mi\">x<\/span><\/span><span id=\"MathJax-Span-42410\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42411\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42412\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42413\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42414\" class=\"mi\">p<\/span><span id=\"MathJax-Span-42415\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fx(t)=pt<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-2004-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42416\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42417\" class=\"mrow\"><span id=\"MathJax-Span-42418\" class=\"semantics\"><span id=\"MathJax-Span-42419\" class=\"mrow\"><span id=\"MathJax-Span-42420\" class=\"mrow\"><span id=\"MathJax-Span-42421\" class=\"msub\"><span id=\"MathJax-Span-42422\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42423\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-42424\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42425\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42426\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42427\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42428\" class=\"mi\">n<\/span><span id=\"MathJax-Span-42429\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42430\" class=\"mi\">q<\/span><span id=\"MathJax-Span-42431\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fy(t)=n+qt<\/span><\/span>\u00a0where\u00a0<em>p<\/em>,\u00a0<em>q<\/em>, and\u00a0<em>n<\/em>\u00a0are constants. Find the position\u00a0<span id=\"MathJax-Element-2005-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42432\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42433\" class=\"mrow\"><span id=\"MathJax-Span-42434\" class=\"semantics\"><span id=\"MathJax-Span-42435\" class=\"mrow\"><span id=\"MathJax-Span-42436\" class=\"mrow\"><span id=\"MathJax-Span-42437\" class=\"mstyle\"><span id=\"MathJax-Span-42438\" class=\"mrow\"><span id=\"MathJax-Span-42439\" class=\"mover\"><span id=\"MathJax-Span-42440\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42441\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42442\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42443\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42444\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)<\/span><\/span>\u00a0and velocity\u00a0<span id=\"MathJax-Element-2006-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42445\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42446\" class=\"mrow\"><span id=\"MathJax-Span-42447\" class=\"semantics\"><span id=\"MathJax-Span-42448\" class=\"mrow\"><span id=\"MathJax-Span-42449\" class=\"mrow\"><span id=\"MathJax-Span-42450\" class=\"mstyle\"><span id=\"MathJax-Span-42451\" class=\"mrow\"><span id=\"MathJax-Span-42452\" class=\"mover\"><span id=\"MathJax-Span-42453\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42454\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42455\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42456\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42457\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v\u2192(t)<\/span><\/span>\u00a0as functions of time\u00a0<em>t<\/em>.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035661242\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035639346\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035661242-solution\">105<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035662853\">A 2.0-kg object has a velocity of\u00a0<span id=\"MathJax-Element-2007-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42458\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42459\" class=\"mrow\"><span id=\"MathJax-Span-42460\" class=\"semantics\"><span id=\"MathJax-Span-42461\" class=\"mrow\"><span id=\"MathJax-Span-42462\" class=\"mrow\"><span id=\"MathJax-Span-42463\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42464\" class=\"mstyle\"><span id=\"MathJax-Span-42465\" class=\"mrow\"><span id=\"MathJax-Span-42466\" class=\"mover\"><span id=\"MathJax-Span-42467\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42468\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42469\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42470\" class=\"mtext\">m\/s<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">4.0i^m\/s<\/span><\/span>\u00a0at\u00a0<span id=\"MathJax-Element-2008-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42471\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42472\" class=\"mrow\"><span id=\"MathJax-Span-42473\" class=\"semantics\"><span id=\"MathJax-Span-42474\" class=\"mrow\"><span id=\"MathJax-Span-42475\" class=\"mrow\"><span id=\"MathJax-Span-42476\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42477\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42478\" class=\"mn\">0<\/span><span id=\"MathJax-Span-42479\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0.<\/span><\/span>\u00a0A constant resultant force of\u00a0<span id=\"MathJax-Element-2009-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42480\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42481\" class=\"mrow\"><span id=\"MathJax-Span-42482\" class=\"semantics\"><span id=\"MathJax-Span-42483\" class=\"mrow\"><span id=\"MathJax-Span-42484\" class=\"mrow\"><span id=\"MathJax-Span-42485\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42486\" class=\"mn\">2.0<\/span><span id=\"MathJax-Span-42487\" class=\"mstyle\"><span id=\"MathJax-Span-42488\" class=\"mrow\"><span id=\"MathJax-Span-42489\" class=\"mover\"><span id=\"MathJax-Span-42490\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42491\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42492\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42493\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42494\" class=\"mstyle\"><span id=\"MathJax-Span-42495\" class=\"mrow\"><span id=\"MathJax-Span-42496\" class=\"mover\"><span id=\"MathJax-Span-42497\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42498\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42499\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42500\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42501\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(2.0i^+4.0j^)N<\/span><\/span>\u00a0then acts on the object for 3.0 s. What is the magnitude of the object\u2019s velocity at the end of the 3.0-s interval?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035975760\" class=\"\"><section>\r\n<div id=\"fs-id1165039234284\"><span class=\"os-number\">106<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039323978\">A 1.5-kg mass has an acceleration of\u00a0<span id=\"MathJax-Element-2010-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42502\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42503\" class=\"mrow\"><span id=\"MathJax-Span-42504\" class=\"semantics\"><span id=\"MathJax-Span-42505\" class=\"mrow\"><span id=\"MathJax-Span-42506\" class=\"mrow\"><span id=\"MathJax-Span-42507\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42508\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42509\" class=\"mstyle\"><span id=\"MathJax-Span-42510\" class=\"mrow\"><span id=\"MathJax-Span-42511\" class=\"mover\"><span id=\"MathJax-Span-42512\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42513\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42514\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-42515\" class=\"mn\">3.0<\/span><span id=\"MathJax-Span-42516\" class=\"mstyle\"><span id=\"MathJax-Span-42517\" class=\"mrow\"><span id=\"MathJax-Span-42518\" class=\"mover\"><span id=\"MathJax-Span-42519\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42520\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42521\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42522\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42523\" class=\"msup\"><span id=\"MathJax-Span-42524\" class=\"mrow\"><span id=\"MathJax-Span-42525\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-42526\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42527\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(4.0i^\u22123.0j^)m\/s2.<\/span><\/span>\u00a0Only two forces act on the mass. If one of the forces is\u00a0<span id=\"MathJax-Element-2011-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42528\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42529\" class=\"mrow\"><span id=\"MathJax-Span-42530\" class=\"semantics\"><span id=\"MathJax-Span-42531\" class=\"mrow\"><span id=\"MathJax-Span-42532\" class=\"mrow\"><span id=\"MathJax-Span-42533\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42534\" class=\"mn\">2.0<\/span><span id=\"MathJax-Span-42535\" class=\"mstyle\"><span id=\"MathJax-Span-42536\" class=\"mrow\"><span id=\"MathJax-Span-42537\" class=\"mover\"><span id=\"MathJax-Span-42538\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42539\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42540\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-42541\" class=\"mn\">1.4<\/span><span id=\"MathJax-Span-42542\" class=\"mstyle\"><span id=\"MathJax-Span-42543\" class=\"mrow\"><span id=\"MathJax-Span-42544\" class=\"mover\"><span id=\"MathJax-Span-42545\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42546\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42547\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42548\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42549\" class=\"mtext\">N,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(2.0i^\u22121.4j^)N,<\/span><\/span>\u00a0what is the magnitude of the other force?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035669542\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165036075410\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035669542-solution\">107<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039345662\">A box is dropped onto a conveyor belt moving at 3.4 m\/s. If the coefficient of friction between the box and the belt is 0.27, how long will it take before the box moves without slipping?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039426530\" class=\"\"><section>\r\n<div id=\"fs-id1165033371416\"><span class=\"os-number\">108<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039273532\">Shown below is a 10.0-kg block being pushed by a horizontal force\u00a0<span id=\"MathJax-Element-2012-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42550\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42551\" class=\"mrow\"><span id=\"MathJax-Span-42552\" class=\"semantics\"><span id=\"MathJax-Span-42553\" class=\"mrow\"><span id=\"MathJax-Span-42554\" class=\"mstyle\"><span id=\"MathJax-Span-42555\" class=\"mrow\"><span id=\"MathJax-Span-42556\" class=\"mover\"><span id=\"MathJax-Span-42557\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42558\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192<\/span><\/span>\u00a0of magnitude 200.0 N. The coefficient of kinetic friction between the two surfaces is 0.50. Find the acceleration of the block.<\/p>\r\n\r\n<span id=\"fs-id1165039091495\"><img id=\"98156\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/e4781e956b67a53c0d88ed7057fa766873a7b064\" alt=\"An illustration of a 10.0 kilogram block being pushed into a slope by a horizontal force F. The slope angles up and to the right at an angle of 30 degrees to the horizontal and the force F points to the right.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035865609\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165038974227\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035865609-solution\">109<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036162461\">As shown below, the mass of block 1 is\u00a0<span id=\"MathJax-Element-2013-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42559\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42560\" class=\"mrow\"><span id=\"MathJax-Span-42561\" class=\"semantics\"><span id=\"MathJax-Span-42562\" class=\"mrow\"><span id=\"MathJax-Span-42563\" class=\"mrow\"><span id=\"MathJax-Span-42564\" class=\"msub\"><span id=\"MathJax-Span-42565\" class=\"mi\">m<\/span><span id=\"MathJax-Span-42566\" class=\"mn\">1<\/span><\/span><span id=\"MathJax-Span-42567\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42568\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42569\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42570\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">m1=4.0kg,<\/span><\/span>\u00a0while the mass of block 2 is\u00a0<span id=\"MathJax-Element-2014-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42571\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42572\" class=\"mrow\"><span id=\"MathJax-Span-42573\" class=\"semantics\"><span id=\"MathJax-Span-42574\" class=\"mrow\"><span id=\"MathJax-Span-42575\" class=\"mrow\"><span id=\"MathJax-Span-42576\" class=\"msub\"><span id=\"MathJax-Span-42577\" class=\"mi\">m<\/span><span id=\"MathJax-Span-42578\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42579\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42580\" class=\"mn\">8.0<\/span><span id=\"MathJax-Span-42581\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42582\" class=\"mtext\">kg<\/span><span id=\"MathJax-Span-42583\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">m2=8.0kg.<\/span><\/span>\u00a0The coefficient of friction between\u00a0<span id=\"MathJax-Element-2015-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42584\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42585\" class=\"mrow\"><span id=\"MathJax-Span-42586\" class=\"semantics\"><span id=\"MathJax-Span-42587\" class=\"mrow\"><span id=\"MathJax-Span-42588\" class=\"mrow\"><span id=\"MathJax-Span-42589\" class=\"msub\"><span id=\"MathJax-Span-42590\" class=\"mi\">m<\/span><span id=\"MathJax-Span-42591\" class=\"mn\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">m1<\/span><\/span>\u00a0and the inclined surface is\u00a0<span id=\"MathJax-Element-2016-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42592\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42593\" class=\"mrow\"><span id=\"MathJax-Span-42594\" class=\"semantics\"><span id=\"MathJax-Span-42595\" class=\"mrow\"><span id=\"MathJax-Span-42596\" class=\"mrow\"><span id=\"MathJax-Span-42597\" class=\"msub\"><span id=\"MathJax-Span-42598\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-42599\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-42600\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42601\" class=\"mn\">0.40<\/span><span id=\"MathJax-Span-42602\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bck=0.40.<\/span><\/span>\u00a0What is the acceleration of the system?<\/p>\r\n\r\n<span id=\"fs-id1165035730256\"><img id=\"38422\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/5fc65f8faf919d36f9b0e584c5d6e311986f76a4\" alt=\"Block 1 is on a ramp inclined up and to the right at an angle of 37 degrees above the horizontal. It is connected to a string that passes over a pulley at the top of the ramp, then hangs straight down and connects to  block 2. Block 2 is not in contact with the ramp.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039114252\" class=\"\"><section>\r\n<div id=\"fs-id1165033370832\"><span class=\"os-number\">110<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035946630\">A student is attempting to move a 30-kg mini-fridge into her dorm room. During a moment of inattention, the mini-fridge slides down a 35 degree incline at constant speed when she applies a force of 25 N acting up and parallel to the incline. What is the coefficient of kinetic friction between the fridge and the surface of the incline?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039077028\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039398265\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039077028-solution\">111<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035640986\">A crate of mass 100.0 kg rests on a rough surface inclined at an angle of\u00a0<span id=\"MathJax-Element-2017-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42603\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42604\" class=\"mrow\"><span id=\"MathJax-Span-42605\" class=\"semantics\"><span id=\"MathJax-Span-42606\" class=\"mrow\"><span id=\"MathJax-Span-42607\" class=\"mrow\"><span id=\"MathJax-Span-42608\" class=\"mn\">37.0<\/span><span id=\"MathJax-Span-42609\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">37.0\u00b0<\/span><\/span>\u00a0with the horizontal. A massless rope to which a force can be applied parallel to the surface is attached to the crate and leads to the top of the incline. In its present state, the crate is just ready to slip and start to move down the plane. The coefficient of friction is\u00a0<span id=\"MathJax-Element-2018-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42610\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42611\" class=\"mrow\"><span id=\"MathJax-Span-42612\" class=\"semantics\"><span id=\"MathJax-Span-42613\" class=\"mrow\"><span id=\"MathJax-Span-42614\" class=\"mrow\"><span id=\"MathJax-Span-42615\" class=\"mn\">80<\/span><span id=\"MathJax-Span-42616\" class=\"mi\">%<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">80%<\/span><\/span>\u00a0of that for the static case. (a) What is the coefficient of static friction? (b) What is the maximum force that can be applied upward along the plane on the rope and not move the block? (c) With a slightly greater applied force, the block will slide up the plane. Once it begins to move, what is its acceleration and what reduced force is necessary to keep it moving upward at constant speed? (d) If the block is given a slight nudge to get it started down the plane, what will be its acceleration in that direction? (e) Once the block begins to slide downward, what upward force on the rope is required to keep the block from accelerating downward?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039470299\" class=\"\"><section>\r\n<div id=\"fs-id1165039297163\"><span class=\"os-number\">112<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039079533\">A car is moving at high speed along a highway when the driver makes an emergency braking. The wheels become locked (stop rolling), and the resulting skid marks are 32.0 meters long. If the coefficient of kinetic friction between tires and road is 0.550, and the acceleration was constant during braking, how fast was the car going when the wheels became locked?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039446833\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035610527\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039446833-solution\">113<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039376043\">A crate having mass 50.0 kg falls horizontally off the back of the flatbed truck, which is traveling at 100 km\/h. Find the value of the coefficient of kinetic friction between the road and crate if the crate slides 50 m on the road in coming to rest. The initial speed of the crate is the same as the truck, 100 km\/h.<\/p>\r\n\r\n<span id=\"fs-id1165036158315\"><img id=\"33868\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/49bf81ed6f402b3d4be4e95fab22f4fab9ce3ade\" alt=\"The figure shows a truck moving to the right at 100 kilometers per hour and a 50 kilogram crate on the ground behind the truck.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039111122\" class=\"\"><section>\r\n<div id=\"fs-id1165039074000\"><span class=\"os-number\">114<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039392702\">A 15-kg sled is pulled across a horizontal, snow-covered surface by a force applied to a rope at 30 degrees with the horizontal. The coefficient of kinetic friction between the sled and the snow is 0.20. (a) If the force is 33 N, what is the horizontal acceleration of the sled? (b) What must the force be in order to pull the sled at constant velocity?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039446792\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165036146754\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039446792-solution\">115<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039422312\">A 30.0-g ball at the end of a string is swung in a vertical circle with a radius of 25.0 cm. The rotational velocity is 200.0 cm\/s. Find the tension in the string: (a) at the top of the circle, (b) at the bottom of the circle, and (c) at a distance of 12.5 cm from the center of the circle\u00a0<span id=\"MathJax-Element-2019-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42617\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42618\" class=\"mrow\"><span id=\"MathJax-Span-42619\" class=\"semantics\"><span id=\"MathJax-Span-42620\" class=\"mrow\"><span id=\"MathJax-Span-42621\" class=\"mrow\"><span id=\"MathJax-Span-42622\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42623\" class=\"mrow\"><span id=\"MathJax-Span-42624\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42625\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42626\" class=\"mn\">12.5<\/span><span id=\"MathJax-Span-42627\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42628\" class=\"mtext\">cm<\/span><\/span><span id=\"MathJax-Span-42629\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42630\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(r=12.5cm).<\/span><\/span><\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035619639\" class=\"\"><section>\r\n<div id=\"fs-id1165036154120\"><span class=\"os-number\">116<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039003350\">A particle of mass 0.50 kg starts moves through a circular path in the\u00a0<em>xy<\/em>-plane with a position given by\u00a0<span id=\"MathJax-Element-2020-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42631\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42632\" class=\"mrow\"><span id=\"MathJax-Span-42633\" class=\"semantics\"><span id=\"MathJax-Span-42634\" class=\"mrow\"><span id=\"MathJax-Span-42635\" class=\"mrow\"><span id=\"MathJax-Span-42636\" class=\"mstyle\"><span id=\"MathJax-Span-42637\" class=\"mrow\"><span id=\"MathJax-Span-42638\" class=\"mover\"><span id=\"MathJax-Span-42639\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42640\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42641\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42642\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42643\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42644\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42645\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42646\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42647\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42648\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-42649\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42650\" class=\"mn\">3<\/span><span id=\"MathJax-Span-42651\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42652\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42653\" class=\"mstyle\"><span id=\"MathJax-Span-42654\" class=\"mrow\"><span id=\"MathJax-Span-42655\" class=\"mover\"><span id=\"MathJax-Span-42656\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42657\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42658\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42659\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42660\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42661\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42662\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-42663\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42664\" class=\"mn\">3<\/span><span id=\"MathJax-Span-42665\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42666\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42667\" class=\"mstyle\"><span id=\"MathJax-Span-42668\" class=\"mrow\"><span id=\"MathJax-Span-42669\" class=\"mover\"><span id=\"MathJax-Span-42670\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42671\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)=(4.0cos3t)i^+(4.0sin3t)j^<\/span><\/span>\u00a0where\u00a0<em>r<\/em>\u00a0is in meters and\u00a0<em>t<\/em>\u00a0is in seconds. (a) Find the velocity and acceleration vectors as functions of time. (b) Show that the acceleration vector always points toward the center of the circle (and thus represents centripetal acceleration). (c) Find the centripetal force vector as a function of time.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039274986\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039371171\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039274986-solution\">117<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035869114\">A stunt cyclist rides on the interior of a cylinder 12 m in radius. The coefficient of static friction between the tires and the wall is 0.68. Find the value of the minimum speed for the cyclist to perform the stunt.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035662464\" class=\"\"><section>\r\n<div id=\"fs-id1165039401204\"><span class=\"os-number\">118<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039046569\">When a body of mass 0.25 kg is attached to a vertical massless spring, it is extended 5.0 cm from its unstretched length of 4.0 cm. The body and spring are placed on a horizontal frictionless surface and rotated about the held end of the spring at 2.0 rev\/s. How far is the spring stretched?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035654187\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165038993163\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035654187-solution\">119<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039297494\">Railroad tracks follow a circular curve of radius 500.0 m and are banked at an angle of\u00a0<span id=\"MathJax-Element-2021-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42672\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42673\" class=\"mrow\"><span id=\"MathJax-Span-42674\" class=\"semantics\"><span id=\"MathJax-Span-42675\" class=\"mrow\"><span id=\"MathJax-Span-42676\" class=\"mrow\"><span id=\"MathJax-Span-42677\" class=\"mn\">5.00<\/span><span id=\"MathJax-Span-42678\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.00\u00b0<\/span><\/span>. For trains of what speed are these tracks designed?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039317700\" class=\"\"><section>\r\n<div id=\"fs-id1165039297475\"><span class=\"os-number\">120<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035733788\">A plumb bob hangs from the roof of a railroad car. The car rounds a circular track of radius 300.0 m at a speed of 90.0 km\/h. At what angle relative to the vertical does the plumb bob hang?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036081646\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039300224\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036081646-solution\">121<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035870165\">An airplane flies at 120.0 m\/s and banks at a\u00a0<span id=\"MathJax-Element-2022-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42679\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42680\" class=\"mrow\"><span id=\"MathJax-Span-42681\" class=\"semantics\"><span id=\"MathJax-Span-42682\" class=\"mrow\"><span id=\"MathJax-Span-42683\" class=\"mrow\"><span id=\"MathJax-Span-42684\" class=\"mn\">30<\/span><span id=\"MathJax-Span-42685\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">30\u00b0<\/span><\/span>\u00a0angle. If its mass is\u00a0<span id=\"MathJax-Element-2023-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42686\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42687\" class=\"mrow\"><span id=\"MathJax-Span-42688\" class=\"semantics\"><span id=\"MathJax-Span-42689\" class=\"mrow\"><span id=\"MathJax-Span-42690\" class=\"mrow\"><span id=\"MathJax-Span-42691\" class=\"mn\">2.50<\/span><span id=\"MathJax-Span-42692\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42693\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42694\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42695\" class=\"msup\"><span id=\"MathJax-Span-42696\" class=\"mrow\"><span id=\"MathJax-Span-42697\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42698\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42699\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42700\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.50\u00d7103kg,<\/span><\/span>\u00a0(a) what is the magnitude of the lift force? (b) what is the radius of the turn?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039443997\" class=\"\"><section>\r\n<div id=\"fs-id1165035633482\"><span class=\"os-number\">122<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036147875\">The position of a particle is given by\u00a0<span id=\"MathJax-Element-2024-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42701\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42702\" class=\"mrow\"><span id=\"MathJax-Span-42703\" class=\"semantics\"><span id=\"MathJax-Span-42704\" class=\"mrow\"><span id=\"MathJax-Span-42705\" class=\"mrow\"><span id=\"MathJax-Span-42706\" class=\"mstyle\"><span id=\"MathJax-Span-42707\" class=\"mrow\"><span id=\"MathJax-Span-42708\" class=\"mover\"><span id=\"MathJax-Span-42709\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42710\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42711\" class=\"mrow\"><span id=\"MathJax-Span-42712\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42713\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42714\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42715\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42716\" class=\"mi\">A<\/span><span id=\"MathJax-Span-42717\" class=\"mrow\"><span id=\"MathJax-Span-42718\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42719\" class=\"mrow\"><span id=\"MathJax-Span-42720\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-42721\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42722\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42723\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42724\" class=\"mstyle\"><span id=\"MathJax-Span-42725\" class=\"mrow\"><span id=\"MathJax-Span-42726\" class=\"mover\"><span id=\"MathJax-Span-42727\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42728\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42729\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42730\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-42731\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42732\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42733\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42734\" class=\"mstyle\"><span id=\"MathJax-Span-42735\" class=\"mrow\"><span id=\"MathJax-Span-42736\" class=\"mover\"><span id=\"MathJax-Span-42737\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42738\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42739\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42740\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)=A(cos\u03c9ti^+sin\u03c9tj^),<\/span><\/span>\u00a0where\u00a0<span id=\"MathJax-Element-2025-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42741\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42742\" class=\"mrow\"><span id=\"MathJax-Span-42743\" class=\"semantics\"><span id=\"MathJax-Span-42744\" class=\"mrow\"><span id=\"MathJax-Span-42745\" class=\"mi\">\u03c9<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03c9<\/span><\/span>\u00a0is a constant. (a) Show that the particle moves in a circle of radius\u00a0<em>A<\/em>. (b) Calculate\u00a0<span id=\"MathJax-Element-2026-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42746\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42747\" class=\"mrow\"><span id=\"MathJax-Span-42748\" class=\"semantics\"><span id=\"MathJax-Span-42749\" class=\"mrow\"><span id=\"MathJax-Span-42750\" class=\"mrow\"><span id=\"MathJax-Span-42751\" class=\"mrow\"><span id=\"MathJax-Span-42752\" class=\"mrow\"><span id=\"MathJax-Span-42753\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42754\" class=\"mstyle\"><span id=\"MathJax-Span-42755\" class=\"mrow\"><span id=\"MathJax-Span-42756\" class=\"mover\"><span id=\"MathJax-Span-42757\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42758\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42759\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-42760\" class=\"mrow\"><span id=\"MathJax-Span-42761\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42762\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">dr\u2192\/dt<\/span><\/span>\u00a0and then show that the speed of the particle is a constant\u00a0<span id=\"MathJax-Element-2027-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42763\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42764\" class=\"mrow\"><span id=\"MathJax-Span-42765\" class=\"semantics\"><span id=\"MathJax-Span-42766\" class=\"mrow\"><span id=\"MathJax-Span-42767\" class=\"mrow\"><span id=\"MathJax-Span-42768\" class=\"msub\"><span id=\"MathJax-Span-42769\" class=\"mi\">A<\/span><span id=\"MathJax-Span-42770\" class=\"mi\">\u03c9<\/span><\/span><span id=\"MathJax-Span-42771\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">A\u03c9.<\/span><\/span>\u00a0(c) Determine\u00a0<span id=\"MathJax-Element-2028-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42772\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42773\" class=\"mrow\"><span id=\"MathJax-Span-42774\" class=\"semantics\"><span id=\"MathJax-Span-42775\" class=\"mrow\"><span id=\"MathJax-Span-42776\" class=\"mrow\"><span id=\"MathJax-Span-42777\" class=\"mrow\"><span id=\"MathJax-Span-42778\" class=\"mrow\"><span id=\"MathJax-Span-42779\" class=\"msup\"><span id=\"MathJax-Span-42780\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42781\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42782\" class=\"mstyle\"><span id=\"MathJax-Span-42783\" class=\"mrow\"><span id=\"MathJax-Span-42784\" class=\"mover\"><span id=\"MathJax-Span-42785\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42786\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42787\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-42788\" class=\"mrow\"><span id=\"MathJax-Span-42789\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42790\" class=\"msup\"><span id=\"MathJax-Span-42791\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42792\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">d2r\u2192\/dt2<\/span><\/span>\u00a0and show that\u00a0<em>a<\/em>\u00a0is given by<span id=\"MathJax-Element-2029-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42793\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42794\" class=\"mrow\"><span id=\"MathJax-Span-42795\" class=\"semantics\"><span id=\"MathJax-Span-42796\" class=\"mrow\"><span id=\"MathJax-Span-42797\" class=\"mrow\"><span id=\"MathJax-Span-42798\" class=\"msub\"><span id=\"MathJax-Span-42799\" class=\"mi\">a<\/span><span id=\"MathJax-Span-42800\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-42801\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42802\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42803\" class=\"msup\"><span id=\"MathJax-Span-42804\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42805\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42806\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">ac=r\u03c92.<\/span><\/span>\u00a0(d) Calculate the centripetal force on the particle. [<em>Hint<\/em>: For (b) and (c), you will need to use\u00a0<span id=\"MathJax-Element-2030-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42807\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42808\" class=\"mrow\"><span id=\"MathJax-Span-42809\" class=\"semantics\"><span id=\"MathJax-Span-42810\" class=\"mrow\"><span id=\"MathJax-Span-42811\" class=\"mrow\"><span id=\"MathJax-Span-42812\" class=\"mrow\"><span id=\"MathJax-Span-42813\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42814\" class=\"mrow\"><span id=\"MathJax-Span-42815\" class=\"mrow\"><span id=\"MathJax-Span-42816\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42817\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-42818\" class=\"mrow\"><span id=\"MathJax-Span-42819\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42820\" class=\"mi\">t<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42821\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42822\" class=\"mrow\"><span id=\"MathJax-Span-42823\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42824\" class=\"mrow\"><span id=\"MathJax-Span-42825\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-42826\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42827\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42828\" class=\"mi\">t<\/span><\/span><span id=\"MathJax-Span-42829\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42830\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42831\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-42832\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42833\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42834\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-42835\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42836\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42837\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(d\/dt)(cos\u03c9t)=\u2212\u03c9sin\u03c9t<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-2031-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42838\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42839\" class=\"mrow\"><span id=\"MathJax-Span-42840\" class=\"semantics\"><span id=\"MathJax-Span-42841\" class=\"mrow\"><span id=\"MathJax-Span-42842\" class=\"mrow\"><span id=\"MathJax-Span-42843\" class=\"mrow\"><span id=\"MathJax-Span-42844\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42845\" class=\"mrow\"><span id=\"MathJax-Span-42846\" class=\"mrow\"><span id=\"MathJax-Span-42847\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42848\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-42849\" class=\"mrow\"><span id=\"MathJax-Span-42850\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42851\" class=\"mi\">t<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42852\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42853\" class=\"mrow\"><span id=\"MathJax-Span-42854\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42855\" class=\"mrow\"><span id=\"MathJax-Span-42856\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-42857\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42858\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42859\" class=\"mi\">t<\/span><\/span><span id=\"MathJax-Span-42860\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42861\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42862\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42863\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42864\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-42865\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42866\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42867\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42868\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(d\/dt)(sin\u03c9t)=\u03c9cos\u03c9t.<\/span><\/span><\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039504554\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035635058\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039504554-solution\">123<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039073034\">Two blocks connected by a string are pulled across a horizontal surface by a force applied to one of the blocks, as shown below. The coefficient of kinetic friction between the blocks and the surface is 0.25. If each block has an acceleration of\u00a0<span id=\"MathJax-Element-2032-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42869\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42870\" class=\"mrow\"><span id=\"MathJax-Span-42871\" class=\"semantics\"><span id=\"MathJax-Span-42872\" class=\"mrow\"><span id=\"MathJax-Span-42873\" class=\"mrow\"><span id=\"MathJax-Span-42874\" class=\"mn\">2.0<\/span><span id=\"MathJax-Span-42875\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42876\" class=\"msup\"><span id=\"MathJax-Span-42877\" class=\"mrow\"><span id=\"MathJax-Span-42878\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-42879\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.0m\/s2<\/span><\/span>\u00a0to the right, what is the magnitude\u00a0<em>F<\/em>\u00a0of the applied force?<\/p>\r\n\r\n<span id=\"fs-id1165035715506\"><img id=\"6962\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/c78d2dc779dd8110971b21760e2a152a36e2e166\" alt=\"Two blocks, 1.0 kilograms on the left and 3.0 kilograms on the right, are connected by a string and are on a horizontal surface. Force F acts on the 3.0 kilogram mass and points up and to the right at a angle of 60 degrees above the horizontal.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039479037\" class=\"\"><section>\r\n<div id=\"fs-id1165035646533\"><span class=\"os-number\">124<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035646536\">As shown below, the coefficient of kinetic friction between the surface and the larger block is 0.20, and the coefficient of kinetic friction between the surface and the smaller block is 0.30. If\u00a0<span id=\"MathJax-Element-2033-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42880\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42881\" class=\"mrow\"><span id=\"MathJax-Span-42882\" class=\"semantics\"><span id=\"MathJax-Span-42883\" class=\"mrow\"><span id=\"MathJax-Span-42884\" class=\"mrow\"><span id=\"MathJax-Span-42885\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42886\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42887\" class=\"mn\">10<\/span><span id=\"MathJax-Span-42888\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42889\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F=10N<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-2034-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42890\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42891\" class=\"mrow\"><span id=\"MathJax-Span-42892\" class=\"semantics\"><span id=\"MathJax-Span-42893\" class=\"mrow\"><span id=\"MathJax-Span-42894\" class=\"mrow\"><span id=\"MathJax-Span-42895\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42896\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42897\" class=\"mn\">1.0<\/span><span id=\"MathJax-Span-42898\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42899\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=1.0kg<\/span><\/span>, what is the tension in the connecting string?<\/p>\r\n\r\n<span id=\"fs-id1165039351827\"><img id=\"62415\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/a503dc53b79ac38ef0f48b88d87623571f75b4e8\" alt=\"Two blocks, 2 M on the left and M on the right, are connected by a string and are on a horizontal surface. Force F acts on M and points to the right.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165036007610\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035664399\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036007610-solution\">125<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035664401\">In the figure, the coefficient of kinetic friction between the surface and the blocks is\u00a0<span id=\"MathJax-Element-2035-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42900\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42901\" class=\"mrow\"><span id=\"MathJax-Span-42902\" class=\"semantics\"><span id=\"MathJax-Span-42903\" class=\"mrow\"><span id=\"MathJax-Span-42904\" class=\"mrow\"><span id=\"MathJax-Span-42905\" class=\"msub\"><span id=\"MathJax-Span-42906\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-42907\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-42908\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bck.<\/span><\/span>\u00a0If\u00a0<span id=\"MathJax-Element-2036-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42909\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42910\" class=\"mrow\"><span id=\"MathJax-Span-42911\" class=\"semantics\"><span id=\"MathJax-Span-42912\" class=\"mrow\"><span id=\"MathJax-Span-42913\" class=\"mrow\"><span id=\"MathJax-Span-42914\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42915\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42916\" class=\"mn\">1.0<\/span><span id=\"MathJax-Span-42917\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42918\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=1.0kg,<\/span><\/span>\u00a0find an expression for the magnitude of the acceleration of either block (in terms of\u00a0<em>F<\/em>,\u00a0<span id=\"MathJax-Element-2037-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42919\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42920\" class=\"mrow\"><span id=\"MathJax-Span-42921\" class=\"semantics\"><span id=\"MathJax-Span-42922\" class=\"mrow\"><span id=\"MathJax-Span-42923\" class=\"mrow\"><span id=\"MathJax-Span-42924\" class=\"msub\"><span id=\"MathJax-Span-42925\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-42926\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-42927\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bck,<\/span><\/span>\u00a0and\u00a0<em>g<\/em>).<\/p>\r\n\r\n<span id=\"fs-id1165035748378\"><img id=\"53635\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/58bc9a491f5e85eeb1c061f73130320572ec1165\" alt=\"Two blocks, M on the left and 3 M on the right, are connected by a string and are on a horizontal surface. The following forces are indicated: f sub k 2 acting on M and pointing to the right, f sub k 1 acting on 3 M and pointing to the right, F acting on 3 M and pointing to the left, N sub 2 acting on M and pointing up, N sub 1 acting on 3 M and pointing up, M g acting on M and pointing down, , 3 M g acting on 3 M and pointing down.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035662213\" class=\"\"><section>\r\n<div id=\"fs-id1165039209290\"><span class=\"os-number\">126<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039209292\">Two blocks are stacked as shown below, and rest on a frictionless surface. There is friction between the two blocks (coefficient of friction\u00a0<span id=\"MathJax-Element-2038-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42928\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42929\" class=\"mrow\"><span id=\"MathJax-Span-42930\" class=\"semantics\"><span id=\"MathJax-Span-42931\" class=\"mrow\"><span id=\"MathJax-Span-42932\" class=\"mi\">\u03bc<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bc<\/span><\/span>). An external force is applied to the top block at an angle\u00a0<span id=\"MathJax-Element-2039-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42933\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42934\" class=\"mrow\"><span id=\"MathJax-Span-42935\" class=\"semantics\"><span id=\"MathJax-Span-42936\" class=\"mrow\"><span id=\"MathJax-Span-42937\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0with the horizontal. What is the maximum force\u00a0<em>F<\/em>\u00a0that can be applied for the two blocks to move together?<\/p>\r\n\r\n<span id=\"fs-id1165035758924\"><img id=\"48921\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/713b7cb5d8a0235c6ef30aef5f07c19e68d9106d\" alt=\"Rectangular block M sub 2 is on a horizontal surface. Rectangular block M sub 1 is on top of block M sub 2. A force F pushes on block M sub 1. Force F is directed down and to the right, at a angle theta to the horizontal.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039496071\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039443360\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039496071-solution\">127<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039433646\">A box rests on the (horizontal) back of a truck. The coefficient of static friction between the box and the surface on which it rests is 0.24. What maximum distance can the truck travel (starting from rest and moving horizontally with constant acceleration) in 3.0 s without having the box slide?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039232615\" class=\"\"><section>\r\n<div id=\"fs-id1165039257182\"><span class=\"os-number\">128<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039070262\">A double-incline plane is shown below. The coefficient of friction on the left surface is 0.30, and on the right surface 0.16. Calculate the acceleration of the system.<\/p>\r\n\r\n<\/div>\r\n<div><span id=\"fs-id1165035627255\"><img id=\"32363\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/bc9f831a81c4a58a2915254a9632088e78bfa1b0\" alt=\"Two carts connected by a string passing over a pulley are on either side of a double inclined plane. The string passes over a pulley attached to the top of the double incline. On the left, the incline makes an angle of 37 degrees with the horizontal and the cart on that side has mass 10 kilograms. On the right, the incline makes an angle of 53 degrees with the horizontal and the cart on that side has mass 15 kilograms.\" \/><\/span><\/div>\r\n<div><\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<div class=\"os-review-challenge-container\">\r\n<h3><span class=\"os-text\">Challenge Problems<\/span><\/h3>\r\n<section id=\"fs-id1165039325516\" class=\"review-challenge\">\r\n<div id=\"fs-id1165039026859\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035723079\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039026859-solution\">129<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165036007334\">In a later chapter, you will find that the weight of a particle varies with altitude such that\u00a0<span id=\"MathJax-Element-2040-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42938\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42939\" class=\"mrow\"><span id=\"MathJax-Span-42940\" class=\"semantics\"><span id=\"MathJax-Span-42941\" class=\"mrow\"><span id=\"MathJax-Span-42942\" class=\"mrow\"><span id=\"MathJax-Span-42943\" class=\"mi\">w<\/span><span id=\"MathJax-Span-42944\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42945\" class=\"mfrac\"><span id=\"MathJax-Span-42946\" class=\"mrow\"><span id=\"MathJax-Span-42947\" class=\"mi\">m<\/span><span id=\"MathJax-Span-42948\" class=\"mi\">g<\/span><span id=\"MathJax-Span-42949\" class=\"msub\"><span id=\"MathJax-Span-42950\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42951\" class=\"mn\">0<\/span><\/span><span id=\"MathJax-Span-42952\" class=\"msup\"><span id=\"MathJax-Span-42953\" class=\"mrow\"><\/span><span id=\"MathJax-Span-42954\" class=\"mn\">2<\/span><\/span><\/span><span id=\"MathJax-Span-42955\" class=\"mrow\"><span id=\"MathJax-Span-42956\" class=\"msup\"><span id=\"MathJax-Span-42957\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42958\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">w=mgr02r2<\/span><\/span>\u00a0where\u00a0<span id=\"MathJax-Element-2041-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42959\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42960\" class=\"mrow\"><span id=\"MathJax-Span-42961\" class=\"semantics\"><span id=\"MathJax-Span-42962\" class=\"mrow\"><span id=\"MathJax-Span-42963\" class=\"mrow\"><span id=\"MathJax-Span-42964\" class=\"msub\"><span id=\"MathJax-Span-42965\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42966\" class=\"mn\">0<\/span><\/span><span id=\"MathJax-Span-42967\" class=\"msup\"><span id=\"MathJax-Span-42968\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r0<\/span><\/span>\u00a0is the radius of Earth and\u00a0<em>r<\/em>\u00a0is the distance from Earth\u2019s center. If the particle is fired vertically with velocity\u00a0<span id=\"MathJax-Element-2042-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42969\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42970\" class=\"mrow\"><span id=\"MathJax-Span-42971\" class=\"semantics\"><span id=\"MathJax-Span-42972\" class=\"mrow\"><span id=\"MathJax-Span-42973\" class=\"mrow\"><span id=\"MathJax-Span-42974\" class=\"msub\"><span id=\"MathJax-Span-42975\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42976\" class=\"mn\">0<\/span><\/span><span id=\"MathJax-Span-42977\" class=\"msup\"><span id=\"MathJax-Span-42978\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v0<\/span><\/span>\u00a0from Earth\u2019s surface, determine its velocity as a function of position\u00a0<em>r<\/em>. (<em>Hint:<\/em>\u00a0use\u00a0<span id=\"MathJax-Element-2043-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42979\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42980\" class=\"mrow\"><span id=\"MathJax-Span-42981\" class=\"semantics\"><span id=\"MathJax-Span-42982\" class=\"mrow\"><span id=\"MathJax-Span-42983\" class=\"mrow\"><span id=\"MathJax-Span-42984\" class=\"msup\"><span id=\"MathJax-Span-42985\" class=\"mi\">a<\/span><\/span><span id=\"MathJax-Span-42986\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42987\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42988\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42989\" class=\"msup\"><span id=\"MathJax-Span-42990\" class=\"mi\">v<\/span><\/span><span id=\"MathJax-Span-42991\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42992\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42993\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">adr=vdv,<\/span><\/span>\u00a0the rearrangement mentioned in the text.)<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039315132\" class=\"\"><section>\r\n<div id=\"fs-id1165039315134\"><span class=\"os-number\">130<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035926222\">A large centrifuge, like the one shown below, is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries. (a) At what angular velocity is the centripetal acceleration 10<em>g<\/em>\u00a0if the rider is 15.0 m from the center of rotation? (b) The rider\u2019s cage hangs on a pivot at the end of the arm, allowing it to swing outward during rotation as shown in the bottom accompanying figure. At what angle\u00a0<span id=\"MathJax-Element-2044-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42994\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42995\" class=\"mrow\"><span id=\"MathJax-Span-42996\" class=\"semantics\"><span id=\"MathJax-Span-42997\" class=\"mrow\"><span id=\"MathJax-Span-42998\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0below the horizontal will the cage hang when the centripetal acceleration is 10<em>g<\/em>? (<em>Hint:<\/em>\u00a0The arm supplies centripetal force and supports the weight of the cage. Draw a free-body diagram of the forces to see what the angle\u00a0<span id=\"MathJax-Element-2045-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42999\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43000\" class=\"mrow\"><span id=\"MathJax-Span-43001\" class=\"semantics\"><span id=\"MathJax-Span-43002\" class=\"mrow\"><span id=\"MathJax-Span-43003\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0should be.)<\/p>\r\n\r\n<span id=\"fs-id1165039487271\"><img id=\"64783\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/94622e0aba8bdb935d2dde3c8cc055dec14c39bb\" alt=\"(a) A photograph of a high g training centrifuge. The astronaut sits in a cage at the end of a long arm that rotates in a horizontal plane. (b) An illustration of a top view of the centrifuge along with an illustration of the forces. The free body diagram shows the weight, w, pointing vertically down and the force F sub arm pointing up and to the left. The forces are then shown rearranged to form a right triangle. F sub arm is the hypotenuse of the triangle pointing up and left, w is the vertical side pointing down, and F sub c is the base pointing to the left. The F sub c arrow is then shown separately with the notation that vector F sub c equals F sub net.\" \/><\/span><\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035646117\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035772540\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035646117-solution\">131<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035772542\">A car of mass 1000.0 kg is traveling along a level road at 100.0 km\/h when its brakes are applied. Calculate the stopping distance if the coefficient of kinetic friction of the tires is 0.500. Neglect air resistance. (<em>Hint:<\/em>\u00a0since the distance traveled is of interest rather than the time,\u00a0<em>x<\/em>\u00a0is the desired independent variable and not\u00a0<em>t<\/em>. Use the Chain Rule to change the variable:\u00a0<span id=\"MathJax-Element-2046-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43004\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43005\" class=\"mrow\"><span id=\"MathJax-Span-43006\" class=\"semantics\"><span id=\"MathJax-Span-43007\" class=\"mrow\"><span id=\"MathJax-Span-43008\" class=\"mrow\"><span id=\"MathJax-Span-43009\" class=\"mfrac\"><span id=\"MathJax-Span-43010\" class=\"mrow\"><span id=\"MathJax-Span-43011\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43012\" class=\"mi\">v<\/span><\/span><span id=\"MathJax-Span-43013\" class=\"mrow\"><span id=\"MathJax-Span-43014\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43015\" class=\"mi\">t<\/span><\/span><\/span><span id=\"MathJax-Span-43016\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43017\" class=\"mfrac\"><span id=\"MathJax-Span-43018\" class=\"mrow\"><span id=\"MathJax-Span-43019\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43020\" class=\"mi\">v<\/span><\/span><span id=\"MathJax-Span-43021\" class=\"mrow\"><span id=\"MathJax-Span-43022\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43023\" class=\"mi\">x<\/span><\/span><\/span><span id=\"MathJax-Span-43024\" class=\"mspace\"><\/span><span id=\"MathJax-Span-43025\" class=\"mfrac\"><span id=\"MathJax-Span-43026\" class=\"mrow\"><span id=\"MathJax-Span-43027\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43028\" class=\"mi\">x<\/span><\/span><span id=\"MathJax-Span-43029\" class=\"mrow\"><span id=\"MathJax-Span-43030\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43031\" class=\"mi\">t<\/span><\/span><\/span><span id=\"MathJax-Span-43032\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43033\" class=\"mi\">v<\/span><span id=\"MathJax-Span-43034\" class=\"mfrac\"><span id=\"MathJax-Span-43035\" class=\"mrow\"><span id=\"MathJax-Span-43036\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43037\" class=\"mi\">v<\/span><\/span><span id=\"MathJax-Span-43038\" class=\"mrow\"><span id=\"MathJax-Span-43039\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43040\" class=\"mi\">x<\/span><\/span><\/span><span id=\"MathJax-Span-43041\" class=\"mo\">.<\/span><span id=\"MathJax-Span-43042\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">dvdt=dvdxdxdt=vdvdx.)<\/span><\/span><\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035679412\" class=\"\"><section>\r\n<div id=\"fs-id1165039494317\"><span class=\"os-number\">132<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039331601\">An airplane flying at 200.0 m\/s makes a turn that takes 4.0 min. What bank angle is required? What is the percentage increase in the perceived weight of the passengers?<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165039416120\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165035975002\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039416120-solution\">133<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035975004\">A skydiver is at an altitude of 1520 m. After 10.0 seconds of free fall, he opens his parachute and finds that the air resistance,\u00a0<span id=\"MathJax-Element-2047-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43043\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43044\" class=\"mrow\"><span id=\"MathJax-Span-43045\" class=\"semantics\"><span id=\"MathJax-Span-43046\" class=\"mrow\"><span id=\"MathJax-Span-43047\" class=\"mrow\"><span id=\"MathJax-Span-43048\" class=\"msub\"><span id=\"MathJax-Span-43049\" class=\"mi\">F<\/span><span id=\"MathJax-Span-43050\" class=\"mtext\">D<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">FD<\/span><\/span>, is given by the formula\u00a0<span id=\"MathJax-Element-2048-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43051\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43052\" class=\"mrow\"><span id=\"MathJax-Span-43053\" class=\"semantics\"><span id=\"MathJax-Span-43054\" class=\"mrow\"><span id=\"MathJax-Span-43055\" class=\"mrow\"><span id=\"MathJax-Span-43056\" class=\"msub\"><span id=\"MathJax-Span-43057\" class=\"mi\">F<\/span><span id=\"MathJax-Span-43058\" class=\"mtext\">D<\/span><\/span><span id=\"MathJax-Span-43059\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43060\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-43061\" class=\"mi\">b<\/span><span id=\"MathJax-Span-43062\" class=\"mi\">v<\/span><span id=\"MathJax-Span-43063\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">FD=\u2212bv,<\/span><\/span>\u00a0where\u00a0<em>b<\/em>\u00a0is a constant and\u00a0<em>v<\/em>\u00a0is the velocity. If\u00a0<span id=\"MathJax-Element-2049-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43064\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43065\" class=\"mrow\"><span id=\"MathJax-Span-43066\" class=\"semantics\"><span id=\"MathJax-Span-43067\" class=\"mrow\"><span id=\"MathJax-Span-43068\" class=\"mrow\"><span id=\"MathJax-Span-43069\" class=\"mi\">b<\/span><span id=\"MathJax-Span-43070\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43071\" class=\"mn\">0.750<\/span><span id=\"MathJax-Span-43072\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">b=0.750,<\/span><\/span>\u00a0and the mass of the skydiver is 82.0 kg, first set up differential equations for the velocity and the position, and then find: (a) the speed of the skydiver when the parachute opens, (b) the distance fallen before the parachute opens, (c) the terminal velocity after the parachute opens (find the limiting velocity), and (d) the time the skydiver is in the air after the parachute opens.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035694730\" class=\"\"><section>\r\n<div id=\"fs-id1165035654245\"><span class=\"os-number\">134<\/span><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165035654247\">In a television commercial, a small, spherical bead of mass 4.00 g is released from rest at\u00a0<span id=\"MathJax-Element-2050-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43073\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43074\" class=\"mrow\"><span id=\"MathJax-Span-43075\" class=\"semantics\"><span id=\"MathJax-Span-43076\" class=\"mrow\"><span id=\"MathJax-Span-43077\" class=\"mrow\"><span id=\"MathJax-Span-43078\" class=\"mi\">t<\/span><span id=\"MathJax-Span-43079\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43080\" class=\"mn\">0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0<\/span><\/span>\u00a0in a bottle of liquid shampoo. The terminal speed is observed to be 2.00 cm\/s. Find (a) the value of the constant\u00a0<em>b<\/em>\u00a0in the equation\u00a0<span id=\"MathJax-Element-2051-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43081\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43082\" class=\"mrow\"><span id=\"MathJax-Span-43083\" class=\"semantics\"><span id=\"MathJax-Span-43084\" class=\"mrow\"><span id=\"MathJax-Span-43085\" class=\"mrow\"><span id=\"MathJax-Span-43086\" class=\"mi\">v<\/span><span id=\"MathJax-Span-43087\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43088\" class=\"mfrac\"><span id=\"MathJax-Span-43089\" class=\"mrow\"><span id=\"MathJax-Span-43090\" class=\"mi\">m<\/span><span id=\"MathJax-Span-43091\" class=\"mi\">g<\/span><\/span><span id=\"MathJax-Span-43092\" class=\"mi\">b<\/span><\/span><span id=\"MathJax-Span-43093\" class=\"mo\">(<\/span><span id=\"MathJax-Span-43094\" class=\"mn\">1<\/span><span id=\"MathJax-Span-43095\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-43096\" class=\"msup\"><span id=\"MathJax-Span-43097\" class=\"mi\">e<\/span><span id=\"MathJax-Span-43098\" class=\"mrow\"><span id=\"MathJax-Span-43099\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-43100\" class=\"mrow\"><span id=\"MathJax-Span-43101\" class=\"mrow\"><span id=\"MathJax-Span-43102\" class=\"mi\">b<\/span><span id=\"MathJax-Span-43103\" class=\"mi\">t<\/span><\/span><span id=\"MathJax-Span-43104\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-43105\" class=\"mi\">m<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-43106\" class=\"mo\">)<\/span><span id=\"MathJax-Span-43107\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v=mgb(1\u2212e\u2212bt\/m),<\/span><\/span>\u00a0and (b) the value of the resistive force when the bead reaches terminal speed.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165035685642\" class=\"os-hasSolution\"><section>\r\n<div id=\"fs-id1165039511826\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035685642-solution\">135<\/a><span class=\"os-divider\">.<\/span>\r\n<p id=\"fs-id1165039511828\">A boater and motor boat are at rest on a lake. Together, they have mass 200.0 kg. If the thrust of the motor is a constant force of 40.0 N in the direction of motion, and if the resistive force of the water is numerically equivalent to 2 times the speed\u00a0<em>v<\/em>of the boat, set up and solve the differential equation to find: (a) the velocity of the boat at time\u00a0<em>t<\/em>; (b) the limiting velocity (the velocity after a long time has passed).<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>","rendered":"<div class=\"os-glossary-container\">\n<div class=\"textbox key-takeaways\">\n<h3><span class=\"os-text\">Key Terms<\/span><\/h3>\n<dl id=\"fs-id1165039331406\">\n<dt id=\"33239\"><strong>banked curve<\/strong><\/dt>\n<dd id=\"fs-id1165039440814\">curve in a road that is sloping in a manner that helps a vehicle negotiate the curve<\/dd>\n<\/dl>\n<dl id=\"fs-id1165039353624\">\n<dt id=\"31765\"><strong>centripetal force<\/strong><\/dt>\n<dd id=\"fs-id1165039106544\">any net force causing uniform circular motion<\/dd>\n<\/dl>\n<dl id=\"fs-id1165039329539\">\n<dt id=\"46238\"><strong>Coriolis force<\/strong><\/dt>\n<dd id=\"fs-id1165039425267\">inertial force causing the apparent deflection of moving objects when viewed in a rotating frame of reference<\/dd>\n<\/dl>\n<dl id=\"fs-id1165039331836\">\n<dt id=\"47300\"><strong>drag force<\/strong><\/dt>\n<dd id=\"fs-id1165039450582\">force that always opposes the motion of an object in a fluid; unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid<\/dd>\n<\/dl>\n<dl id=\"fs-id1165038031500\">\n<dt id=\"72218\"><strong>friction<\/strong><\/dt>\n<dd id=\"fs-id1165037166731\">force that opposes relative motion or attempts at motion between systems in contact<\/dd>\n<\/dl>\n<dl id=\"fs-id1165039255178\">\n<dt id=\"19445\"><strong>ideal banking<\/strong><\/dt>\n<dd id=\"fs-id1165035708390\">sloping of a curve in a road, where the angle of the slope allows the vehicle to negotiate the curve at a certain speed without the aid of friction between the tires and the road; the net external force on the vehicle equals the horizontal centripetal force in the absence of friction<\/dd>\n<\/dl>\n<dl id=\"fs-id1165035708392\">\n<dt id=\"82109\"><strong>inertial force<\/strong><\/dt>\n<dd id=\"fs-id1165039075055\">force that has no physical origin<\/dd>\n<\/dl>\n<dl id=\"fs-id1165038337854\">\n<dt id=\"77544\"><strong>kinetic friction<\/strong><\/dt>\n<dd id=\"fs-id1165036846768\">force that opposes the motion of two systems that are in contact and moving relative to each other<\/dd>\n<\/dl>\n<dl id=\"fs-id1165039512549\">\n<dt id=\"89165\"><strong>noninertial frame of reference<\/strong><\/dt>\n<dd id=\"fs-id1165039234658\">accelerated frame of reference<\/dd>\n<\/dl>\n<dl id=\"fs-id1165036846773\">\n<dt id=\"36669\"><strong>static friction<\/strong><\/dt>\n<dd id=\"fs-id1165038327277\">force that opposes the motion of two systems that are in contact and are not moving relative to each other<\/dd>\n<\/dl>\n<dl id=\"fs-id1165039337073\">\n<dt id=\"65315\"><strong>terminal velocity<\/strong><\/dt>\n<dd id=\"fs-id1165035673837\">constant velocity achieved by a falling object, which occurs when the weight of the object is balanced by the upward drag force<\/dd>\n<\/dl>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"os-key-equations-container\">\n<div class=\"textbox shaded\">\n<h3><span class=\"os-text\">Key Equations<\/span><\/h3>\n<section id=\"fs-id1165035743931\" class=\"key-equations\">\n<table id=\"fs-id1170902765576\" class=\"unnumbered unstyled\" summary=\"This table gives the following formulae: Magnitude of static friction, f subscript s is less than or equal to mu subscript s N; Magnitude of kinetic friction, f subscript k equal to mu subscript k N; Centripetal force, F subscript C equal to m v squared by r or F subscript C equal to m r omega squared; Ideal angle of a banked curve, tan theta equal to v squared by rg; Drag force, F subscript D equal to half C rho A v squared; Stokes\u2019 law, F subscript S equal to 6 pi r eta v.\">\n<tbody>\n<tr valign=\"top\">\n<td>Magnitude of static friction<\/td>\n<td><span id=\"MathJax-Element-1890-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40924\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40925\" class=\"mrow\"><span id=\"MathJax-Span-40926\" class=\"semantics\"><span id=\"MathJax-Span-40927\" class=\"mrow\"><span id=\"MathJax-Span-40928\" class=\"mrow\"><span id=\"MathJax-Span-40929\" class=\"msub\"><span id=\"MathJax-Span-40930\" class=\"mi\">f<\/span><span id=\"MathJax-Span-40931\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-40932\" class=\"mo\">\u2264<\/span><span id=\"MathJax-Span-40933\" class=\"msub\"><span id=\"MathJax-Span-40934\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-40935\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-40936\" class=\"mi\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">fs\u2264\u03bcsN<\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Magnitude of kinetic friction<\/td>\n<td><span id=\"MathJax-Element-1891-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40937\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40938\" class=\"mrow\"><span id=\"MathJax-Span-40939\" class=\"semantics\"><span id=\"MathJax-Span-40940\" class=\"mrow\"><span id=\"MathJax-Span-40941\" class=\"mrow\"><span id=\"MathJax-Span-40942\" class=\"msub\"><span id=\"MathJax-Span-40943\" class=\"mi\">f<\/span><span id=\"MathJax-Span-40944\" class=\"mi\">k<\/span><\/span><span id=\"MathJax-Span-40945\" class=\"mo\">=<\/span><span id=\"MathJax-Span-40946\" class=\"msub\"><span id=\"MathJax-Span-40947\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-40948\" class=\"mi\">k<\/span><\/span><span id=\"MathJax-Span-40949\" class=\"mi\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">fk=\u03bckN<\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Centripetal force<\/td>\n<td><span id=\"MathJax-Element-1892-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40950\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40951\" class=\"mrow\"><span id=\"MathJax-Span-40952\" class=\"semantics\"><span id=\"MathJax-Span-40953\" class=\"mrow\"><span id=\"MathJax-Span-40954\" class=\"mrow\"><span id=\"MathJax-Span-40955\" class=\"msub\"><span id=\"MathJax-Span-40956\" class=\"mi\">F<\/span><span id=\"MathJax-Span-40957\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-40958\" class=\"mo\">=<\/span><span id=\"MathJax-Span-40959\" class=\"mi\">m<\/span><span id=\"MathJax-Span-40960\" class=\"mfrac\"><span id=\"MathJax-Span-40961\" class=\"mrow\"><span id=\"MathJax-Span-40962\" class=\"msup\"><span id=\"MathJax-Span-40963\" class=\"mi\">v<\/span><span id=\"MathJax-Span-40964\" class=\"mn\">2<\/span><\/span><\/span><span id=\"MathJax-Span-40965\" class=\"mi\">r<\/span><\/span><span id=\"MathJax-Span-40966\" class=\"mspace\"><\/span><span id=\"MathJax-Span-40967\" class=\"mtext\">or<\/span><span id=\"MathJax-Span-40968\" class=\"mspace\"><\/span><span id=\"MathJax-Span-40969\" class=\"msub\"><span id=\"MathJax-Span-40970\" class=\"mi\">F<\/span><span id=\"MathJax-Span-40971\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-40972\" class=\"mo\">=<\/span><span id=\"MathJax-Span-40973\" class=\"mi\">m<\/span><span id=\"MathJax-Span-40974\" class=\"mi\">r<\/span><span id=\"MathJax-Span-40975\" class=\"msup\"><span id=\"MathJax-Span-40976\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-40977\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fc=mv2rorFc=mr\u03c92<\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Ideal angle of a banked curve<\/td>\n<td><span id=\"MathJax-Element-1893-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40978\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40979\" class=\"mrow\"><span id=\"MathJax-Span-40980\" class=\"semantics\"><span id=\"MathJax-Span-40981\" class=\"mrow\"><span id=\"MathJax-Span-40982\" class=\"mrow\"><span id=\"MathJax-Span-40983\" class=\"mtext\">tan<\/span><span id=\"MathJax-Span-40984\" class=\"mspace\"><\/span><span id=\"MathJax-Span-40985\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-40986\" class=\"mo\">=<\/span><span id=\"MathJax-Span-40987\" class=\"mfrac\"><span id=\"MathJax-Span-40988\" class=\"mrow\"><span id=\"MathJax-Span-40989\" class=\"msup\"><span id=\"MathJax-Span-40990\" class=\"mi\">v<\/span><span id=\"MathJax-Span-40991\" class=\"mn\">2<\/span><\/span><\/span><span id=\"MathJax-Span-40992\" class=\"mrow\"><span id=\"MathJax-Span-40993\" class=\"mi\">r<\/span><span id=\"MathJax-Span-40994\" class=\"mi\">g<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">tan\u03b8=v2rg<\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Drag force<\/td>\n<td><span id=\"MathJax-Element-1894-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-40995\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-40996\" class=\"mrow\"><span id=\"MathJax-Span-40997\" class=\"semantics\"><span id=\"MathJax-Span-40998\" class=\"mrow\"><span id=\"MathJax-Span-40999\" class=\"mrow\"><span id=\"MathJax-Span-41000\" class=\"msub\"><span id=\"MathJax-Span-41001\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41002\" class=\"mi\">D<\/span><\/span><span id=\"MathJax-Span-41003\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41004\" class=\"mfrac\"><span id=\"MathJax-Span-41005\" class=\"mn\">1<\/span><span id=\"MathJax-Span-41006\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-41007\" class=\"mi\">C<\/span><span id=\"MathJax-Span-41008\" class=\"mi\">\u03c1<\/span><span id=\"MathJax-Span-41009\" class=\"mi\">A<\/span><span id=\"MathJax-Span-41010\" class=\"msup\"><span id=\"MathJax-Span-41011\" class=\"mi\">v<\/span><span id=\"MathJax-Span-41012\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">FD=12C\u03c1Av2<\/span><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td>Stokes\u2019 law<\/td>\n<td><span id=\"MathJax-Element-1895-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41013\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41014\" class=\"mrow\"><span id=\"MathJax-Span-41015\" class=\"semantics\"><span id=\"MathJax-Span-41016\" class=\"mrow\"><span id=\"MathJax-Span-41017\" class=\"mrow\"><span id=\"MathJax-Span-41018\" class=\"msub\"><span id=\"MathJax-Span-41019\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41020\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41021\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41022\" class=\"mn\">6<\/span><span id=\"MathJax-Span-41023\" class=\"mi\">\u03c0<\/span><span id=\"MathJax-Span-41024\" class=\"mi\">r<\/span><span id=\"MathJax-Span-41025\" class=\"mi\">\u03b7<\/span><span id=\"MathJax-Span-41026\" class=\"mi\">v<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fs=6\u03c0r\u03b7v<\/span><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"os-key-concepts-container\">\n<div class=\"textbox\">\n<h3><span class=\"os-text\">Summary<\/span><\/h3>\n<div class=\"os-key-concepts\">\n<div class=\"os-section-area\">\n<section id=\"fs-id1165036775374\" class=\"key-concepts\">\n<h4 id=\"59779_copy_1\"><span class=\"os-number\">6.1<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Solving Problems with Newton\u2019s Laws<\/span><\/h4>\n<ul id=\"fs-id1165036852024\">\n<li>Newton\u2019s laws of motion can be applied in numerous situations to solve motion problems.<\/li>\n<li>Some problems contain multiple force vectors acting in different directions on an object. Be sure to draw diagrams, resolve all force vectors into horizontal and vertical components, and draw a free-body diagram. Always analyze the direction in which an object accelerates so that you can determine whether\u00a0<span id=\"MathJax-Element-1896-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41027\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41028\" class=\"mrow\"><span id=\"MathJax-Span-41029\" class=\"semantics\"><span id=\"MathJax-Span-41030\" class=\"mrow\"><span id=\"MathJax-Span-41031\" class=\"mrow\"><span id=\"MathJax-Span-41032\" class=\"msub\"><span id=\"MathJax-Span-41033\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41034\" class=\"mrow\"><span id=\"MathJax-Span-41035\" class=\"mtext\">net<\/span><\/span><\/span><span id=\"MathJax-Span-41036\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41037\" class=\"mi\">m<\/span><span id=\"MathJax-Span-41038\" class=\"mi\">a<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fnet=ma<\/span><\/span>\u00a0or\u00a0<span id=\"MathJax-Element-1897-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41039\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41040\" class=\"mrow\"><span id=\"MathJax-Span-41041\" class=\"semantics\"><span id=\"MathJax-Span-41042\" class=\"mrow\"><span id=\"MathJax-Span-41043\" class=\"mrow\"><span id=\"MathJax-Span-41044\" class=\"msub\"><span id=\"MathJax-Span-41045\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41046\" class=\"mrow\"><span id=\"MathJax-Span-41047\" class=\"mtext\">net<\/span><\/span><\/span><span id=\"MathJax-Span-41048\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41049\" class=\"mn\">0<\/span><span id=\"MathJax-Span-41050\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fnet=0.<\/span><\/span><\/li>\n<li>The normal force on an object is not always equal in magnitude to the weight of the object. If an object is accelerating vertically, the normal force is less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force is always less than the full weight of the object.<\/li>\n<li>Some problems contain several physical quantities, such as forces, acceleration, velocity, or position. You can apply concepts from kinematics and dynamics to solve these problems.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<div class=\"os-section-area\">\n<section id=\"fs-id1165037218490\" class=\"key-concepts\">\n<h4 id=\"11340_copy_1\"><span class=\"os-number\">6.2<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Friction<\/span><\/h4>\n<ul id=\"fs-id1165038192411\">\n<li>Friction is a contact force that opposes the motion or attempted motion between two systems. Simple friction is proportional to the normal force\u00a0<em>N<\/em>\u00a0supporting the two systems.<\/li>\n<li>The magnitude of static friction force between two materials stationary relative to each other is determined using the coefficient of static friction, which depends on both materials.<\/li>\n<li>The kinetic friction force between two materials moving relative to each other is determined using the coefficient of kinetic friction, which also depends on both materials and is always less than the coefficient of static friction.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<div class=\"os-section-area\">\n<section id=\"fs-id1165039291121\" class=\"key-concepts\">\n<h4 id=\"71765_copy_1\"><span class=\"os-number\">6.3<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Centripetal Force<\/span><\/h4>\n<ul id=\"fs-id1165038989422\">\n<li>Centripetal force\u00a0<span id=\"MathJax-Element-1898-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41051\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41052\" class=\"mrow\"><span id=\"MathJax-Span-41053\" class=\"semantics\"><span id=\"MathJax-Span-41054\" class=\"mrow\"><span id=\"MathJax-Span-41055\" class=\"mrow\"><span id=\"MathJax-Span-41056\" class=\"msub\"><span id=\"MathJax-Span-41057\" class=\"mstyle\"><span id=\"MathJax-Span-41058\" class=\"mrow\"><span id=\"MathJax-Span-41059\" class=\"mover\"><span id=\"MathJax-Span-41060\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41061\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41062\" class=\"mtext\">c<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192c<\/span><\/span>\u00a0is a \u201ccenter-seeking\u201d force that always points toward the center of rotation. It is perpendicular to linear velocity and has the magnitude\n<div id=\"40902\"><\/div>\n<div id=\"fs-id1165035627257\" class=\"unnumbered\">\n<div class=\"MathJax_Display\"><span id=\"MathJax-Element-1899-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41063\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41064\" class=\"mrow\"><span id=\"MathJax-Span-41065\" class=\"semantics\"><span id=\"MathJax-Span-41066\" class=\"mrow\"><span id=\"MathJax-Span-41067\" class=\"mrow\"><span id=\"MathJax-Span-41068\" class=\"msub\"><span id=\"MathJax-Span-41069\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41070\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-41071\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41072\" class=\"mi\">m<\/span><span id=\"MathJax-Span-41073\" class=\"msub\"><span id=\"MathJax-Span-41074\" class=\"mi\">a<\/span><span id=\"MathJax-Span-41075\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-41076\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML MJX_Assistive_MathML_Block\" role=\"presentation\">Fc=mac.<\/span><\/span><\/div>\n<\/div>\n<\/li>\n<li>Rotating and accelerated frames of reference are noninertial. Inertial forces, such as the Coriolis force, are needed to explain motion in such frames.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<div class=\"os-section-area\">\n<section id=\"fs-id1165039277022\" class=\"key-concepts\">\n<h4 id=\"47362_copy_1\"><span class=\"os-number\">6.4<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Drag Force and Terminal Speed<\/span><\/h4>\n<ul id=\"fs-id1165035867590\">\n<li>Drag forces acting on an object moving in a fluid oppose the motion. For larger objects (such as a baseball) moving at a velocity in air, the drag force is determined using the drag coefficient (typical values are given in\u00a0<a class=\"autogenerated-content\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:36a2674d-4cdb-4dfd-a2b0-5c63cc5a969b@5#fs-id1165035723394\">Table 6.2<\/a>), the area of the object facing the fluid, and the fluid density.<\/li>\n<li>For small objects (such as a bacterium) moving in a denser medium (such as water), the drag force is given by Stokes\u2019 law.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"os-review-conceptual-questions-container\">\n<div class=\"textbox learning-objectives\">\n<h3><span class=\"os-text\">Conceptual Questions<\/span><\/h3>\n<div class=\"os-review-conceptual-questions\">\n<div class=\"os-section-area\">\n<section id=\"fs-id1165036735565\" class=\"review-conceptual-questions\">\n<h4 id=\"59779_copy_2\"><span class=\"os-number\">6.1<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Solving Problems with Newton\u2019s Laws<\/span><\/h4>\n<div id=\"fs-id1165036983215\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165036983217\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036983215-solution\">1<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036983219\">To simulate the apparent weightlessness of space orbit, astronauts are trained in the hold of a cargo aircraft that is accelerating downward at\u00a0<em>g<\/em>. Why do they appear to be weightless, as measured by standing on a bathroom scale, in this accelerated frame of reference? Is there any difference between their apparent weightlessness in orbit and in the aircraft?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"os-section-area\">\n<section id=\"fs-id1165037183984\" class=\"review-conceptual-questions\">\n<h4 id=\"11340_copy_2\"><span class=\"os-number\">6.2<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Friction<\/span><\/h4>\n<div id=\"fs-id1165037088661\" class=\"\">\n<section>\n<div id=\"fs-id1165038383771\"><span class=\"os-number\">2<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038383773\">The glue on a piece of tape can exert forces. Can these forces be a type of simple friction? Explain, considering especially that tape can stick to vertical walls and even to ceilings.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038006273\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037982195\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038006273-solution\">3<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037982197\">When you learn to drive, you discover that you need to let up slightly on the brake pedal as you come to a stop or the car will stop with a jerk. Explain this in terms of the relationship between static and kinetic friction.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038331619\" class=\"\">\n<section>\n<div id=\"fs-id1165038331621\"><span class=\"os-number\">4<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036884400\">When you push a piece of chalk across a chalkboard, it sometimes screeches because it rapidly alternates between slipping and sticking to the board. Describe this process in more detail, in particular, explaining how it is related to the fact that kinetic friction is less than static friction. (The same slip-grab process occurs when tires screech on pavement.)<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038360288\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165038360290\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038360288-solution\">5<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037062457\">A physics major is cooking breakfast when she notices that the frictional force between her steel spatula and Teflon frying pan is only 0.200 N. Knowing the coefficient of kinetic friction between the two materials, she quickly calculates the normal force. What is it?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"os-section-area\">\n<section id=\"fs-id1165039026731\" class=\"review-conceptual-questions\">\n<h4 id=\"71765_copy_2\"><span class=\"os-number\">6.3<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Centripetal Force<\/span><\/h4>\n<div id=\"fs-id1165039075190\" class=\"\">\n<section>\n<div id=\"fs-id1165035760150\"><span class=\"os-number\">6<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035718736\">If you wish to reduce the stress (which is related to centripetal force) on high-speed tires, would you use large- or small-diameter tires? Explain.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039326296\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035715382\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039326296-solution\">7<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038959750\">Define centripetal force. Can any type of force (for example, tension, gravitational force, friction, and so on) be a centripetal force? Can any combination of forces be a centripetal force?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039510688\" class=\"\">\n<section>\n<div id=\"fs-id1165038990778\"><span class=\"os-number\">8<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039496835\">If centripetal force is directed toward the center, why do you feel that you are \u2018thrown\u2019 away from the center as a car goes around a curve? Explain.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039003212\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039384949\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039003212-solution\">9<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039113906\">Race car drivers routinely cut corners, as shown below (Path 2). Explain how this allows the curve to be taken at the greatest speed.<\/p>\n<p><span id=\"fs-id1165039497191\"><img decoding=\"async\" id=\"69252\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/41b71f50ff2da85e0b90b8a9acf6f61e6febe734\" alt=\"Two paths are shown inside a race track through a ninety degree curve. Two cars, a red and a blue one,  and their paths of travel are shown. The blue car is making a tight turn on path one, which is the inside path along the track. The red car is shown overtaking the first car, while taking a wider turn and crossing in front of the blue car into the inside path and then back out of it.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039393190\" class=\"\">\n<section>\n<div id=\"fs-id1165039073338\"><span class=\"os-number\">10<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035726450\">Many amusement parks have rides that make vertical loops like the one shown below. For safety, the cars are attached to the rails in such a way that they cannot fall off. If the car goes over the top at just the right speed, gravity alone will supply the centripetal force. What other force acts and what is its direction if:<\/p>\n<p id=\"fs-id1165039284952\">(a) The car goes over the top at faster than this speed?<\/p>\n<p id=\"fs-id1165039000642\">(b) The car goes over the top at slower than this speed?<\/p>\n<p><span id=\"fs-id1165039287645\"><img decoding=\"async\" id=\"79571\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/2cd9784e5ca0ebf1365ff2205c09c89539af5f76\" alt=\"A photo of a roller coaster with a vertical loop. The loop has a tighter curvature at the top than at the bottom, making an inverted teardrop shape.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039264324\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039066680\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039264324-solution\">11<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035685329\">What causes water to be removed from clothes in a spin-dryer?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039302968\" class=\"\">\n<section>\n<div id=\"fs-id1165039458614\"><span class=\"os-number\">12<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035703891\">As a skater forms a circle, what force is responsible for making his turn? Use a free-body diagram in your answer.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039099187\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039108353\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039099187-solution\">13<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039483406\">Suppose a child is riding on a merry-go-round at a distance about halfway between its center and edge. She has a lunch box resting on wax paper, so that there is very little friction between it and the merry-go-round. Which path shown below will the lunch box take when she lets go? The lunch box leaves a trail in the dust on the merry-go-round. Is that trail straight, curved to the left, or curved to the right? Explain your answer.<\/p>\n<p><span id=\"fs-id1165039286742\"><img decoding=\"async\" id=\"98583\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/12198ae05fdad562467c8f2e7576986bd851dc72\" alt=\"An illustration of the circular base of a merry-go-round with a single horse and child on it. The angular velocity, omega, is clockwise, shown here with an arrow. A point P is shown near the horse, on a circle concentric with the merry-go-round. Three arrows are shown coming out of point P, depicting the three possible path of the lunch box. Path A curves into the circle, to the right from the perspective of the box. Path B is straight, tangent to the circle. Path C curves to the left from the perspective of the box, out of the circle.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039122580\" class=\"\">\n<section>\n<div id=\"fs-id1165035708917\"><span class=\"os-number\">14<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039376236\">Do you feel yourself thrown to either side when you negotiate a curve that is ideally banked for your car\u2019s speed? What is the direction of the force exerted on you by the car seat?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035731478\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039297560\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035731478-solution\">15<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039098694\">Suppose a mass is moving in a circular path on a frictionless table as shown below. In Earth\u2019s frame of reference, there is no centrifugal force pulling the mass away from the center of rotation, yet there is a force stretching the string attaching the mass to the nail. Using concepts related to centripetal force and Newton\u2019s third law, explain what force stretches the string, identifying its physical origin.<\/p>\n<p><span id=\"fs-id1165039121255\"><img decoding=\"async\" id=\"27241\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/5965080d38ed2d7862b7dcc73e44dc80aa0950df\" alt=\"An illustration of a mass moving in a circular path on a table. The mass is attached to a string that is pinned at the center of the circle to the table at the other end.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039064683\" class=\"\">\n<section>\n<div id=\"fs-id1165039419790\"><span class=\"os-number\">16<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036155941\">When a toilet is flushed or a sink is drained, the water (and other material) begins to rotate about the drain on the way down. Assuming no initial rotation and a flow initially directly straight toward the drain, explain what causes the rotation and which direction it has in the Northern Hemisphere. (Note that this is a small effect and in most toilets the rotation is caused by directional water jets.) Would the direction of rotation reverse if water were forced up the drain?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039347286\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039021031\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039347286-solution\">17<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039264802\">A car rounds a curve and encounters a patch of ice with a very low coefficient of kinetic fiction. The car slides off the road. Describe the path of the car as it leaves the road.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039315339\" class=\"\">\n<section>\n<div id=\"fs-id1165039293336\"><span class=\"os-number\">18<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039376622\">In one amusement park ride, riders enter a large vertical barrel and stand against the wall on its horizontal floor. The barrel is spun up and the floor drops away. Riders feel as if they are pinned to the wall by a force something like the gravitational force. This is an inertial force sensed and used by the riders to explain events in the rotating frame of reference of the barrel. Explain in an inertial frame of reference (Earth is nearly one) what pins the riders to the wall, and identify all forces acting on them.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039098388\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039087924\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039098388-solution\">19<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038974400\">Two friends are having a conversation. Anna says a satellite in orbit is in free fall because the satellite keeps falling toward Earth. Tom says a satellite in orbit is not in free fall because the acceleration due to gravity is not\u00a0<span id=\"MathJax-Element-1900-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41077\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41078\" class=\"mrow\"><span id=\"MathJax-Span-41079\" class=\"semantics\"><span id=\"MathJax-Span-41080\" class=\"mrow\"><span id=\"MathJax-Span-41081\" class=\"mrow\"><span id=\"MathJax-Span-41082\" class=\"mn\">9.80<\/span><span id=\"MathJax-Span-41083\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41084\" class=\"msup\"><span id=\"MathJax-Span-41085\" class=\"mrow\"><span id=\"MathJax-Span-41086\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41087\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">9.80m\/s2<\/span><\/span>. Who do you agree with and why?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039224165\" class=\"\">\n<section>\n<div id=\"fs-id1165035756498\"><span class=\"os-number\">20<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039027867\">A nonrotating frame of reference placed at the center of the Sun is very nearly an inertial one. Why is it not exactly an inertial frame?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"os-section-area\">\n<section id=\"fs-id1165035772600\" class=\"review-conceptual-questions\">\n<h4 id=\"47362_copy_2\"><span class=\"os-number\">6.4<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Drag Force and Terminal Speed<\/span><\/h4>\n<div id=\"fs-id1165036157277\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165038999340\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036157277-solution\">21<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036144143\">Athletes such as swimmers and bicyclists wear body suits in competition. Formulate a list of pros and cons of such suits.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039276994\" class=\"\">\n<section>\n<div id=\"fs-id1165039512659\"><span class=\"os-number\">22<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039079000\">Two expressions were used for the drag force experienced by a moving object in a liquid. One depended upon the speed, while the other was proportional to the square of the speed. In which types of motion would each of these expressions be more applicable than the other one?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036143556\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165036148102\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036143556-solution\">23<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035639634\">As cars travel, oil and gasoline leaks onto the road surface. If a light rain falls, what does this do to the control of the car? Does a heavy rain make any difference?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035977406\" class=\"\">\n<section>\n<div id=\"fs-id1165039391020\"><span class=\"os-number\">24<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039291834\">Why can a squirrel jump from a tree branch to the ground and run away undamaged, while a human could break a bone in such a fall?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"os-review-problems-container\">\n<div class=\"textbox exercises\">\n<div class=\"os-review-problems-container\">\n<h3><span class=\"os-text\">Problems<\/span><\/h3>\n<div class=\"os-review-problems\">\n<div class=\"os-section-area\">\n<section id=\"fs-id1165036776336\" class=\"review-problems\">\n<h4 id=\"59779_copy_3\"><span class=\"os-number\">6.1<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Solving Problems with Newton\u2019s Laws<\/span><\/h4>\n<div id=\"fs-id1165037186198\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037186200\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037186198-solution\">25<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037186202\">A 30.0-kg girl in a swing is pushed to one side and held at rest by a horizontal force\u00a0<span id=\"MathJax-Element-1901-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41088\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41089\" class=\"mrow\"><span id=\"MathJax-Span-41090\" class=\"semantics\"><span id=\"MathJax-Span-41091\" class=\"mrow\"><span id=\"MathJax-Span-41092\" class=\"mstyle\"><span id=\"MathJax-Span-41093\" class=\"mrow\"><span id=\"MathJax-Span-41094\" class=\"mover\"><span id=\"MathJax-Span-41095\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41096\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192<\/span><\/span>\u00a0so that the swing ropes are\u00a0<span id=\"MathJax-Element-1902-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41097\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41098\" class=\"mrow\"><span id=\"MathJax-Span-41099\" class=\"semantics\"><span id=\"MathJax-Span-41100\" class=\"mrow\"><span id=\"MathJax-Span-41101\" class=\"mrow\"><span id=\"MathJax-Span-41102\" class=\"mn\">30.0<\/span><span id=\"MathJax-Span-41103\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">30.0\u00b0<\/span><\/span>with respect to the vertical. (a) Calculate the tension in each of the two ropes supporting the swing under these conditions. (b) Calculate the magnitude of\u00a0<span id=\"MathJax-Element-1903-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41104\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41105\" class=\"mrow\"><span id=\"MathJax-Span-41106\" class=\"semantics\"><span id=\"MathJax-Span-41107\" class=\"mrow\"><span id=\"MathJax-Span-41108\" class=\"mrow\"><span id=\"MathJax-Span-41109\" class=\"mstyle\"><span id=\"MathJax-Span-41110\" class=\"mrow\"><span id=\"MathJax-Span-41111\" class=\"mover\"><span id=\"MathJax-Span-41112\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41113\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41114\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192.<\/span><\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037155160\" class=\"\">\n<section>\n<div id=\"fs-id1165037155163\"><span class=\"os-number\">26<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037155165\">Find the tension in each of the three cables supporting the traffic light if it weighs 2.00 \u00d7 10<sup>2<\/sup>\u00a0N.<\/p>\n<p><span id=\"fs-id1165037155169\"><img decoding=\"async\" id=\"7843\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/6712294d8942dbcf6876aea4500ec11115ca2cc9\" alt=\"A sketch of a traffic light suspended by a cable that is in turn suspended from two other cables is shown. Tension T sub 3 is the tension in the cable connecting the traffic light to the upper cables. Tension T sub one is the tension in the upper cable pulling up and to the left, making a 41 degree angle with the horizontal. Tension T sub two is the tension pulling up and to the right, making a 63 degree angle with the horizontal. Force vector w equal to 200 Newtons pulls vertically downward on the traffic light.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037151556\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037151559\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037151556-solution\">27<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037151561\">Three forces act on an object, considered to be a particle, which moves with constant velocity\u00a0<span id=\"MathJax-Element-1904-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41115\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41116\" class=\"mrow\"><span id=\"MathJax-Span-41117\" class=\"semantics\"><span id=\"MathJax-Span-41118\" class=\"mrow\"><span id=\"MathJax-Span-41119\" class=\"mrow\"><span id=\"MathJax-Span-41120\" class=\"mi\">v<\/span><span id=\"MathJax-Span-41121\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41122\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41123\" class=\"mn\">3<\/span><span id=\"MathJax-Span-41124\" class=\"mstyle\"><span id=\"MathJax-Span-41125\" class=\"mrow\"><span id=\"MathJax-Span-41126\" class=\"mover\"><span id=\"MathJax-Span-41127\" class=\"mi\">i<\/span><span id=\"MathJax-Span-41128\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41129\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-41130\" class=\"mn\">2<\/span><span id=\"MathJax-Span-41131\" class=\"mstyle\"><span id=\"MathJax-Span-41132\" class=\"mrow\"><span id=\"MathJax-Span-41133\" class=\"mover\"><span id=\"MathJax-Span-41134\" class=\"mi\">j<\/span><span id=\"MathJax-Span-41135\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41136\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41137\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41138\" class=\"mtext\">m\/s<\/span><span id=\"MathJax-Span-41139\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v=(3i^\u22122j^)m\/s.<\/span><\/span>\u00a0Two of the forces are\u00a0<span id=\"MathJax-Element-1905-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41140\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41141\" class=\"mrow\"><span id=\"MathJax-Span-41142\" class=\"semantics\"><span id=\"MathJax-Span-41143\" class=\"mrow\"><span id=\"MathJax-Span-41144\" class=\"mrow\"><span id=\"MathJax-Span-41145\" class=\"msub\"><span id=\"MathJax-Span-41146\" class=\"mstyle\"><span id=\"MathJax-Span-41147\" class=\"mrow\"><span id=\"MathJax-Span-41148\" class=\"mover\"><span id=\"MathJax-Span-41149\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41150\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41151\" class=\"mn\">1<\/span><\/span><span id=\"MathJax-Span-41152\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41153\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41154\" class=\"mn\">3<\/span><span id=\"MathJax-Span-41155\" class=\"mstyle\"><span id=\"MathJax-Span-41156\" class=\"mrow\"><span id=\"MathJax-Span-41157\" class=\"mover\"><span id=\"MathJax-Span-41158\" class=\"mi\">i<\/span><span id=\"MathJax-Span-41159\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41160\" class=\"mo\">+<\/span><span id=\"MathJax-Span-41161\" class=\"mn\">5<\/span><span id=\"MathJax-Span-41162\" class=\"mstyle\"><span id=\"MathJax-Span-41163\" class=\"mrow\"><span id=\"MathJax-Span-41164\" class=\"mover\"><span id=\"MathJax-Span-41165\" class=\"mi\">j<\/span><span id=\"MathJax-Span-41166\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41167\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-41168\" class=\"mn\">6<\/span><span id=\"MathJax-Span-41169\" class=\"mstyle\"><span id=\"MathJax-Span-41170\" class=\"mrow\"><span id=\"MathJax-Span-41171\" class=\"mover\"><span id=\"MathJax-Span-41172\" class=\"mi\">k<\/span><span id=\"MathJax-Span-41173\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41174\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41175\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41176\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u21921=(3i^+5j^\u22126k^)N<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1906-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41177\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41178\" class=\"mrow\"><span id=\"MathJax-Span-41179\" class=\"semantics\"><span id=\"MathJax-Span-41180\" class=\"mrow\"><span id=\"MathJax-Span-41181\" class=\"mrow\"><span id=\"MathJax-Span-41182\" class=\"msub\"><span id=\"MathJax-Span-41183\" class=\"mstyle\"><span id=\"MathJax-Span-41184\" class=\"mrow\"><span id=\"MathJax-Span-41185\" class=\"mover\"><span id=\"MathJax-Span-41186\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41187\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41188\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-41189\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41190\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41191\" class=\"mn\">4<\/span><span id=\"MathJax-Span-41192\" class=\"mstyle\"><span id=\"MathJax-Span-41193\" class=\"mrow\"><span id=\"MathJax-Span-41194\" class=\"mover\"><span id=\"MathJax-Span-41195\" class=\"mi\">i<\/span><span id=\"MathJax-Span-41196\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41197\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-41198\" class=\"mn\">7<\/span><span id=\"MathJax-Span-41199\" class=\"mstyle\"><span id=\"MathJax-Span-41200\" class=\"mrow\"><span id=\"MathJax-Span-41201\" class=\"mover\"><span id=\"MathJax-Span-41202\" class=\"mi\">j<\/span><span id=\"MathJax-Span-41203\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41204\" class=\"mo\">+<\/span><span id=\"MathJax-Span-41205\" class=\"mn\">2<\/span><span id=\"MathJax-Span-41206\" class=\"mstyle\"><span id=\"MathJax-Span-41207\" class=\"mrow\"><span id=\"MathJax-Span-41208\" class=\"mover\"><span id=\"MathJax-Span-41209\" class=\"mi\">k<\/span><span id=\"MathJax-Span-41210\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-41211\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41212\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41213\" class=\"mtext\">N<\/span><span id=\"MathJax-Span-41214\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u21922=(4i^\u22127j^+2k^)N.<\/span><\/span>\u00a0Find the third force.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037154683\" class=\"\">\n<section>\n<div id=\"fs-id1165037154685\"><span class=\"os-number\">28<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037154687\">A flea jumps by exerting a force of\u00a0<span id=\"MathJax-Element-1907-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41215\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41216\" class=\"mrow\"><span id=\"MathJax-Span-41217\" class=\"semantics\"><span id=\"MathJax-Span-41218\" class=\"mrow\"><span id=\"MathJax-Span-41219\" class=\"mrow\"><span id=\"MathJax-Span-41220\" class=\"mn\">1.20<\/span><span id=\"MathJax-Span-41221\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41222\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41223\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41224\" class=\"msup\"><span id=\"MathJax-Span-41225\" class=\"mrow\"><span id=\"MathJax-Span-41226\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41227\" class=\"mrow\"><span id=\"MathJax-Span-41228\" class=\"mn\">\u22125<\/span><\/span><\/span><span id=\"MathJax-Span-41229\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41230\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.20\u00d710\u22125N<\/span><\/span>\u00a0straight down on the ground. A breeze blowing on the flea parallel to the ground exerts a force of\u00a0<span id=\"MathJax-Element-1908-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41231\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41232\" class=\"mrow\"><span id=\"MathJax-Span-41233\" class=\"semantics\"><span id=\"MathJax-Span-41234\" class=\"mrow\"><span id=\"MathJax-Span-41235\" class=\"mrow\"><span id=\"MathJax-Span-41236\" class=\"mn\">0.500<\/span><span id=\"MathJax-Span-41237\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41238\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41239\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41240\" class=\"msup\"><span id=\"MathJax-Span-41241\" class=\"mrow\"><span id=\"MathJax-Span-41242\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41243\" class=\"mrow\"><span id=\"MathJax-Span-41244\" class=\"mn\">\u22126<\/span><\/span><\/span><span id=\"MathJax-Span-41245\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41246\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.500\u00d710\u22126N<\/span><\/span>\u00a0on the flea while the flea is still in contact with the ground. Find the direction and magnitude of the acceleration of the flea if its mass is\u00a0<span id=\"MathJax-Element-1909-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41247\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41248\" class=\"mrow\"><span id=\"MathJax-Span-41249\" class=\"semantics\"><span id=\"MathJax-Span-41250\" class=\"mrow\"><span id=\"MathJax-Span-41251\" class=\"mrow\"><span id=\"MathJax-Span-41252\" class=\"mn\">6.00<\/span><span id=\"MathJax-Span-41253\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41254\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41255\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41256\" class=\"msup\"><span id=\"MathJax-Span-41257\" class=\"mrow\"><span id=\"MathJax-Span-41258\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41259\" class=\"mrow\"><span id=\"MathJax-Span-41260\" class=\"mn\">\u22127<\/span><\/span><\/span><span id=\"MathJax-Span-41261\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41262\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">6.00\u00d710\u22127kg<\/span><\/span>. Do not neglect the gravitational force.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036901084\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165036901086\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036901084-solution\">29<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036901088\">Two muscles in the back of the leg pull upward on the Achilles tendon, as shown below. (These muscles are called the medial and lateral heads of the gastrocnemius muscle.) Find the magnitude and direction of the total force on the Achilles tendon. What type of movement could be caused by this force?<\/p>\n<p><span id=\"fs-id1165037262483\"><img decoding=\"async\" id=\"64788\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/9c6d40ff32e9ba34cbb98d3100d0d990d42af261\" alt=\"An Achilles tendon is shown in the figure with two forces exerted on it by the lateral and medial heads of the gastrocnemius muscle. F sub one, equal to two hundred Newtons, is shown as a vector making an angle twenty degrees to the right of vertical, and F sub two, equal to two hundred Newtons, is shown making an angle of twenty degrees left of vertical.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038018567\" class=\"\">\n<section>\n<div id=\"fs-id1165036834090\"><span class=\"os-number\">30<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036834092\">After a mishap, a 76.0-kg circus performer clings to a trapeze, which is being pulled to the side by another circus artist, as shown here. Calculate the tension in the two ropes if the person is momentarily motionless. Include a free-body diagram in your solution.<\/p>\n<p><span id=\"fs-id1165036834098\"><img decoding=\"async\" id=\"52754\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/feaad2318d1eed2fdee8dae38210d0b706a006a1\" alt=\"A circus performer hanging from a trapeze is being pulled to the right by another performer using a rope. Her weight is shown by a vector w acting vertically downward. The trapeze rope exerts a tension, T sub one, up and to the left, making an angle of fifteen degrees with the vertical. The second performer pulls with tension T sub two, making an angle of ten degrees above the positive x direction.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037271925\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037271927\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037271925-solution\">31<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037271929\">A 35.0-kg dolphin decelerates from 12.0 to 7.50 m\/s in 2.30 s to join another dolphin in play. What average force was exerted to slow the first dolphin if it was moving horizontally? (The gravitational force is balanced by the buoyant force of the water.)<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037017931\" class=\"\">\n<section>\n<div id=\"fs-id1165037017933\"><span class=\"os-number\">32<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037017935\">When starting a foot race, a 70.0-kg sprinter exerts an average force of 650 N backward on the ground for 0.800 s. (a) What is his final speed? (b) How far does he travel?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037985556\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037985558\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037985556-solution\">33<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037985560\">A large rocket has a mass of\u00a0<span id=\"MathJax-Element-1910-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41263\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41264\" class=\"mrow\"><span id=\"MathJax-Span-41265\" class=\"semantics\"><span id=\"MathJax-Span-41266\" class=\"mrow\"><span id=\"MathJax-Span-41267\" class=\"mrow\"><span id=\"MathJax-Span-41268\" class=\"mn\">2.00<\/span><span id=\"MathJax-Span-41269\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41270\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41271\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41272\" class=\"msup\"><span id=\"MathJax-Span-41273\" class=\"mrow\"><span id=\"MathJax-Span-41274\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41275\" class=\"mn\">6<\/span><\/span><span id=\"MathJax-Span-41276\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41277\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.00\u00d7106kg<\/span><\/span>\u00a0at takeoff, and its engines produce a thrust of\u00a0<span id=\"MathJax-Element-1911-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41278\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41279\" class=\"mrow\"><span id=\"MathJax-Span-41280\" class=\"semantics\"><span id=\"MathJax-Span-41281\" class=\"mrow\"><span id=\"MathJax-Span-41282\" class=\"mrow\"><span id=\"MathJax-Span-41283\" class=\"mn\">3.50<\/span><span id=\"MathJax-Span-41284\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41285\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41286\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41287\" class=\"msup\"><span id=\"MathJax-Span-41288\" class=\"mrow\"><span id=\"MathJax-Span-41289\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41290\" class=\"mn\">7<\/span><\/span><span id=\"MathJax-Span-41291\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41292\" class=\"mtext\">N<\/span><span id=\"MathJax-Span-41293\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.50\u00d7107N.<\/span><\/span>\u00a0(a) Find its initial acceleration if it takes off vertically. (b) How long does it take to reach a velocity of 120 km\/h straight up, assuming constant mass and thrust?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037221234\" class=\"\">\n<section>\n<div id=\"fs-id1165037221236\"><span class=\"os-number\">34<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037221238\">A basketball player jumps straight up for a ball. To do this, he lowers his body 0.300 m and then accelerates through this distance by forcefully straightening his legs. This player leaves the floor with a vertical velocity sufficient to carry him 0.900 m above the floor. (a) Calculate his velocity when he leaves the floor. (b) Calculate his acceleration while he is straightening his legs. He goes from zero to the velocity found in (a) in a distance of 0.300 m. (c) Calculate the force he exerts on the floor to do this, given that his mass is 110.0 kg.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037223036\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037223038\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037223036-solution\">35<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037223040\">A 2.50-kg fireworks shell is fired straight up from a mortar and reaches a height of 110.0 m. (a) Neglecting air resistance (a poor assumption, but we will make it for this example), calculate the shell\u2019s velocity when it leaves the mortar. (b) The mortar itself is a tube 0.450 m long. Calculate the average acceleration of the shell in the tube as it goes from zero to the velocity found in (a). (c) What is the average force on the shell in the mortar? Express your answer in newtons and as a ratio to the weight of the shell.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037183422\" class=\"\">\n<section>\n<div id=\"fs-id1165037183424\"><span class=\"os-number\">36<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037183426\">A 0.500-kg potato is fired at an angle of\u00a0<span id=\"MathJax-Element-1912-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41294\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41295\" class=\"mrow\"><span id=\"MathJax-Span-41296\" class=\"semantics\"><span id=\"MathJax-Span-41297\" class=\"mrow\"><span id=\"MathJax-Span-41298\" class=\"mrow\"><span id=\"MathJax-Span-41299\" class=\"mn\">80.0<\/span><span id=\"MathJax-Span-41300\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">80.0\u00b0<\/span><\/span>\u00a0above the horizontal from a PVC pipe used as a \u201cpotato gun\u201d and reaches a height of 110.0 m. (a) Neglecting air resistance, calculate the potato\u2019s velocity when it leaves the gun. (b) The gun itself is a tube 0.450 m long. Calculate the average acceleration of the potato in the tube as it goes from zero to the velocity found in (a). (c) What is the average force on the potato in the gun? Express your answer in newtons and as a ratio to the weight of the potato.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036784614\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165036784616\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036784614-solution\">37<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036784618\">An elevator filled with passengers has a mass of\u00a0<span id=\"MathJax-Element-1913-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41301\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41302\" class=\"mrow\"><span id=\"MathJax-Span-41303\" class=\"semantics\"><span id=\"MathJax-Span-41304\" class=\"mrow\"><span id=\"MathJax-Span-41305\" class=\"mrow\"><span id=\"MathJax-Span-41306\" class=\"mn\">1.70<\/span><span id=\"MathJax-Span-41307\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41308\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41309\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41310\" class=\"msup\"><span id=\"MathJax-Span-41311\" class=\"mrow\"><span id=\"MathJax-Span-41312\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41313\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41314\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41315\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.70\u00d7103kg<\/span><\/span>. (a) The elevator accelerates upward from rest at a rate of\u00a0<span id=\"MathJax-Element-1914-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41316\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41317\" class=\"mrow\"><span id=\"MathJax-Span-41318\" class=\"semantics\"><span id=\"MathJax-Span-41319\" class=\"mrow\"><span id=\"MathJax-Span-41320\" class=\"mrow\"><span id=\"MathJax-Span-41321\" class=\"mn\">1.20<\/span><span id=\"MathJax-Span-41322\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41323\" class=\"msup\"><span id=\"MathJax-Span-41324\" class=\"mrow\"><span id=\"MathJax-Span-41325\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41326\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.20m\/s2<\/span><\/span>\u00a0for 1.50 s. Calculate the tension in the cable supporting the elevator. (b) The elevator continues upward at constant velocity for 8.50 s. What is the tension in the cable during this time? (c) The elevator decelerates at a rate of\u00a0<span id=\"MathJax-Element-1915-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41327\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41328\" class=\"mrow\"><span id=\"MathJax-Span-41329\" class=\"semantics\"><span id=\"MathJax-Span-41330\" class=\"mrow\"><span id=\"MathJax-Span-41331\" class=\"mrow\"><span id=\"MathJax-Span-41332\" class=\"mn\">0.600<\/span><span id=\"MathJax-Span-41333\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41334\" class=\"msup\"><span id=\"MathJax-Span-41335\" class=\"mrow\"><span id=\"MathJax-Span-41336\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41337\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.600m\/s2<\/span><\/span>\u00a0for 3.00 s. What is the tension in the cable during deceleration? (d) How high has the elevator moved above its original starting point, and what is its final velocity?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038203868\" class=\"\">\n<section>\n<div id=\"fs-id1165038203870\"><span class=\"os-number\">38<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038203872\">A 20.0-g ball hangs from the roof of a freight car by a string. When the freight car begins to move, the string makes an angle of\u00a0<span id=\"MathJax-Element-1916-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41338\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41339\" class=\"mrow\"><span id=\"MathJax-Span-41340\" class=\"semantics\"><span id=\"MathJax-Span-41341\" class=\"mrow\"><span id=\"MathJax-Span-41342\" class=\"mrow\"><span id=\"MathJax-Span-41343\" class=\"mn\">35.0<\/span><span id=\"MathJax-Span-41344\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">35.0\u00b0<\/span><\/span>\u00a0with the vertical. (a) What is the acceleration of the freight car? (b) What is the tension in the string?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036887553\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165036887555\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036887553-solution\">39<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036887557\">A student\u2019s backpack, full of textbooks, is hung from a spring scale attached to the ceiling of an elevator. When the elevator is accelerating downward at\u00a0<span id=\"MathJax-Element-1917-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41345\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41346\" class=\"mrow\"><span id=\"MathJax-Span-41347\" class=\"semantics\"><span id=\"MathJax-Span-41348\" class=\"mrow\"><span id=\"MathJax-Span-41349\" class=\"mrow\"><span id=\"MathJax-Span-41350\" class=\"mn\">3.8<\/span><span id=\"MathJax-Span-41351\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41352\" class=\"msup\"><span id=\"MathJax-Span-41353\" class=\"mrow\"><span id=\"MathJax-Span-41354\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41355\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.8m\/s2<\/span><\/span>, the scale reads 60 N. (a) What is the mass of the backpack? (b) What does the scale read if the elevator moves upward while slowing down at a rate\u00a0<span id=\"MathJax-Element-1918-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41356\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41357\" class=\"mrow\"><span id=\"MathJax-Span-41358\" class=\"semantics\"><span id=\"MathJax-Span-41359\" class=\"mrow\"><span id=\"MathJax-Span-41360\" class=\"mrow\"><span id=\"MathJax-Span-41361\" class=\"mn\">3.8<\/span><span id=\"MathJax-Span-41362\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41363\" class=\"msup\"><span id=\"MathJax-Span-41364\" class=\"mrow\"><span id=\"MathJax-Span-41365\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41366\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.8m\/s2<\/span><\/span>? (c) What does the scale read if the elevator moves upward at constant velocity? (d) If the elevator had no brakes and the cable supporting it were to break loose so that the elevator could fall freely, what would the spring scale read?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038183857\" class=\"\">\n<section>\n<div id=\"fs-id1165038183859\"><span class=\"os-number\">40<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038183861\">A service elevator takes a load of garbage, mass 10.0 kg, from a floor of a skyscraper under construction, down to ground level, accelerating downward at a rate of\u00a0<span id=\"MathJax-Element-1919-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41367\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41368\" class=\"mrow\"><span id=\"MathJax-Span-41369\" class=\"semantics\"><span id=\"MathJax-Span-41370\" class=\"mrow\"><span id=\"MathJax-Span-41371\" class=\"mrow\"><span id=\"MathJax-Span-41372\" class=\"mn\">1.2<\/span><span id=\"MathJax-Span-41373\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41374\" class=\"msup\"><span id=\"MathJax-Span-41375\" class=\"mrow\"><span id=\"MathJax-Span-41376\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-41377\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.2m\/s2<\/span><\/span>. Find the magnitude of the force the garbage exerts on the floor of the service elevator?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037265785\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037265787\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037265785-solution\">41<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037265789\">A roller coaster car starts from rest at the top of a track 30.0 m long and inclined at\u00a0<span id=\"MathJax-Element-1920-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41378\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41379\" class=\"mrow\"><span id=\"MathJax-Span-41380\" class=\"semantics\"><span id=\"MathJax-Span-41381\" class=\"mrow\"><span id=\"MathJax-Span-41382\" class=\"mrow\"><span id=\"MathJax-Span-41383\" class=\"mn\">20.0<\/span><span id=\"MathJax-Span-41384\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">20.0\u00b0<\/span><\/span>\u00a0to the horizontal. Assume that friction can be ignored. (a) What is the acceleration of the car? (b) How much time elapses before it reaches the bottom of the track?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036842411\" class=\"\">\n<section>\n<div id=\"fs-id1165036842413\"><span class=\"os-number\">42<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036842415\">The device shown below is the Atwood\u2019s machine considered in\u00a0<a class=\"autogenerated-content\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e811c980-cf62-4f66-8fef-c399eb88dd2a@5#fs-id1165037923338\">Example 6.5<\/a>. Assuming that the masses of the string and the frictionless pulley are negligible, (a) find an equation for the acceleration of the two blocks; (b) find an equation for the tension in the string; and (c) find both the acceleration and tension when block 1 has mass 2.00 kg and block 2 has mass 4.00 kg.<\/p>\n<p><span id=\"fs-id1165037265194\"><img decoding=\"async\" id=\"30115\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/6ae358105f6272d7737ebb4e55bf3014cc04b8bd\" alt=\"An Atwood machine consisting of masses suspended on either side of a pulley by a string passing over the pulley is shown. Mass m sub 1 is on the left and mass m sub 2 is on the right.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036779432\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165036779434\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036779432-solution\">43<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036779436\">Two blocks are connected by a massless rope as shown below. The mass of the block on the table is 4.0 kg and the hanging mass is 1.0 kg. The table and the pulley are frictionless. (a) Find the acceleration of the system. (b) Find the tension in the rope. (c) Find the speed with which the hanging mass hits the floor if it starts from rest and is initially located 1.0 m from the floor.<\/p>\n<p><span id=\"fs-id1165036963759\"><img decoding=\"async\" id=\"11684\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/282733f27565e7ecb908869cf2f96f5759d35fe0\" alt=\"Block m sub 1 is on a horizontal table. It is connected to a string that passes over a pulley at the edge of the table. The string then hangs straight down and connects to block m sub 2, which is not in contact with the table. Block m sub 1 has acceleration a sub 1 directed to the right. Block m sub 2 has acceleration a sub 2 directed downward.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038010136\" class=\"\">\n<section>\n<div id=\"fs-id1165038010139\"><span class=\"os-number\">44<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038010141\">Shown below are two carts connected by a cord that passes over a small frictionless pulley. Each cart rolls freely with negligible friction. Calculate the acceleration of the carts and the tension in the cord.<\/p>\n<p><span id=\"fs-id1165038010146\"><img decoding=\"async\" id=\"57774\" src=\"https:\/\/cnx.org\/resources\/0e76b273a6a0729619717c847a9f7a6f50bebfdf\" alt=\"Two carts connected by a string passing over a pulley are on either side of a double inclined plane. The string passes over a pulley attached to the top of the double incline. On the left, the incline makes an angle of 37 degrees with the horizontal and the cart on that side has mass 10 kilograms. On the right, the incline makes an angle of 53 degrees with the horizontal and the cart on that side has mass 15 kilograms.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037268006\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037063523\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037268006-solution\">45<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037063525\">A 2.00 kg block (mass 1) and a 4.00 kg block (mass 2) are connected by a light string as shown; the inclination of the ramp is\u00a0<span id=\"MathJax-Element-1921-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41385\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41386\" class=\"mrow\"><span id=\"MathJax-Span-41387\" class=\"semantics\"><span id=\"MathJax-Span-41388\" class=\"mrow\"><span id=\"MathJax-Span-41389\" class=\"mrow\"><span id=\"MathJax-Span-41390\" class=\"mn\">40.0<\/span><span id=\"MathJax-Span-41391\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">40.0\u00b0<\/span><\/span>. Friction is negligible. What is (a) the acceleration of each block and (b) the tension in the string?<\/p>\n<p><span id=\"fs-id1165037063537\"><img decoding=\"async\" id=\"59880\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/62dac4eaa2d3fa779fad11d99af3ebfe423000ca\" alt=\"Block 1 is on a ramp inclined up and to the right at an angle of 40 degrees above the horizontal. It is connected to a string that passes over a pulley at the top of the ramp, then hangs straight down and connects to block 2. Block 2 is not in contact with the ramp.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"os-section-area\">\n<section id=\"fs-id1165036748145\" class=\"review-problems\">\n<h4 id=\"11340_copy_3\"><span class=\"os-number\">6.2<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Friction<\/span><\/h4>\n<div id=\"fs-id1165037064164\" class=\"\">\n<section>\n<div id=\"fs-id1165037064166\"><span class=\"os-number\">46<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036783994\">(a) When rebuilding his car\u2019s engine, a physics major must exert\u00a0<span id=\"MathJax-Element-1922-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41392\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41393\" class=\"mrow\"><span id=\"MathJax-Span-41394\" class=\"semantics\"><span id=\"MathJax-Span-41395\" class=\"mrow\"><span id=\"MathJax-Span-41396\" class=\"mrow\"><span id=\"MathJax-Span-41397\" class=\"mn\">3.00<\/span><span id=\"MathJax-Span-41398\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41399\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41400\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41401\" class=\"msup\"><span id=\"MathJax-Span-41402\" class=\"mrow\"><span id=\"MathJax-Span-41403\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41404\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.00\u00d7102<\/span><\/span>\u00a0N of force to insert a dry steel piston into a steel cylinder. What is the normal force between the piston and cylinder? (b) What force would he have to exert if the steel parts were oiled?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037233270\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037233273\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037233270-solution\">47<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037262674\">(a) What is the maximum frictional force in the knee joint of a person who supports 66.0 kg of her mass on that knee? (b) During strenuous exercise, it is possible to exert forces to the joints that are easily 10 times greater than the weight being supported. What is the maximum force of friction under such conditions? The frictional forces in joints are relatively small in all circumstances except when the joints deteriorate, such as from injury or arthritis. Increased frictional forces can cause further damage and pain.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037046771\" class=\"\">\n<section>\n<div id=\"fs-id1165037046773\"><span class=\"os-number\">48<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036846790\">Suppose you have a 120-kg wooden crate resting on a wood floor, with coefficient of static friction 0.500 between these wood surfaces. (a) What maximum force can you exert horizontally on the crate without moving it? (b) If you continue to exert this force once the crate starts to slip, what will its acceleration then be? The coefficient of sliding friction is known to be 0.300 for this situation.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037027752\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165038360212\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037027752-solution\">49<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038360214\">(a) If half of the weight of a small\u00a0<span id=\"MathJax-Element-1923-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41405\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41406\" class=\"mrow\"><span id=\"MathJax-Span-41407\" class=\"semantics\"><span id=\"MathJax-Span-41408\" class=\"mrow\"><span id=\"MathJax-Span-41409\" class=\"mrow\"><span id=\"MathJax-Span-41410\" class=\"mn\">1.00<\/span><span id=\"MathJax-Span-41411\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41412\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41413\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41414\" class=\"msup\"><span id=\"MathJax-Span-41415\" class=\"mrow\"><span id=\"MathJax-Span-41416\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41417\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41418\" class=\"mtext\">-kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.00\u00d7103-kg<\/span><\/span>\u00a0utility truck is supported by its two drive wheels, what is the maximum acceleration it can achieve on dry concrete? (b) Will a metal cabinet lying on the wooden bed of the truck slip if it accelerates at this rate? (c) Solve both problems assuming the truck has four-wheel drive.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037263604\" class=\"\">\n<section>\n<div id=\"fs-id1165038155161\"><span class=\"os-number\">50<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038155163\">A team of eight dogs pulls a sled with waxed wood runners on wet snow (mush!). The dogs have average masses of 19.0 kg, and the loaded sled with its rider has a mass of 210 kg. (a) Calculate the acceleration of the dogs starting from rest if each dog exerts an average force of 185 N backward on the snow. (b) Calculate the force in the coupling between the dogs and the sled.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038313967\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165038313969\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038313967-solution\">51<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036785328\">Consider the 65.0-kg ice skater being pushed by two others shown below. (a) Find the direction and magnitude of\u00a0<span id=\"MathJax-Element-1924-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41419\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41420\" class=\"mrow\"><span id=\"MathJax-Span-41421\" class=\"semantics\"><span id=\"MathJax-Span-41422\" class=\"mrow\"><span id=\"MathJax-Span-41423\" class=\"mrow\"><span id=\"MathJax-Span-41424\" class=\"msub\"><span id=\"MathJax-Span-41425\" class=\"mstyle\"><span id=\"MathJax-Span-41426\" class=\"mrow\"><span id=\"MathJax-Span-41427\" class=\"mi\">F<\/span><\/span><\/span><span id=\"MathJax-Span-41428\" class=\"mrow\"><span id=\"MathJax-Span-41429\" class=\"mtext\">tot<\/span><\/span><\/span><span id=\"MathJax-Span-41430\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Ftot,<\/span><\/span>\u00a0the total force exerted on her by the others, given that the magnitudes\u00a0<span id=\"MathJax-Element-1925-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41431\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41432\" class=\"mrow\"><span id=\"MathJax-Span-41433\" class=\"semantics\"><span id=\"MathJax-Span-41434\" class=\"mrow\"><span id=\"MathJax-Span-41435\" class=\"mrow\"><span id=\"MathJax-Span-41436\" class=\"msub\"><span id=\"MathJax-Span-41437\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41438\" class=\"mn\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F1<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1926-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41439\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41440\" class=\"mrow\"><span id=\"MathJax-Span-41441\" class=\"semantics\"><span id=\"MathJax-Span-41442\" class=\"mrow\"><span id=\"MathJax-Span-41443\" class=\"mrow\"><span id=\"MathJax-Span-41444\" class=\"msub\"><span id=\"MathJax-Span-41445\" class=\"mi\">F<\/span><span id=\"MathJax-Span-41446\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F2<\/span><\/span>\u00a0are 26.4 N and 18.6 N, respectively. (b) What is her initial acceleration if she is initially stationary and wearing steel-bladed skates that point in the direction of\u00a0<span id=\"MathJax-Element-1927-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41447\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41448\" class=\"mrow\"><span id=\"MathJax-Span-41449\" class=\"semantics\"><span id=\"MathJax-Span-41450\" class=\"mrow\"><span id=\"MathJax-Span-41451\" class=\"mrow\"><span id=\"MathJax-Span-41452\" class=\"msub\"><span id=\"MathJax-Span-41453\" class=\"mstyle\"><span id=\"MathJax-Span-41454\" class=\"mrow\"><span id=\"MathJax-Span-41455\" class=\"mi\">F<\/span><\/span><\/span><span id=\"MathJax-Span-41456\" class=\"mrow\"><span id=\"MathJax-Span-41457\" class=\"mtext\">tot<\/span><\/span><\/span><span id=\"MathJax-Span-41458\" class=\"mo\">?<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Ftot?<\/span><\/span>\u00a0(c) What is her acceleration assuming she is already moving in the direction of\u00a0<span id=\"MathJax-Element-1928-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41459\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41460\" class=\"mrow\"><span id=\"MathJax-Span-41461\" class=\"semantics\"><span id=\"MathJax-Span-41462\" class=\"mrow\"><span id=\"MathJax-Span-41463\" class=\"mrow\"><span id=\"MathJax-Span-41464\" class=\"msub\"><span id=\"MathJax-Span-41465\" class=\"mstyle\"><span id=\"MathJax-Span-41466\" class=\"mrow\"><span id=\"MathJax-Span-41467\" class=\"mi\">F<\/span><\/span><\/span><span id=\"MathJax-Span-41468\" class=\"mrow\"><span id=\"MathJax-Span-41469\" class=\"mtext\">tot<\/span><\/span><\/span><span id=\"MathJax-Span-41470\" class=\"mo\">?<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Ftot?<\/span><\/span>\u00a0(Remember that friction always acts in the direction opposite that of motion or attempted motion between surfaces in contact.)<\/p>\n<p><span id=\"fs-id1165037171810\"><img decoding=\"async\" id=\"52665\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/00c2f606486ab5e6defc00a56a9f5c43a94541b4\" alt=\"(a) Overhead view of two ice skaters pushing on a third. One skater pushes with a force F one, represented by an arrow pointing to the right, and a second skater pushes with a force F two, represented by an arrow pointing up. Vector F one and vector F two are along the arms of the two skaters acting on the third skater. A vector diagram is shown in the form of a right triangle in which the base is vector F one pointing to the right, and perpendicular to F one is vector F two pointing up. The resultant vector is shown by the hypotenuse pointing up and to the right and is labeled as vector F sub tot. (b) Free body diagram showing only the forces F sub one and F sub 2 acting on the skater.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037014648\" class=\"\">\n<section>\n<div id=\"fs-id1165036965255\"><span class=\"os-number\">52<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036965257\">Show that the acceleration of any object down a frictionless incline that makes an angle\u00a0<span id=\"MathJax-Element-1929-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41471\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41472\" class=\"mrow\"><span id=\"MathJax-Span-41473\" class=\"semantics\"><span id=\"MathJax-Span-41474\" class=\"mrow\"><span id=\"MathJax-Span-41475\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0with the horizontal is\u00a0<span id=\"MathJax-Element-1930-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41476\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41477\" class=\"mrow\"><span id=\"MathJax-Span-41478\" class=\"semantics\"><span id=\"MathJax-Span-41479\" class=\"mrow\"><span id=\"MathJax-Span-41480\" class=\"mrow\"><span id=\"MathJax-Span-41481\" class=\"mi\">a<\/span><span id=\"MathJax-Span-41482\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41483\" class=\"mi\">g<\/span><span id=\"MathJax-Span-41484\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41485\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-41486\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41487\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">a=gsin\u03b8<\/span><\/span>. (Note that this acceleration is independent of mass.)<\/p>\n<p><span id=\"fs-id1165038017726\"><img decoding=\"async\" id=\"98748\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/76d33d47d237582d2d0c36930d628dab7a69e64d\" alt=\"An illustration of  block on  a slope. The slope angles down and to the right at an angle of theta degrees to the horizontal. The block has an acceleration a parallel to the slope, toward its bottom. The following forces are shown: N perpendicular to the slope and pointing out of it, and w which equals m times g vertically down. An x y coordinate system is shown tilted so that positive x is downslope, parallel to the surface, and positive y is perpendicular to the slope, pointing out of the surface.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038010694\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165038244292\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038010694-solution\">53<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038244294\">Show that the acceleration of any object down an incline where friction behaves simply (that is, where\u00a0<span id=\"MathJax-Element-1931-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41488\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41489\" class=\"mrow\"><span id=\"MathJax-Span-41490\" class=\"semantics\"><span id=\"MathJax-Span-41491\" class=\"mrow\"><span id=\"MathJax-Span-41492\" class=\"mrow\"><span id=\"MathJax-Span-41493\" class=\"msub\"><span id=\"MathJax-Span-41494\" class=\"mi\">f<\/span><span id=\"MathJax-Span-41495\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-41496\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41497\" class=\"msub\"><span id=\"MathJax-Span-41498\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41499\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-41500\" class=\"mi\">N<\/span><span id=\"MathJax-Span-41501\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">fk=\u03bckN)<\/span><\/span>\u00a0is\u00a0<span id=\"MathJax-Element-1932-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41502\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41503\" class=\"mrow\"><span id=\"MathJax-Span-41504\" class=\"semantics\"><span id=\"MathJax-Span-41505\" class=\"mrow\"><span id=\"MathJax-Span-41506\" class=\"mrow\"><span id=\"MathJax-Span-41507\" class=\"mi\">a<\/span><span id=\"MathJax-Span-41508\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41509\" class=\"mi\">g<\/span><span id=\"MathJax-Span-41510\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41511\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-41512\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41513\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-41514\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-41515\" class=\"msub\"><span id=\"MathJax-Span-41516\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41517\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-41518\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41519\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-41520\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41521\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-41522\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41523\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">a=g(sin\u03b8\u2212\u03bckcos\u03b8).<\/span><\/span>\u00a0Note that the acceleration is independent of mass and reduces to the expression found in the previous problem when friction becomes negligibly small\u00a0<span id=\"MathJax-Element-1933-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41524\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41525\" class=\"mrow\"><span id=\"MathJax-Span-41526\" class=\"semantics\"><span id=\"MathJax-Span-41527\" class=\"mrow\"><span id=\"MathJax-Span-41528\" class=\"mrow\"><span id=\"MathJax-Span-41529\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41530\" class=\"msub\"><span id=\"MathJax-Span-41531\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41532\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-41533\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41534\" class=\"mn\">0<\/span><span id=\"MathJax-Span-41535\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41536\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(\u03bck=0).<\/span><\/span><\/p>\n<p><span id=\"fs-id1165036982731\"><img decoding=\"async\" id=\"58987\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/617457a351deabedf57f3137a869928efd7ad8e0\" alt=\"An illustration of  block on  a slope. The slope angles down and to the right at an angle of theta degrees to the horizontal. The block has an acceleration, a, parallel to the slope, toward its bottom. The following forces are shown:  f in a direction parallel to the slope toward its top, N perpendicular to the slope and pointing out of it, w sub x in a direction parallel to the slope toward its bottom, and w sub y perpendicular to the slope and pointing into it. An x y coordinate system is shown tilted so that positive x is downslope, parallel to the surface, and positive y is perpendicular to the slope, pointing out of the surface.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037218975\" class=\"\">\n<section>\n<div id=\"fs-id1165037218978\"><span class=\"os-number\">54<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036859341\">Calculate the deceleration of a snow boarder going up a\u00a0<span id=\"MathJax-Element-1934-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41537\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41538\" class=\"mrow\"><span id=\"MathJax-Span-41539\" class=\"semantics\"><span id=\"MathJax-Span-41540\" class=\"mrow\"><span id=\"MathJax-Span-41541\" class=\"mrow\"><span id=\"MathJax-Span-41542\" class=\"mn\">5.00<\/span><span id=\"MathJax-Span-41543\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.00\u00b0<\/span><\/span>\u00a0slope, assuming the coefficient of friction for waxed wood on wet snow. The result of the preceding problem may be useful, but be careful to consider the fact that the snow boarder is going uphill.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037987755\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037987757\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037987755-solution\">55<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037159582\">A machine at a post office sends packages out a chute and down a ramp to be loaded into delivery vehicles. (a) Calculate the acceleration of a box heading down a\u00a0<span id=\"MathJax-Element-1935-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41544\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41545\" class=\"mrow\"><span id=\"MathJax-Span-41546\" class=\"semantics\"><span id=\"MathJax-Span-41547\" class=\"mrow\"><span id=\"MathJax-Span-41548\" class=\"mrow\"><span id=\"MathJax-Span-41549\" class=\"mn\">10.0<\/span><span id=\"MathJax-Span-41550\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">10.0\u00b0<\/span><\/span>\u00a0slope, assuming the coefficient of friction for a parcel on waxed wood is 0.100. (b) Find the angle of the slope down which this box could move at a constant velocity. You can neglect air resistance in both parts.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037229369\" class=\"\">\n<section>\n<div id=\"fs-id1165037229371\"><span class=\"os-number\">56<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038250675\">If an object is to rest on an incline without slipping, then friction must equal the component of the weight of the object parallel to the incline. This requires greater and greater friction for steeper slopes. Show that the maximum angle of an incline above the horizontal for which an object will not slide down is\u00a0<span id=\"MathJax-Element-1936-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41551\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41552\" class=\"mrow\"><span id=\"MathJax-Span-41553\" class=\"semantics\"><span id=\"MathJax-Span-41554\" class=\"mrow\"><span id=\"MathJax-Span-41555\" class=\"mrow\"><span id=\"MathJax-Span-41556\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-41557\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41558\" class=\"msup\"><span id=\"MathJax-Span-41559\" class=\"mrow\"><span id=\"MathJax-Span-41560\" class=\"mtext\">tan<\/span><\/span><span id=\"MathJax-Span-41561\" class=\"mrow\"><span id=\"MathJax-Span-41562\" class=\"mn\">\u22121<\/span><\/span><\/span><span id=\"MathJax-Span-41563\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41564\" class=\"msub\"><span id=\"MathJax-Span-41565\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41566\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41567\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8=tan\u22121\u03bcs.<\/span><\/span>\u00a0You may use the result of the previous problem. Assume that\u00a0<span id=\"MathJax-Element-1937-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41568\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41569\" class=\"mrow\"><span id=\"MathJax-Span-41570\" class=\"semantics\"><span id=\"MathJax-Span-41571\" class=\"mrow\"><span id=\"MathJax-Span-41572\" class=\"mrow\"><span id=\"MathJax-Span-41573\" class=\"mi\">a<\/span><span id=\"MathJax-Span-41574\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41575\" class=\"mn\">0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">a=0<\/span><\/span>\u00a0and that static friction has reached its maximum value.<\/p>\n<p><span id=\"fs-id1165038163415\"><img decoding=\"async\" id=\"3620\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/d38c3860133768d93151529b68829c3123e41e0c\" alt=\"An illustration of  a block mass m on  a slope. The slope angles up and to the right at an angle of theta degrees to the horizontal. The mass feels force w sub parallel in a direction parallel to the slope toward its bottom, and f in a direction parallel to the slope toward its top.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037112519\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037112521\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037112519-solution\">57<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037112524\">Calculate the maximum acceleration of a car that is heading down a\u00a0<span id=\"MathJax-Element-1938-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41576\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41577\" class=\"mrow\"><span id=\"MathJax-Span-41578\" class=\"semantics\"><span id=\"MathJax-Span-41579\" class=\"mrow\"><span id=\"MathJax-Span-41580\" class=\"mrow\"><span id=\"MathJax-Span-41581\" class=\"mn\">6.00<\/span><span id=\"MathJax-Span-41582\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">6.00\u00b0<\/span><\/span>\u00a0slope (one that makes an angle of\u00a0<span id=\"MathJax-Element-1939-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41583\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41584\" class=\"mrow\"><span id=\"MathJax-Span-41585\" class=\"semantics\"><span id=\"MathJax-Span-41586\" class=\"mrow\"><span id=\"MathJax-Span-41587\" class=\"mrow\"><span id=\"MathJax-Span-41588\" class=\"mn\">6.00<\/span><span id=\"MathJax-Span-41589\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">6.00\u00b0<\/span><\/span>\u00a0with the horizontal) under the following road conditions. You may assume that the weight of the car is evenly distributed on all four tires and that the coefficient of static friction is involved\u2014that is, the tires are not allowed to slip during the deceleration. (Ignore rolling.) Calculate for a car: (a) On dry concrete. (b) On wet concrete. (c) On ice, assuming that\u00a0<span id=\"MathJax-Element-1940-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41590\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41591\" class=\"mrow\"><span id=\"MathJax-Span-41592\" class=\"semantics\"><span id=\"MathJax-Span-41593\" class=\"mrow\"><span id=\"MathJax-Span-41594\" class=\"mrow\"><span id=\"MathJax-Span-41595\" class=\"msub\"><span id=\"MathJax-Span-41596\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41597\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41598\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41599\" class=\"mn\">0.100<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bcs=0.100<\/span><\/span>, the same as for shoes on ice.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038247675\" class=\"\">\n<section>\n<div id=\"fs-id1165038247677\"><span class=\"os-number\">58<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038247679\">Calculate the maximum acceleration of a car that is heading up a\u00a0<span id=\"MathJax-Element-1941-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41600\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41601\" class=\"mrow\"><span id=\"MathJax-Span-41602\" class=\"semantics\"><span id=\"MathJax-Span-41603\" class=\"mrow\"><span id=\"MathJax-Span-41604\" class=\"mrow\"><span id=\"MathJax-Span-41605\" class=\"mn\">4.00<\/span><span id=\"MathJax-Span-41606\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">4.00\u00b0<\/span><\/span>\u00a0slope (one that makes an angle of\u00a0<span id=\"MathJax-Element-1942-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41607\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41608\" class=\"mrow\"><span id=\"MathJax-Span-41609\" class=\"semantics\"><span id=\"MathJax-Span-41610\" class=\"mrow\"><span id=\"MathJax-Span-41611\" class=\"mrow\"><span id=\"MathJax-Span-41612\" class=\"mn\">4.00<\/span><span id=\"MathJax-Span-41613\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">4.00\u00b0<\/span><\/span>\u00a0with the horizontal) under the following road conditions. Assume that only half the weight of the car is supported by the two drive wheels and that the coefficient of static friction is involved\u2014that is, the tires are not allowed to slip during the acceleration. (Ignore rolling.) (a) On dry concrete. (b) On wet concrete. (c) On ice, assuming that\u00a0<span id=\"MathJax-Element-1943-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41614\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41615\" class=\"mrow\"><span id=\"MathJax-Span-41616\" class=\"semantics\"><span id=\"MathJax-Span-41617\" class=\"mrow\"><span id=\"MathJax-Span-41618\" class=\"mrow\"><span id=\"MathJax-Span-41619\" class=\"msub\"><span id=\"MathJax-Span-41620\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-41621\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41622\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41623\" class=\"mn\">0.100<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bcs=0.100<\/span><\/span>, the same as for shoes on ice.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038248967\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037089573\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038248967-solution\">59<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037089575\">Repeat the preceding problem for a car with four-wheel drive.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036967840\" class=\"\">\n<section>\n<div id=\"fs-id1165038009903\"><span class=\"os-number\">60<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038009905\">A freight train consists of two\u00a0<span id=\"MathJax-Element-1944-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41624\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41625\" class=\"mrow\"><span id=\"MathJax-Span-41626\" class=\"semantics\"><span id=\"MathJax-Span-41627\" class=\"mrow\"><span id=\"MathJax-Span-41628\" class=\"mrow\"><span id=\"MathJax-Span-41629\" class=\"mn\">8.00<\/span><span id=\"MathJax-Span-41630\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41631\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41632\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41633\" class=\"msup\"><span id=\"MathJax-Span-41634\" class=\"mrow\"><span id=\"MathJax-Span-41635\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41636\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-41637\" class=\"mtext\">-kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">8.00\u00d7105-kg<\/span><\/span>\u00a0engines and 45 cars with average masses of\u00a0<span id=\"MathJax-Element-1945-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41638\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41639\" class=\"mrow\"><span id=\"MathJax-Span-41640\" class=\"semantics\"><span id=\"MathJax-Span-41641\" class=\"mrow\"><span id=\"MathJax-Span-41642\" class=\"mrow\"><span id=\"MathJax-Span-41643\" class=\"mn\">5.50<\/span><span id=\"MathJax-Span-41644\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41645\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41646\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41647\" class=\"msup\"><span id=\"MathJax-Span-41648\" class=\"mrow\"><span id=\"MathJax-Span-41649\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41650\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-41651\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41652\" class=\"mtext\">kg<\/span><span id=\"MathJax-Span-41653\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.50\u00d7105kg.<\/span><\/span>\u00a0(a) What force must each engine exert backward on the track to accelerate the train at a rate of\u00a0<span id=\"MathJax-Element-1946-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41654\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41655\" class=\"mrow\"><span id=\"MathJax-Span-41656\" class=\"semantics\"><span id=\"MathJax-Span-41657\" class=\"mrow\"><span id=\"MathJax-Span-41658\" class=\"mrow\"><span id=\"MathJax-Span-41659\" class=\"mn\">5.00<\/span><span id=\"MathJax-Span-41660\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41661\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41662\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41663\" class=\"msup\"><span id=\"MathJax-Span-41664\" class=\"mrow\"><span id=\"MathJax-Span-41665\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41666\" class=\"mrow\"><span id=\"MathJax-Span-41667\" class=\"mn\">\u22122<\/span><\/span><\/span><span id=\"MathJax-Span-41668\" class=\"mtext\">m<\/span><span id=\"MathJax-Span-41669\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-41670\" class=\"msup\"><span id=\"MathJax-Span-41671\" class=\"mtext\">s<\/span><span id=\"MathJax-Span-41672\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.00\u00d710\u22122m\/s2<\/span><\/span>\u00a0if the force of friction is\u00a0<span id=\"MathJax-Element-1947-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41673\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41674\" class=\"mrow\"><span id=\"MathJax-Span-41675\" class=\"semantics\"><span id=\"MathJax-Span-41676\" class=\"mrow\"><span id=\"MathJax-Span-41677\" class=\"mrow\"><span id=\"MathJax-Span-41678\" class=\"mn\">7.50<\/span><span id=\"MathJax-Span-41679\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41680\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41681\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41682\" class=\"msup\"><span id=\"MathJax-Span-41683\" class=\"mrow\"><span id=\"MathJax-Span-41684\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41685\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-41686\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">7.50\u00d7105N<\/span><\/span>, assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently, trains are very energy-efficient transportation systems. (b) What is the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038191800\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165038191802\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165038191800-solution\">61<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038191805\">Consider the 52.0-kg mountain climber shown below. (a) Find the tension in the rope and the force that the mountain climber must exert with her feet on the vertical rock face to remain stationary. Assume that the force is exerted parallel to her legs. Also, assume negligible force exerted by her arms. (b) What is the minimum coefficient of friction between her shoes and the cliff?<\/p>\n<p><span id=\"fs-id1165037089825\"><img decoding=\"async\" id=\"80081\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/9d7620350be4ecabd3c0406466d44890a59c9ec6\" alt=\"A mountain climber is drawn leaning away from the rock face with her feet against the rock face. The rope extends up from the climber  at an angle of 31 degrees to the vertical. The climbers legs are straight and make an angle of fifteen degrees with the rock face. The force vector F sub T starts at the harness and points away from the climber, along the rope. The force vector F sub legs starts at climber\u2019s feet and points away from the rock, parallel to her legs.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036765363\" class=\"\">\n<section>\n<div id=\"fs-id1165038132570\"><span class=\"os-number\">62<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038132572\">A contestant in a winter sporting event pushes a 45.0-kg block of ice across a frozen lake as shown below. (a) Calculate the minimum force\u00a0<em>F<\/em>\u00a0he must exert to get the block moving. (b) What is its acceleration once it starts to move, if that force is maintained?<\/p>\n<p><span id=\"fs-id1165037166007\"><img decoding=\"async\" id=\"53067\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/9e698d2bb8ad6d6b3aa5ea3f56df6b354680c701\" alt=\"A block of ice is being pushed with a force F that is directed at an angle of twenty five degrees below the horizontal.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165037150306\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165037150308\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165037150306-solution\">63<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165037843834\">The contestant now pulls the block of ice with a rope over his shoulder at the same angle above the horizontal as shown below. Calculate the minimum force\u00a0<em>F<\/em>\u00a0he must exert to get the block moving. (b) What is its acceleration once it starts to move, if that force is maintained?<\/p>\n<p><span id=\"fs-id1165037063269\"><img decoding=\"async\" id=\"64358\" src=\"https:\/\/cnx.org\/resources\/1f0c563cc53fe02196a93993dfc155d7532aa7de\" alt=\"A block of ice is being pulled with a force F that is directed at an angle of twenty five degrees above the horizontal.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165034581253\" class=\"\">\n<section>\n<div id=\"fs-id1165034581255\"><span class=\"os-number\">64<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038000706\">At a post office, a parcel that is a 20.0-kg box slides down a ramp inclined at\u00a0<span id=\"MathJax-Element-1948-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41687\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41688\" class=\"mrow\"><span id=\"MathJax-Span-41689\" class=\"semantics\"><span id=\"MathJax-Span-41690\" class=\"mrow\"><span id=\"MathJax-Span-41691\" class=\"mrow\"><span id=\"MathJax-Span-41692\" class=\"mn\">30.0<\/span><span id=\"MathJax-Span-41693\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">30.0\u00b0<\/span><\/span>\u00a0with the horizontal. The coefficient of kinetic friction between the box and plane is 0.0300. (a) Find the acceleration of the box. (b) Find the velocity of the box as it reaches the end of the plane, if the length of the plane is 2 m and the box starts at rest.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"os-section-area\">\n<section id=\"fs-id1165035734743\" class=\"review-problems\">\n<h4 id=\"71765_copy_3\"><span class=\"os-number\">6.3<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Centripetal Force<\/span><\/h4>\n<div id=\"fs-id1165039087650\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039479306\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039087650-solution\">65<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035682097\">(a) A 22.0-kg child is riding a playground merry-go-round that is rotating at 40.0 rev\/min. What centripetal force is exerted if he is 1.25 m from its center? (b) What centripetal force is exerted if the merry-go-round rotates at 3.00 rev\/min and he is 8.00 m from its center? (c) Compare each force with his weight.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039100614\" class=\"\">\n<section>\n<div id=\"fs-id1165039242547\"><span class=\"os-number\">66<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039209500\">Calculate the centripetal force on the end of a 100-m (radius) wind turbine blade that is rotating at 0.5 rev\/s. Assume the mass is 4 kg.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039098944\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035636247\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039098944-solution\">67<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039438998\">What is the ideal banking angle for a gentle turn of 1.20-km radius on a highway with a 105 km\/h speed limit (about 65 mi\/h), assuming everyone travels at the limit?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036159224\" class=\"\">\n<section>\n<div id=\"fs-id1165039085659\"><span class=\"os-number\">68<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165038974988\">What is the ideal speed to take a 100.0-m-radius curve banked at a\u00a0<span id=\"MathJax-Element-1949-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41694\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41695\" class=\"mrow\"><span id=\"MathJax-Span-41696\" class=\"semantics\"><span id=\"MathJax-Span-41697\" class=\"mrow\"><span id=\"MathJax-Span-41698\" class=\"mrow\"><span id=\"MathJax-Span-41699\" class=\"mn\">20.0<\/span><span id=\"MathJax-Span-41700\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">20.0\u00b0<\/span><\/span>\u00a0angle?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039399485\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035619493\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039399485-solution\">69<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039104015\">(a) What is the radius of a bobsled turn banked at\u00a0<span id=\"MathJax-Element-1950-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41701\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41702\" class=\"mrow\"><span id=\"MathJax-Span-41703\" class=\"semantics\"><span id=\"MathJax-Span-41704\" class=\"mrow\"><span id=\"MathJax-Span-41705\" class=\"mrow\"><span id=\"MathJax-Span-41706\" class=\"mn\">75.0<\/span><span id=\"MathJax-Span-41707\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">75.0\u00b0<\/span><\/span>\u00a0and taken at 30.0 m\/s, assuming it is ideally banked? (b) Calculate the centripetal acceleration. (c) Does this acceleration seem large to you?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039426342\" class=\"\">\n<section>\n<div id=\"fs-id1165038981681\"><span class=\"os-number\">70<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039276298\">Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen below. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components\u2014friction parallel to the road (this must supply the centripetal force) and the vertical normal force (which must equal the system\u2019s weight). (a) Show that\u00a0<span id=\"MathJax-Element-1951-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41708\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41709\" class=\"mrow\"><span id=\"MathJax-Span-41710\" class=\"semantics\"><span id=\"MathJax-Span-41711\" class=\"mrow\"><span id=\"MathJax-Span-41712\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0(as defined as shown) is related to the speed\u00a0<em>v<\/em>\u00a0and radius of curvature\u00a0<em>r<\/em>\u00a0of the turn in the same way as for an ideally banked roadway\u2014that is,\u00a0<span id=\"MathJax-Element-1952-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41713\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41714\" class=\"mrow\"><span id=\"MathJax-Span-41715\" class=\"semantics\"><span id=\"MathJax-Span-41716\" class=\"mrow\"><span id=\"MathJax-Span-41717\" class=\"mrow\"><span id=\"MathJax-Span-41718\" class=\"mi\">\u03b8<\/span><span id=\"MathJax-Span-41719\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41720\" class=\"msup\"><span id=\"MathJax-Span-41721\" class=\"mrow\"><span id=\"MathJax-Span-41722\" class=\"mtext\">tan<\/span><\/span><span id=\"MathJax-Span-41723\" class=\"mrow\"><span id=\"MathJax-Span-41724\" class=\"mn\">\u22121<\/span><\/span><\/span><span id=\"MathJax-Span-41725\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41726\" class=\"msup\"><span id=\"MathJax-Span-41727\" class=\"mi\">v<\/span><span id=\"MathJax-Span-41728\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-41729\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-41730\" class=\"mi\">r<\/span><span id=\"MathJax-Span-41731\" class=\"mi\">g<\/span><span id=\"MathJax-Span-41732\" class=\"mo\">)<\/span><span id=\"MathJax-Span-41733\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8=tan\u22121(v2\/rg).<\/span><\/span>\u00a0(b) Calculate\u00a0<span id=\"MathJax-Element-1953-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41734\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41735\" class=\"mrow\"><span id=\"MathJax-Span-41736\" class=\"semantics\"><span id=\"MathJax-Span-41737\" class=\"mrow\"><span id=\"MathJax-Span-41738\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0for a 12.0-m\/s turn of radius 30.0 m (as in a race).<\/p>\n<p><span id=\"fs-id1165039285032\"><img decoding=\"async\" id=\"37715\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/adc368237be4c9d5ccf02856a9df08c8af510b82\" alt=\"The figure is an illustration of a man riding a bicycle, viewed from the front. The rider and bike are tilted to the right at an angle theta to the vertical. Three force vectors are shown as solid line arrows. One is from the bottom of the front wheel to the right showing the centripetal force F sub c. A second is from the same point vertically upward showing the force N. The third is from the chest of the rider vertically downward showing his weight, w. An additional broken line arrow from the bottom of the wheel to the chest point, at an angle theta to the right of vertical, is also shown and labeled with force F exerting on it.  The vectors F sub c, w and F form a right triangle whose hypotenuse is F. A free-body diagram is also given above the figure showing vectors w and F. The vector relations F equals the sum of N and F sub c, and N equals w are also given alongside the figure.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039512072\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039115696\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039512072-solution\">71<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039115698\">If a car takes a banked curve at less than the ideal speed, friction is needed to keep it from sliding toward the inside of the curve (a problem on icy mountain roads). (a) Calculate the ideal speed to take a 100.0 m radius curve banked at\u00a0<span id=\"MathJax-Element-1954-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41739\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41740\" class=\"mrow\"><span id=\"MathJax-Span-41741\" class=\"semantics\"><span id=\"MathJax-Span-41742\" class=\"mrow\"><span id=\"MathJax-Span-41743\" class=\"mrow\"><span id=\"MathJax-Span-41744\" class=\"mn\">15.0<\/span><span id=\"MathJax-Span-41745\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">15.0\u00b0<\/span><\/span>. (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km\/h?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039045013\" class=\"\">\n<section>\n<div id=\"fs-id1165039045015\"><span class=\"os-number\">72<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039125365\">Modern roller coasters have vertical loops like the one shown here. The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into their seats. (a) What is the speed of the roller coaster at the top of the loop if the radius of curvature there is 15.0 m and the downward acceleration of the car is 1.50\u00a0<em>g<\/em>? (b) How high above the top of the loop must the roller coaster start from rest, assuming negligible friction? (c) If it actually starts 5.00 m higher than your answer to (b), how much energy did it lose to friction? Its mass is\u00a0<span id=\"MathJax-Element-1955-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41746\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41747\" class=\"mrow\"><span id=\"MathJax-Span-41748\" class=\"semantics\"><span id=\"MathJax-Span-41749\" class=\"mrow\"><span id=\"MathJax-Span-41750\" class=\"mrow\"><span id=\"MathJax-Span-41751\" class=\"mn\">1.50<\/span><span id=\"MathJax-Span-41752\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41753\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41754\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41755\" class=\"msup\"><span id=\"MathJax-Span-41756\" class=\"mrow\"><span id=\"MathJax-Span-41757\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41758\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41759\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41760\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.50\u00d7103kg<\/span><\/span>.<\/p>\n<p><span id=\"fs-id1165035733795\"><img decoding=\"async\" id=\"48524\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/0cd33aa8c952fc54cd87c3388a9be18abd783fe7\" alt=\"An illustration of a loop of a roller. The radius of curvature is smaller at the top than on the sides and bottom. The radius of the loop at the tom is shown and labeled as r sub minimum. The radius at the lowest part of the loop is labeled as r sub maximum.  The track is on the inside surface of the loop. The motion is indicated by arrows, starting at ground level to the right of the loop, going up inside the loop on the left, then down the inside right of the loop, and out again at ground level on the left. Four location on the track, A, B, C, and D and B, are labeled. Point A is at ground level, to the right of the loop, where the track is straight and horizontal. Point B is part way up the left side of the loop. Point C is part way up the right side of the loop, at the same level as point B. Point D is at ground level, to the left of the loop, where the track is straight and horizontal.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039091562\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039234647\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039091562-solution\">73<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039234649\">A child of mass 40.0 kg is in a roller coaster car that travels in a loop of radius 7.00 m. At point A the speed of the car is 10.0 m\/s, and at point B, the speed is 10.5 m\/s. Assume the child is not holding on and does not wear a seat belt. (a) What is the force of the car seat on the child at point A? (b) What is the force of the car seat on the child at point B? (c) What minimum speed is required to keep the child in his seat at point A?<\/p>\n<p><span id=\"fs-id1165039439426\"><img decoding=\"async\" id=\"12769\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/f93261c5ebe5098833fc3e47126687e2fc34168f\" alt=\"An illustration of a loop of a roller coaster with a child seated in a car approaching the loop. The track is on the inside surface of the loop. Two location on the loop, A and B, are labeled. Point A is at the top of the loop. Point B is down and to the left of A. The angle between the radii to points A and B is thirty degrees.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039445011\" class=\"\">\n<section>\n<div id=\"fs-id1165039445013\"><span class=\"os-number\">74<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039516436\">In the simple Bohr model of the ground state of the hydrogen atom, the electron travels in a circular orbit around a fixed proton. The radius of the orbit is\u00a0<span id=\"MathJax-Element-1956-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41761\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41762\" class=\"mrow\"><span id=\"MathJax-Span-41763\" class=\"semantics\"><span id=\"MathJax-Span-41764\" class=\"mrow\"><span id=\"MathJax-Span-41765\" class=\"mrow\"><span id=\"MathJax-Span-41766\" class=\"mn\">5.28<\/span><span id=\"MathJax-Span-41767\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41768\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41769\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41770\" class=\"msup\"><span id=\"MathJax-Span-41771\" class=\"mrow\"><span id=\"MathJax-Span-41772\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41773\" class=\"mrow\"><span id=\"MathJax-Span-41774\" class=\"mn\">\u221211<\/span><\/span><\/span><span id=\"MathJax-Span-41775\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41776\" class=\"mtext\">m,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.28\u00d710\u221211m,<\/span><\/span>\u00a0and the speed of the electron is\u00a0<span id=\"MathJax-Element-1957-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41777\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41778\" class=\"mrow\"><span id=\"MathJax-Span-41779\" class=\"semantics\"><span id=\"MathJax-Span-41780\" class=\"mrow\"><span id=\"MathJax-Span-41781\" class=\"mrow\"><span id=\"MathJax-Span-41782\" class=\"mn\">2.18<\/span><span id=\"MathJax-Span-41783\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41784\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41785\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41786\" class=\"msup\"><span id=\"MathJax-Span-41787\" class=\"mrow\"><span id=\"MathJax-Span-41788\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41789\" class=\"mn\">6<\/span><\/span><span id=\"MathJax-Span-41790\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41791\" class=\"mrow\"><span id=\"MathJax-Span-41792\" class=\"mtext\">m<\/span><span id=\"MathJax-Span-41793\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-41794\" class=\"mtext\">s<\/span><\/span><span id=\"MathJax-Span-41795\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.18\u00d7106m\/s.<\/span><\/span>\u00a0The mass of an electron is\u00a0<span id=\"MathJax-Element-1958-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41796\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41797\" class=\"mrow\"><span id=\"MathJax-Span-41798\" class=\"semantics\"><span id=\"MathJax-Span-41799\" class=\"mrow\"><span id=\"MathJax-Span-41800\" class=\"mrow\"><span id=\"MathJax-Span-41801\" class=\"mn\">9.11<\/span><span id=\"MathJax-Span-41802\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41803\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41804\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41805\" class=\"msup\"><span id=\"MathJax-Span-41806\" class=\"mrow\"><span id=\"MathJax-Span-41807\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41808\" class=\"mrow\"><span id=\"MathJax-Span-41809\" class=\"mn\">\u221231<\/span><\/span><\/span><span id=\"MathJax-Span-41810\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41811\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">9.11\u00d710\u221231kg<\/span><\/span>. What is the force on the electron?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035610496\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035610498\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035610496-solution\">75<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039421074\">Railroad tracks follow a circular curve of radius 500.0 m and are banked at an angle of\u00a0<span id=\"MathJax-Element-1959-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41812\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41813\" class=\"mrow\"><span id=\"MathJax-Span-41814\" class=\"semantics\"><span id=\"MathJax-Span-41815\" class=\"mrow\"><span id=\"MathJax-Span-41816\" class=\"mrow\"><span id=\"MathJax-Span-41817\" class=\"mn\">5.0<\/span><span id=\"MathJax-Span-41818\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.0\u00b0<\/span><\/span>. For trains of what speed are these tracks designed?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039295768\" class=\"\">\n<section>\n<div id=\"fs-id1165039440552\"><span class=\"os-number\">76<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039440554\">The CERN particle accelerator is circular with a circumference of 7.0 km. (a) What is the acceleration of the protons\u00a0<span id=\"MathJax-Element-1960-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41819\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41820\" class=\"mrow\"><span id=\"MathJax-Span-41821\" class=\"semantics\"><span id=\"MathJax-Span-41822\" class=\"mrow\"><span id=\"MathJax-Span-41823\" class=\"mrow\"><span id=\"MathJax-Span-41824\" class=\"mo\">(<\/span><span id=\"MathJax-Span-41825\" class=\"mi\">m<\/span><span id=\"MathJax-Span-41826\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41827\" class=\"mn\">1.67<\/span><span id=\"MathJax-Span-41828\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41829\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41830\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41831\" class=\"msup\"><span id=\"MathJax-Span-41832\" class=\"mrow\"><span id=\"MathJax-Span-41833\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41834\" class=\"mrow\"><span id=\"MathJax-Span-41835\" class=\"mn\">\u221227<\/span><\/span><\/span><span id=\"MathJax-Span-41836\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41837\" class=\"mtext\">kg<\/span><span id=\"MathJax-Span-41838\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(m=1.67\u00d710\u221227kg)<\/span><\/span>\u00a0that move around the accelerator at\u00a0<span id=\"MathJax-Element-1961-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41839\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41840\" class=\"mrow\"><span id=\"MathJax-Span-41841\" class=\"semantics\"><span id=\"MathJax-Span-41842\" class=\"mrow\"><span id=\"MathJax-Span-41843\" class=\"mrow\"><span id=\"MathJax-Span-41844\" class=\"mn\">5<\/span><span id=\"MathJax-Span-41845\" class=\"mi\">%<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5%<\/span><\/span>\u00a0of the speed of light? (The speed of light is\u00a0<span id=\"MathJax-Element-1962-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41846\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41847\" class=\"mrow\"><span id=\"MathJax-Span-41848\" class=\"semantics\"><span id=\"MathJax-Span-41849\" class=\"mrow\"><span id=\"MathJax-Span-41850\" class=\"mrow\"><span id=\"MathJax-Span-41851\" class=\"mi\">v<\/span><span id=\"MathJax-Span-41852\" class=\"mo\">=<\/span><span id=\"MathJax-Span-41853\" class=\"mn\">3.00<\/span><span id=\"MathJax-Span-41854\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41855\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41856\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41857\" class=\"msup\"><span id=\"MathJax-Span-41858\" class=\"mrow\"><span id=\"MathJax-Span-41859\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41860\" class=\"mn\">8<\/span><\/span><span id=\"MathJax-Span-41861\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41862\" class=\"mtext\">m\/s<\/span><span id=\"MathJax-Span-41863\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v=3.00\u00d7108m\/s.<\/span><\/span>) (b) What is the force on the protons?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039071024\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039237696\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039071024-solution\">77<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039237698\">A car rounds an unbanked curve of radius 65 m. If the coefficient of static friction between the road and car is 0.70, what is the maximum speed at which the car traverse the curve without slipping?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039315651\" class=\"\">\n<section>\n<div id=\"fs-id1165039513150\"><span class=\"os-number\">78<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039513152\">A banked highway is designed for traffic moving at 90.0 km\/h. The radius of the curve is 310 m. What is the angle of banking of the highway?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"os-section-area\">\n<section id=\"fs-id1165039390902\" class=\"review-problems\">\n<h4 id=\"47362_copy_3\"><span class=\"os-number\">6.4<\/span><span class=\"os-divider\">\u00a0<\/span><span class=\"os-text\">Drag Force and Terminal Speed<\/span><\/h4>\n<div id=\"fs-id1165039251723\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039092545\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039251723-solution\">79<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035682260\">The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Find the terminal velocity (in meters per second and kilometers per hour) of an 80.0-kg skydiver falling in a pike (headfirst) position with a surface area of\u00a0<span id=\"MathJax-Element-1963-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41864\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41865\" class=\"mrow\"><span id=\"MathJax-Span-41866\" class=\"semantics\"><span id=\"MathJax-Span-41867\" class=\"mrow\"><span id=\"MathJax-Span-41868\" class=\"mrow\"><span id=\"MathJax-Span-41869\" class=\"mn\">0.140<\/span><span id=\"MathJax-Span-41870\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41871\" class=\"msup\"><span id=\"MathJax-Span-41872\" class=\"mrow\"><span id=\"MathJax-Span-41873\" class=\"mtext\">m<\/span><\/span><span id=\"MathJax-Span-41874\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.140m2<\/span><\/span>.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035906453\" class=\"\">\n<section>\n<div id=\"fs-id1165036153774\"><span class=\"os-number\">80<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035682242\">A 60.0-kg and a 90.0-kg skydiver jump from an airplane at an altitude of\u00a0<span id=\"MathJax-Element-1964-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41875\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41876\" class=\"mrow\"><span id=\"MathJax-Span-41877\" class=\"semantics\"><span id=\"MathJax-Span-41878\" class=\"mrow\"><span id=\"MathJax-Span-41879\" class=\"mrow\"><span id=\"MathJax-Span-41880\" class=\"mn\">6.00<\/span><span id=\"MathJax-Span-41881\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41882\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41883\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41884\" class=\"msup\"><span id=\"MathJax-Span-41885\" class=\"mrow\"><span id=\"MathJax-Span-41886\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41887\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41888\" class=\"mtext\">m<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">6.00\u00d7103m<\/span><\/span>, both falling in the pike position. Make some assumption on their frontal areas and calculate their terminal velocities. How long will it take for each skydiver to reach the ground (assuming the time to reach terminal velocity is small)? Assume all values are accurate to three significant digits.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036158571\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id11650394586140\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036158571-solution\">81<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035663631\">A 560-g squirrel with a surface area of\u00a0<span id=\"MathJax-Element-1965-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41889\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41890\" class=\"mrow\"><span id=\"MathJax-Span-41891\" class=\"semantics\"><span id=\"MathJax-Span-41892\" class=\"mrow\"><span id=\"MathJax-Span-41893\" class=\"mrow\"><span id=\"MathJax-Span-41894\" class=\"mn\">930<\/span><span id=\"MathJax-Span-41895\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41896\" class=\"msup\"><span id=\"MathJax-Span-41897\" class=\"mrow\"><span id=\"MathJax-Span-41898\" class=\"mtext\">cm<\/span><\/span><span id=\"MathJax-Span-41899\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">930cm2<\/span><\/span>\u00a0falls from a 5.0-m tree to the ground. Estimate its terminal velocity. (Use a drag coefficient for a horizontal skydiver.) What will be the velocity of a 56-kg person hitting the ground, assuming no drag contribution in such a short distance?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039111535\" class=\"\">\n<section>\n<div id=\"fs-id1165036148072\"><span class=\"os-number\">82<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036141290\">To maintain a constant speed, the force provided by a car\u2019s engine must equal the drag force plus the force of friction of the road (the rolling resistance). (a) What are the drag forces at 70 km\/h and 100 km\/h for a Toyota Camry? (Drag area is\u00a0<span id=\"MathJax-Element-1966-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41900\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41901\" class=\"mrow\"><span id=\"MathJax-Span-41902\" class=\"semantics\"><span id=\"MathJax-Span-41903\" class=\"mrow\"><span id=\"MathJax-Span-41904\" class=\"mrow\"><span id=\"MathJax-Span-41905\" class=\"mn\">0.70<\/span><span id=\"MathJax-Span-41906\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41907\" class=\"msup\"><span id=\"MathJax-Span-41908\" class=\"mrow\"><span id=\"MathJax-Span-41909\" class=\"mtext\">m<\/span><\/span><span id=\"MathJax-Span-41910\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.70m2<\/span><\/span>) (b) What is the drag force at 70 km\/h and 100 km\/h for a Hummer H2? (Drag area is\u00a0<span id=\"MathJax-Element-1967-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41911\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41912\" class=\"mrow\"><span id=\"MathJax-Span-41913\" class=\"semantics\"><span id=\"MathJax-Span-41914\" class=\"mrow\"><span id=\"MathJax-Span-41915\" class=\"mrow\"><span id=\"MathJax-Span-41916\" class=\"mn\">2.44<\/span><span id=\"MathJax-Span-41917\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41918\" class=\"msup\"><span id=\"MathJax-Span-41919\" class=\"mrow\"><span id=\"MathJax-Span-41920\" class=\"mtext\">m<\/span><\/span><span id=\"MathJax-Span-41921\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-41922\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.44m2)<\/span><\/span>\u00a0Assume all values are accurate to three significant digits.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039477110\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039103252\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039477110-solution\">83<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039312187\">By what factor does the drag force on a car increase as it goes from 65 to 110 km\/h?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035834936\" class=\"\">\n<section>\n<div id=\"fs-id1165036080908\"><span class=\"os-number\">84<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035939537\">Calculate the velocity a spherical rain drop would achieve falling from 5.00 km (a) in the absence of air drag (b) with air drag. Take the size across of the drop to be 4 mm, the density to be\u00a0<span id=\"MathJax-Element-1968-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41923\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41924\" class=\"mrow\"><span id=\"MathJax-Span-41925\" class=\"semantics\"><span id=\"MathJax-Span-41926\" class=\"mrow\"><span id=\"MathJax-Span-41927\" class=\"mrow\"><span id=\"MathJax-Span-41928\" class=\"mn\">1.00<\/span><span id=\"MathJax-Span-41929\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41930\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41931\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41932\" class=\"msup\"><span id=\"MathJax-Span-41933\" class=\"mrow\"><span id=\"MathJax-Span-41934\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41935\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41936\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41937\" class=\"msup\"><span id=\"MathJax-Span-41938\" class=\"mrow\"><span id=\"MathJax-Span-41939\" class=\"mtext\">kg\/m<\/span><\/span><span id=\"MathJax-Span-41940\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.00\u00d7103kg\/m3<\/span><\/span>, and the surface area to be\u00a0<span id=\"MathJax-Element-1969-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41941\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41942\" class=\"mrow\"><span id=\"MathJax-Span-41943\" class=\"semantics\"><span id=\"MathJax-Span-41944\" class=\"mrow\"><span id=\"MathJax-Span-41945\" class=\"mrow\"><span id=\"MathJax-Span-41946\" class=\"mi\">\u03c0<\/span><span id=\"MathJax-Span-41947\" class=\"msup\"><span id=\"MathJax-Span-41948\" class=\"mi\">r<\/span><span id=\"MathJax-Span-41949\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03c0r2<\/span><\/span>.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039434228\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165036152519\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039434228-solution\">85<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035698600\">Using Stokes\u2019 law, verify that the units for viscosity are kilograms per meter per second.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039415482\" class=\"\">\n<section>\n<div id=\"fs-id1165039292226\"><span class=\"os-number\">86<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035730440\">Find the terminal velocity of a spherical bacterium (diameter\u00a0<span id=\"MathJax-Element-1970-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41950\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41951\" class=\"mrow\"><span id=\"MathJax-Span-41952\" class=\"semantics\"><span id=\"MathJax-Span-41953\" class=\"mrow\"><span id=\"MathJax-Span-41954\" class=\"mrow\"><span id=\"MathJax-Span-41955\" class=\"mn\">2.00<\/span><span id=\"MathJax-Span-41956\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41957\" class=\"mtext\">\u03bcm<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.00\u03bcm<\/span><\/span>) falling in water. You will first need to note that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be\u00a0<span id=\"MathJax-Element-1971-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41958\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41959\" class=\"mrow\"><span id=\"MathJax-Span-41960\" class=\"semantics\"><span id=\"MathJax-Span-41961\" class=\"mrow\"><span id=\"MathJax-Span-41962\" class=\"mrow\"><span id=\"MathJax-Span-41963\" class=\"mn\">1.10<\/span><span id=\"MathJax-Span-41964\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41965\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41966\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41967\" class=\"msup\"><span id=\"MathJax-Span-41968\" class=\"mrow\"><span id=\"MathJax-Span-41969\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41970\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41971\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41972\" class=\"msup\"><span id=\"MathJax-Span-41973\" class=\"mrow\"><span id=\"MathJax-Span-41974\" class=\"mtext\">kg\/m<\/span><\/span><span id=\"MathJax-Span-41975\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">1.10\u00d7103kg\/m3<\/span><\/span>.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039396162\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039423411\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039396162-solution\">87<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039028296\">Stokes\u2019 law describes sedimentation of particles in liquids and can be used to measure viscosity. Particles in liquids achieve terminal velocity quickly. One can measure the time it takes for a particle to fall a certain distance and then use Stokes\u2019 law to calculate the viscosity of the liquid. Suppose a steel ball bearing (density\u00a0<span id=\"MathJax-Element-1972-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41976\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41977\" class=\"mrow\"><span id=\"MathJax-Span-41978\" class=\"semantics\"><span id=\"MathJax-Span-41979\" class=\"mrow\"><span id=\"MathJax-Span-41980\" class=\"mrow\"><span id=\"MathJax-Span-41981\" class=\"mn\">7.8<\/span><span id=\"MathJax-Span-41982\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41983\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-41984\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41985\" class=\"msup\"><span id=\"MathJax-Span-41986\" class=\"mrow\"><span id=\"MathJax-Span-41987\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-41988\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-41989\" class=\"mspace\"><\/span><span id=\"MathJax-Span-41990\" class=\"msup\"><span id=\"MathJax-Span-41991\" class=\"mrow\"><span id=\"MathJax-Span-41992\" class=\"mtext\">kg\/m<\/span><\/span><span id=\"MathJax-Span-41993\" class=\"mn\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">7.8\u00d7103kg\/m3<\/span><\/span>, diameter 3.0 mm) is dropped in a container of motor oil. It takes 12 s to fall a distance of 0.60 m. Calculate the viscosity of the oil.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165033377748\" class=\"\">\n<section>\n<div id=\"fs-id1165039192558\"><span class=\"os-number\">88<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036153540\">Suppose that the resistive force of the air on a skydiver can be approximated by\u00a0<span id=\"MathJax-Element-1973-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-41994\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-41995\" class=\"mrow\"><span id=\"MathJax-Span-41996\" class=\"semantics\"><span id=\"MathJax-Span-41997\" class=\"mrow\"><span id=\"MathJax-Span-41998\" class=\"mrow\"><span id=\"MathJax-Span-41999\" class=\"mi\">f<\/span><span id=\"MathJax-Span-42000\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42001\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-42002\" class=\"mi\">b<\/span><span id=\"MathJax-Span-42003\" class=\"msup\"><span id=\"MathJax-Span-42004\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42005\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42006\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">f=\u2212bv2.<\/span><\/span>\u00a0If the terminal velocity of a 50.0-kg skydiver is 60.0 m\/s, what is the value of\u00a0<em>b<\/em>?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035732089\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035938648\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035732089-solution\">89<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036151480\">A small diamond of mass 10.0 g drops from a swimmer\u2019s earring and falls through the water, reaching a terminal velocity of 2.0 m\/s. (a) Assuming the frictional force on the diamond obeys\u00a0<span id=\"MathJax-Element-1974-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42007\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42008\" class=\"mrow\"><span id=\"MathJax-Span-42009\" class=\"semantics\"><span id=\"MathJax-Span-42010\" class=\"mrow\"><span id=\"MathJax-Span-42011\" class=\"mrow\"><span id=\"MathJax-Span-42012\" class=\"mi\">f<\/span><span id=\"MathJax-Span-42013\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42014\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-42015\" class=\"mi\">b<\/span><span id=\"MathJax-Span-42016\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42017\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">f=\u2212bv,<\/span><\/span>\u00a0what is\u00a0<em>b<\/em>? (b) How far does the diamond fall before it reaches 90 percent of its terminal speed?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039345130\" class=\"\">\n<section>\n<div id=\"fs-id1165038989787\"><span class=\"os-number\">90<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165033370856\">(a) What is the final velocity of a car originally traveling at 50.0 km\/h that decelerates at a rate of\u00a0<span id=\"MathJax-Element-1975-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42018\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42019\" class=\"mrow\"><span id=\"MathJax-Span-42020\" class=\"semantics\"><span id=\"MathJax-Span-42021\" class=\"mrow\"><span id=\"MathJax-Span-42022\" class=\"mrow\"><span id=\"MathJax-Span-42023\" class=\"mn\">0.400<\/span><span id=\"MathJax-Span-42024\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42025\" class=\"msup\"><span id=\"MathJax-Span-42026\" class=\"mrow\"><span id=\"MathJax-Span-42027\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-42028\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">0.400m\/s2<\/span><\/span>\u00a0for 50.0 s? Assume a coefficient of friction of 1.0. (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035761606\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035729743\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035761606-solution\">91<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039464292\">A 75.0-kg woman stands on a bathroom scale in an elevator that accelerates from rest to 30.0 m\/s in 2.00 s. (a) Calculate the scale reading in newtons and compare it with her weight. (The scale exerts an upward force on her equal to its reading.) (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039351979\" class=\"\">\n<section>\n<div id=\"fs-id1165039241179\"><span class=\"os-number\">92<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039396054\">(a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked 50.0 m radius curve at 30.0 m\/s. (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036006942\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035863813\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036006942-solution\">93<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039411492\">As shown below, if\u00a0<span id=\"MathJax-Element-1976-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42029\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42030\" class=\"mrow\"><span id=\"MathJax-Span-42031\" class=\"semantics\"><span id=\"MathJax-Span-42032\" class=\"mrow\"><span id=\"MathJax-Span-42033\" class=\"mrow\"><span id=\"MathJax-Span-42034\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42035\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42036\" class=\"mn\">5.50<\/span><span id=\"MathJax-Span-42037\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42038\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=5.50kg,<\/span><\/span>\u00a0what is the tension in string 1?<\/p>\n<p><span id=\"fs-id1165035723285\"><img decoding=\"async\" id=\"16243\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/86a15128bbd2a06be7a45fe7e705ed0472536502\" alt=\"Mass M is suspended from strings 1 and 2. String 1 connects to a wall at a point below and to the left of the mass. String 1 makes an angle of 40 degrees below the horizontal. String 2 connects to a ceiling at a point above and to the right of the mass. String 2 makes an angle of 40 degrees to the right of vertical.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165038979966\" class=\"\">\n<section>\n<div id=\"fs-id1165035641506\"><span class=\"os-number\">94<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035721482\">As shown below, if\u00a0<span id=\"MathJax-Element-1977-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42039\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42040\" class=\"mrow\"><span id=\"MathJax-Span-42041\" class=\"semantics\"><span id=\"MathJax-Span-42042\" class=\"mrow\"><span id=\"MathJax-Span-42043\" class=\"mrow\"><span id=\"MathJax-Span-42044\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42045\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42046\" class=\"mn\">60.0<\/span><span id=\"MathJax-Span-42047\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42048\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F=60.0N<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1978-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42049\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42050\" class=\"mrow\"><span id=\"MathJax-Span-42051\" class=\"semantics\"><span id=\"MathJax-Span-42052\" class=\"mrow\"><span id=\"MathJax-Span-42053\" class=\"mrow\"><span id=\"MathJax-Span-42054\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42055\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42056\" class=\"mn\">4.00<\/span><span id=\"MathJax-Span-42057\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42058\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=4.00kg,<\/span><\/span>\u00a0what is the magnitude of the acceleration of the suspended object? All surfaces are frictionless.<\/p>\n<p><span id=\"fs-id1165039112562\"><img decoding=\"async\" id=\"6391\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/11b895b8d74a85765b176ee0f621a79ec3d459b3\" alt=\"Two blocks are shown. One block, labeled 2 M is on a horizontal table. A force F pulls on the 2 M block up and to the left at an angle of 30 degrees above the horizontal. On the opposite side, the block is connected to a string that pulls it to the right. The string passes over a pulley at edge of the table, then hangs straight down and connects to  the second block, labeled M. Block 2 is not in contact with the ramp.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035615426\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035868420\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035615426-solution\">95<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035939480\">As shown below, if\u00a0<span id=\"MathJax-Element-1979-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42059\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42060\" class=\"mrow\"><span id=\"MathJax-Span-42061\" class=\"semantics\"><span id=\"MathJax-Span-42062\" class=\"mrow\"><span id=\"MathJax-Span-42063\" class=\"mrow\"><span id=\"MathJax-Span-42064\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42065\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42066\" class=\"mn\">6.0<\/span><span id=\"MathJax-Span-42067\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42068\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=6.0kg,<\/span><\/span>\u00a0what is the tension in the connecting string? The pulley and all surfaces are frictionless.<\/p>\n<p><span id=\"fs-id1165039485639\"><img decoding=\"async\" id=\"26047\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/bf04ac78a612bd5be9c5903ce270c7784153c097\" alt=\"Two blocks, both mass M are connected by a string that passes over a pulley between the blocks. The upper block is on a surface that slopes down and to the right at an angle of 30 degrees to the horizontal. The pulley is attached to the corner at the bottom of the slope, where the surface then bends and goes vertically down. The lower mass hangs straight down. It is not in contact with the surface.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039103753\" class=\"\">\n<section>\n<div id=\"fs-id1165035676708\"><span class=\"os-number\">96<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039337931\">A small space probe is released from a spaceship. The space probe has mass 20.0 kg and contains 90.0 kg of fuel. It starts from rest in deep space, from the origin of a coordinate system based on the spaceship, and burns fuel at the rate of 3.00 kg\/s. The engine provides a constant thrust of 120.0 N. (a) Write an expression for the mass of the space probe as a function of time, between 0 and 30 seconds, assuming that the engine ignites fuel beginning at\u00a0<span id=\"MathJax-Element-1980-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42069\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42070\" class=\"mrow\"><span id=\"MathJax-Span-42071\" class=\"semantics\"><span id=\"MathJax-Span-42072\" class=\"mrow\"><span id=\"MathJax-Span-42073\" class=\"mrow\"><span id=\"MathJax-Span-42074\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42075\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42076\" class=\"mn\">0<\/span><span id=\"MathJax-Span-42077\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0.<\/span><\/span>\u00a0(b) What is the velocity after 15.0 s? (c) What is the position of the space probe after 15.0 s, with initial position at the origin? (d) Write an expression for the position as a function of time, for\u00a0<span id=\"MathJax-Element-1981-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42078\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42079\" class=\"mrow\"><span id=\"MathJax-Span-42080\" class=\"semantics\"><span id=\"MathJax-Span-42081\" class=\"mrow\"><span id=\"MathJax-Span-42082\" class=\"mrow\"><span id=\"MathJax-Span-42083\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42084\" class=\"mo\">&gt;<\/span><span id=\"MathJax-Span-42085\" class=\"mn\">30.0<\/span><span id=\"MathJax-Span-42086\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42087\" class=\"mtext\">s<\/span><span id=\"MathJax-Span-42088\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t&gt;30.0s.<\/span><\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039109864\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039367916\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039109864-solution\">97<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035868296\">A half-full recycling bin has mass 3.0 kg and is pushed up a\u00a0<span id=\"MathJax-Element-1982-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42089\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42090\" class=\"mrow\"><span id=\"MathJax-Span-42091\" class=\"semantics\"><span id=\"MathJax-Span-42092\" class=\"mrow\"><span id=\"MathJax-Span-42093\" class=\"mrow\"><span id=\"MathJax-Span-42094\" class=\"mn\">40.0<\/span><span id=\"MathJax-Span-42095\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">40.0\u00b0<\/span><\/span>\u00a0incline with constant speed under the action of a 26-N force acting up and parallel to the incline. The incline has friction. What magnitude force must act up and parallel to the incline for the bin to move down the incline at constant velocity?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039268406\" class=\"\">\n<section>\n<div id=\"fs-id1165035866354\"><span class=\"os-number\">98<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039353641\">A child has mass 6.0 kg and slides down a\u00a0<span id=\"MathJax-Element-1983-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42096\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42097\" class=\"mrow\"><span id=\"MathJax-Span-42098\" class=\"semantics\"><span id=\"MathJax-Span-42099\" class=\"mrow\"><span id=\"MathJax-Span-42100\" class=\"mrow\"><span id=\"MathJax-Span-42101\" class=\"mn\">35<\/span><span id=\"MathJax-Span-42102\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">35\u00b0<\/span><\/span>\u00a0incline with constant speed under the action of a 34-N force acting up and parallel to the incline. What is the coefficient of kinetic friction between the child and the surface of the incline?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"os-review-additional-problems-container\">\n<h3><span class=\"os-text\">Additional Problems<\/span><\/h3>\n<section id=\"fs-id1165035682534\" class=\"review-additional-problems\">\n<div id=\"fs-id1165039234785\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039219140\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039234785-solution\">99<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036160772\">The two barges shown here are coupled by a cable of negligible mass. The mass of the front barge is\u00a0<span id=\"MathJax-Element-1984-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42103\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42104\" class=\"mrow\"><span id=\"MathJax-Span-42105\" class=\"semantics\"><span id=\"MathJax-Span-42106\" class=\"mrow\"><span id=\"MathJax-Span-42107\" class=\"mrow\"><span id=\"MathJax-Span-42108\" class=\"mn\">2.00<\/span><span id=\"MathJax-Span-42109\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42110\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42111\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42112\" class=\"msup\"><span id=\"MathJax-Span-42113\" class=\"mrow\"><span id=\"MathJax-Span-42114\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42115\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42116\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42117\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.00\u00d7103kg<\/span><\/span>\u00a0and the mass of the rear barge is\u00a0<span id=\"MathJax-Element-1985-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42118\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42119\" class=\"mrow\"><span id=\"MathJax-Span-42120\" class=\"semantics\"><span id=\"MathJax-Span-42121\" class=\"mrow\"><span id=\"MathJax-Span-42122\" class=\"mrow\"><span id=\"MathJax-Span-42123\" class=\"mn\">3.00<\/span><span id=\"MathJax-Span-42124\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42125\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42126\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42127\" class=\"msup\"><span id=\"MathJax-Span-42128\" class=\"mrow\"><span id=\"MathJax-Span-42129\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42130\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42131\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42132\" class=\"mtext\">kg<\/span><span id=\"MathJax-Span-42133\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.00\u00d7103kg.<\/span><\/span>\u00a0A tugboat pulls the front barge with a horizontal force of magnitude\u00a0<span id=\"MathJax-Element-1986-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42134\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42135\" class=\"mrow\"><span id=\"MathJax-Span-42136\" class=\"semantics\"><span id=\"MathJax-Span-42137\" class=\"mrow\"><span id=\"MathJax-Span-42138\" class=\"mrow\"><span id=\"MathJax-Span-42139\" class=\"mn\">20.0<\/span><span id=\"MathJax-Span-42140\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42141\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42142\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42143\" class=\"msup\"><span id=\"MathJax-Span-42144\" class=\"mrow\"><span id=\"MathJax-Span-42145\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42146\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42147\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42148\" class=\"mtext\">N,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">20.0\u00d7103N,<\/span><\/span>\u00a0and the frictional forces of the water on the front and rear barges are\u00a0<span id=\"MathJax-Element-1987-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42149\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42150\" class=\"mrow\"><span id=\"MathJax-Span-42151\" class=\"semantics\"><span id=\"MathJax-Span-42152\" class=\"mrow\"><span id=\"MathJax-Span-42153\" class=\"mrow\"><span id=\"MathJax-Span-42154\" class=\"mn\">8.00<\/span><span id=\"MathJax-Span-42155\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42156\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42157\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42158\" class=\"msup\"><span id=\"MathJax-Span-42159\" class=\"mrow\"><span id=\"MathJax-Span-42160\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42161\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42162\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42163\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">8.00\u00d7103N<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1988-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42164\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42165\" class=\"mrow\"><span id=\"MathJax-Span-42166\" class=\"semantics\"><span id=\"MathJax-Span-42167\" class=\"mrow\"><span id=\"MathJax-Span-42168\" class=\"mrow\"><span id=\"MathJax-Span-42169\" class=\"mn\">10.0<\/span><span id=\"MathJax-Span-42170\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42171\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42172\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42173\" class=\"msup\"><span id=\"MathJax-Span-42174\" class=\"mrow\"><span id=\"MathJax-Span-42175\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42176\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42177\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42178\" class=\"mtext\">N,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">10.0\u00d7103N,<\/span><\/span>respectively. Find the horizontal acceleration of the barges and the tension in the connecting cable.<\/p>\n<p><span id=\"fs-id1165033379354\"><img decoding=\"async\" id=\"58353\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/a5325babf6dbc08087b6e367b47c75ad4bd381bc\" alt=\"An illustration showing a tug boat pulling two barges. The barge directly attached to the tug boat has mass 2.00 times 10 to the third kilograms. The barge at the end,  behind the first barge, has mass 3.00 times 10 to the third kilograms.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036035448\" class=\"\">\n<section>\n<div id=\"fs-id1165038990654\"><span class=\"os-number\">100<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035669137\">If the order of the barges of the preceding exercise is reversed so that the tugboat pulls the\u00a0<span id=\"MathJax-Element-1989-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42179\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42180\" class=\"mrow\"><span id=\"MathJax-Span-42181\" class=\"semantics\"><span id=\"MathJax-Span-42182\" class=\"mrow\"><span id=\"MathJax-Span-42183\" class=\"mrow\"><span id=\"MathJax-Span-42184\" class=\"mn\">3.00<\/span><span id=\"MathJax-Span-42185\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42186\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42187\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42188\" class=\"msup\"><span id=\"MathJax-Span-42189\" class=\"mrow\"><span id=\"MathJax-Span-42190\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42191\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42192\" class=\"mtext\">-kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">3.00\u00d7103-kg<\/span><\/span>\u00a0barge with a force of\u00a0<span id=\"MathJax-Element-1990-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42193\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42194\" class=\"mrow\"><span id=\"MathJax-Span-42195\" class=\"semantics\"><span id=\"MathJax-Span-42196\" class=\"mrow\"><span id=\"MathJax-Span-42197\" class=\"mrow\"><span id=\"MathJax-Span-42198\" class=\"mn\">20.0<\/span><span id=\"MathJax-Span-42199\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42200\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42201\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42202\" class=\"msup\"><span id=\"MathJax-Span-42203\" class=\"mrow\"><span id=\"MathJax-Span-42204\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42205\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42206\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42207\" class=\"mtext\">N<\/span><span id=\"MathJax-Span-42208\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">20.0\u00d7103N,<\/span><\/span>\u00a0what are the acceleration of the barges and the tension in the coupling cable?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039027173\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039293063\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039027173-solution\">101<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039296223\">An object with mass\u00a0<em>m<\/em>\u00a0moves along the\u00a0<em>x<\/em>-axis. Its position at any time is given by\u00a0<span id=\"MathJax-Element-1991-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42209\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42210\" class=\"mrow\"><span id=\"MathJax-Span-42211\" class=\"semantics\"><span id=\"MathJax-Span-42212\" class=\"mrow\"><span id=\"MathJax-Span-42213\" class=\"mrow\"><span id=\"MathJax-Span-42214\" class=\"mi\">x<\/span><span id=\"MathJax-Span-42215\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42216\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42217\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42218\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42219\" class=\"mi\">p<\/span><span id=\"MathJax-Span-42220\" class=\"msup\"><span id=\"MathJax-Span-42221\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42222\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42223\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42224\" class=\"mi\">q<\/span><span id=\"MathJax-Span-42225\" class=\"msup\"><span id=\"MathJax-Span-42226\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42227\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">x(t)=pt3+qt2<\/span><\/span>\u00a0where\u00a0<em>p<\/em>\u00a0and\u00a0<em>q<\/em>\u00a0are constants. Find the net force on this object for any time\u00a0<em>t<\/em>.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039335542\" class=\"\">\n<section>\n<div id=\"fs-id1165035792005\"><span class=\"os-number\">102<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039352836\">A helicopter with mass\u00a0<span id=\"MathJax-Element-1992-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42228\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42229\" class=\"mrow\"><span id=\"MathJax-Span-42230\" class=\"semantics\"><span id=\"MathJax-Span-42231\" class=\"mrow\"><span id=\"MathJax-Span-42232\" class=\"mrow\"><span id=\"MathJax-Span-42233\" class=\"mn\">2.35<\/span><span id=\"MathJax-Span-42234\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42235\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42236\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42237\" class=\"msup\"><span id=\"MathJax-Span-42238\" class=\"mrow\"><span id=\"MathJax-Span-42239\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42240\" class=\"mn\">4<\/span><\/span><span id=\"MathJax-Span-42241\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42242\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.35\u00d7104kg<\/span><\/span>\u00a0has a position given by\u00a0<span id=\"MathJax-Element-1993-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42243\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42244\" class=\"mrow\"><span id=\"MathJax-Span-42245\" class=\"semantics\"><span id=\"MathJax-Span-42246\" class=\"mrow\"><span id=\"MathJax-Span-42247\" class=\"mrow\"><span id=\"MathJax-Span-42248\" class=\"mstyle\"><span id=\"MathJax-Span-42249\" class=\"mrow\"><span id=\"MathJax-Span-42250\" class=\"mover\"><span id=\"MathJax-Span-42251\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42252\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42253\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42254\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42255\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42256\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42257\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42258\" class=\"mn\">0.020<\/span><span id=\"MathJax-Span-42259\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42260\" class=\"msup\"><span id=\"MathJax-Span-42261\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42262\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42263\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42264\" class=\"mstyle\"><span id=\"MathJax-Span-42265\" class=\"mrow\"><span id=\"MathJax-Span-42266\" class=\"mover\"><span id=\"MathJax-Span-42267\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42268\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42269\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42270\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42271\" class=\"mn\">2.2<\/span><span id=\"MathJax-Span-42272\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42273\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42274\" class=\"mstyle\"><span id=\"MathJax-Span-42275\" class=\"mrow\"><span id=\"MathJax-Span-42276\" class=\"mover\"><span id=\"MathJax-Span-42277\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42278\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42279\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-42280\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42281\" class=\"mn\">0.060<\/span><span id=\"MathJax-Span-42282\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42283\" class=\"msup\"><span id=\"MathJax-Span-42284\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42285\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42286\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42287\" class=\"mstyle\"><span id=\"MathJax-Span-42288\" class=\"mrow\"><span id=\"MathJax-Span-42289\" class=\"mover\"><span id=\"MathJax-Span-42290\" class=\"mi\">k<\/span><span id=\"MathJax-Span-42291\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42292\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)=(0.020t3)i^+(2.2t)j^\u2212(0.060t2)k^.<\/span><\/span>\u00a0Find the net force on the helicopter at\u00a0<span id=\"MathJax-Element-1994-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42293\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42294\" class=\"mrow\"><span id=\"MathJax-Span-42295\" class=\"semantics\"><span id=\"MathJax-Span-42296\" class=\"mrow\"><span id=\"MathJax-Span-42297\" class=\"mrow\"><span id=\"MathJax-Span-42298\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42299\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42300\" class=\"mn\">3.0<\/span><span id=\"MathJax-Span-42301\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42302\" class=\"mtext\">s<\/span><span id=\"MathJax-Span-42303\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=3.0s.<\/span><\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039453675\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035980452\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039453675-solution\">103<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035688942\">Located at the origin, an electric car of mass\u00a0<em>m<\/em>\u00a0is at rest and in equilibrium. A time dependent force of\u00a0<span id=\"MathJax-Element-1995-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42304\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42305\" class=\"mrow\"><span id=\"MathJax-Span-42306\" class=\"semantics\"><span id=\"MathJax-Span-42307\" class=\"mrow\"><span id=\"MathJax-Span-42308\" class=\"mrow\"><span id=\"MathJax-Span-42309\" class=\"mstyle\"><span id=\"MathJax-Span-42310\" class=\"mrow\"><span id=\"MathJax-Span-42311\" class=\"mover\"><span id=\"MathJax-Span-42312\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42313\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42314\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42315\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42316\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192(t)<\/span><\/span>\u00a0is applied at time\u00a0<span id=\"MathJax-Element-1996-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42317\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42318\" class=\"mrow\"><span id=\"MathJax-Span-42319\" class=\"semantics\"><span id=\"MathJax-Span-42320\" class=\"mrow\"><span id=\"MathJax-Span-42321\" class=\"mrow\"><span id=\"MathJax-Span-42322\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42323\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42324\" class=\"mn\">0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0<\/span><\/span>, and its components are\u00a0<span id=\"MathJax-Element-1997-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42325\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42326\" class=\"mrow\"><span id=\"MathJax-Span-42327\" class=\"semantics\"><span id=\"MathJax-Span-42328\" class=\"mrow\"><span id=\"MathJax-Span-42329\" class=\"mrow\"><span id=\"MathJax-Span-42330\" class=\"msub\"><span id=\"MathJax-Span-42331\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42332\" class=\"mi\">x<\/span><\/span><span id=\"MathJax-Span-42333\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42334\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42335\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42336\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42337\" class=\"mi\">p<\/span><span id=\"MathJax-Span-42338\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42339\" class=\"mi\">n<\/span><span id=\"MathJax-Span-42340\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fx(t)=p+nt<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-1998-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42341\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42342\" class=\"mrow\"><span id=\"MathJax-Span-42343\" class=\"semantics\"><span id=\"MathJax-Span-42344\" class=\"mrow\"><span id=\"MathJax-Span-42345\" class=\"mrow\"><span id=\"MathJax-Span-42346\" class=\"msub\"><span id=\"MathJax-Span-42347\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42348\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-42349\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42350\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42351\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42352\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42353\" class=\"mi\">q<\/span><span id=\"MathJax-Span-42354\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fy(t)=qt<\/span><\/span>\u00a0where\u00a0<em>p<\/em>,\u00a0<em>q<\/em>, and\u00a0<em>n<\/em>\u00a0are constants. Find the position\u00a0<span id=\"MathJax-Element-1999-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42355\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42356\" class=\"mrow\"><span id=\"MathJax-Span-42357\" class=\"semantics\"><span id=\"MathJax-Span-42358\" class=\"mrow\"><span id=\"MathJax-Span-42359\" class=\"mrow\"><span id=\"MathJax-Span-42360\" class=\"mstyle\"><span id=\"MathJax-Span-42361\" class=\"mrow\"><span id=\"MathJax-Span-42362\" class=\"mover\"><span id=\"MathJax-Span-42363\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42364\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42365\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42366\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42367\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)<\/span><\/span>\u00a0and velocity\u00a0<span id=\"MathJax-Element-2000-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42368\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42369\" class=\"mrow\"><span id=\"MathJax-Span-42370\" class=\"semantics\"><span id=\"MathJax-Span-42371\" class=\"mrow\"><span id=\"MathJax-Span-42372\" class=\"mrow\"><span id=\"MathJax-Span-42373\" class=\"mstyle\"><span id=\"MathJax-Span-42374\" class=\"mrow\"><span id=\"MathJax-Span-42375\" class=\"mover\"><span id=\"MathJax-Span-42376\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42377\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42378\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42379\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42380\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v\u2192(t)<\/span><\/span>\u00a0as functions of time\u00a0<em>t<\/em>.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165033378093\" class=\"\">\n<section>\n<div id=\"fs-id1165039402799\"><span class=\"os-number\">104<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035867149\">A particle of mass\u00a0<em>m<\/em>\u00a0is located at the origin. It is at rest and in equilibrium. A time-dependent force of\u00a0<span id=\"MathJax-Element-2001-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42381\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42382\" class=\"mrow\"><span id=\"MathJax-Span-42383\" class=\"semantics\"><span id=\"MathJax-Span-42384\" class=\"mrow\"><span id=\"MathJax-Span-42385\" class=\"mrow\"><span id=\"MathJax-Span-42386\" class=\"mstyle\"><span id=\"MathJax-Span-42387\" class=\"mrow\"><span id=\"MathJax-Span-42388\" class=\"mover\"><span id=\"MathJax-Span-42389\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42390\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42391\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42392\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42393\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192(t)<\/span><\/span>\u00a0is applied at time\u00a0<span id=\"MathJax-Element-2002-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42394\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42395\" class=\"mrow\"><span id=\"MathJax-Span-42396\" class=\"semantics\"><span id=\"MathJax-Span-42397\" class=\"mrow\"><span id=\"MathJax-Span-42398\" class=\"mrow\"><span id=\"MathJax-Span-42399\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42400\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42401\" class=\"mn\">0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0<\/span><\/span>, and its components are\u00a0<span id=\"MathJax-Element-2003-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42402\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42403\" class=\"mrow\"><span id=\"MathJax-Span-42404\" class=\"semantics\"><span id=\"MathJax-Span-42405\" class=\"mrow\"><span id=\"MathJax-Span-42406\" class=\"mrow\"><span id=\"MathJax-Span-42407\" class=\"msub\"><span id=\"MathJax-Span-42408\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42409\" class=\"mi\">x<\/span><\/span><span id=\"MathJax-Span-42410\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42411\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42412\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42413\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42414\" class=\"mi\">p<\/span><span id=\"MathJax-Span-42415\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fx(t)=pt<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-2004-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42416\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42417\" class=\"mrow\"><span id=\"MathJax-Span-42418\" class=\"semantics\"><span id=\"MathJax-Span-42419\" class=\"mrow\"><span id=\"MathJax-Span-42420\" class=\"mrow\"><span id=\"MathJax-Span-42421\" class=\"msub\"><span id=\"MathJax-Span-42422\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42423\" class=\"mi\">y<\/span><\/span><span id=\"MathJax-Span-42424\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42425\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42426\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42427\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42428\" class=\"mi\">n<\/span><span id=\"MathJax-Span-42429\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42430\" class=\"mi\">q<\/span><span id=\"MathJax-Span-42431\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">Fy(t)=n+qt<\/span><\/span>\u00a0where\u00a0<em>p<\/em>,\u00a0<em>q<\/em>, and\u00a0<em>n<\/em>\u00a0are constants. Find the position\u00a0<span id=\"MathJax-Element-2005-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42432\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42433\" class=\"mrow\"><span id=\"MathJax-Span-42434\" class=\"semantics\"><span id=\"MathJax-Span-42435\" class=\"mrow\"><span id=\"MathJax-Span-42436\" class=\"mrow\"><span id=\"MathJax-Span-42437\" class=\"mstyle\"><span id=\"MathJax-Span-42438\" class=\"mrow\"><span id=\"MathJax-Span-42439\" class=\"mover\"><span id=\"MathJax-Span-42440\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42441\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42442\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42443\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42444\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)<\/span><\/span>\u00a0and velocity\u00a0<span id=\"MathJax-Element-2006-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42445\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42446\" class=\"mrow\"><span id=\"MathJax-Span-42447\" class=\"semantics\"><span id=\"MathJax-Span-42448\" class=\"mrow\"><span id=\"MathJax-Span-42449\" class=\"mrow\"><span id=\"MathJax-Span-42450\" class=\"mstyle\"><span id=\"MathJax-Span-42451\" class=\"mrow\"><span id=\"MathJax-Span-42452\" class=\"mover\"><span id=\"MathJax-Span-42453\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42454\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42455\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42456\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42457\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v\u2192(t)<\/span><\/span>\u00a0as functions of time\u00a0<em>t<\/em>.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035661242\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035639346\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035661242-solution\">105<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035662853\">A 2.0-kg object has a velocity of\u00a0<span id=\"MathJax-Element-2007-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42458\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42459\" class=\"mrow\"><span id=\"MathJax-Span-42460\" class=\"semantics\"><span id=\"MathJax-Span-42461\" class=\"mrow\"><span id=\"MathJax-Span-42462\" class=\"mrow\"><span id=\"MathJax-Span-42463\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42464\" class=\"mstyle\"><span id=\"MathJax-Span-42465\" class=\"mrow\"><span id=\"MathJax-Span-42466\" class=\"mover\"><span id=\"MathJax-Span-42467\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42468\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42469\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42470\" class=\"mtext\">m\/s<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">4.0i^m\/s<\/span><\/span>\u00a0at\u00a0<span id=\"MathJax-Element-2008-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42471\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42472\" class=\"mrow\"><span id=\"MathJax-Span-42473\" class=\"semantics\"><span id=\"MathJax-Span-42474\" class=\"mrow\"><span id=\"MathJax-Span-42475\" class=\"mrow\"><span id=\"MathJax-Span-42476\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42477\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42478\" class=\"mn\">0<\/span><span id=\"MathJax-Span-42479\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0.<\/span><\/span>\u00a0A constant resultant force of\u00a0<span id=\"MathJax-Element-2009-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42480\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42481\" class=\"mrow\"><span id=\"MathJax-Span-42482\" class=\"semantics\"><span id=\"MathJax-Span-42483\" class=\"mrow\"><span id=\"MathJax-Span-42484\" class=\"mrow\"><span id=\"MathJax-Span-42485\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42486\" class=\"mn\">2.0<\/span><span id=\"MathJax-Span-42487\" class=\"mstyle\"><span id=\"MathJax-Span-42488\" class=\"mrow\"><span id=\"MathJax-Span-42489\" class=\"mover\"><span id=\"MathJax-Span-42490\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42491\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42492\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42493\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42494\" class=\"mstyle\"><span id=\"MathJax-Span-42495\" class=\"mrow\"><span id=\"MathJax-Span-42496\" class=\"mover\"><span id=\"MathJax-Span-42497\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42498\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42499\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42500\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42501\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(2.0i^+4.0j^)N<\/span><\/span>\u00a0then acts on the object for 3.0 s. What is the magnitude of the object\u2019s velocity at the end of the 3.0-s interval?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035975760\" class=\"\">\n<section>\n<div id=\"fs-id1165039234284\"><span class=\"os-number\">106<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039323978\">A 1.5-kg mass has an acceleration of\u00a0<span id=\"MathJax-Element-2010-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42502\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42503\" class=\"mrow\"><span id=\"MathJax-Span-42504\" class=\"semantics\"><span id=\"MathJax-Span-42505\" class=\"mrow\"><span id=\"MathJax-Span-42506\" class=\"mrow\"><span id=\"MathJax-Span-42507\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42508\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42509\" class=\"mstyle\"><span id=\"MathJax-Span-42510\" class=\"mrow\"><span id=\"MathJax-Span-42511\" class=\"mover\"><span id=\"MathJax-Span-42512\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42513\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42514\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-42515\" class=\"mn\">3.0<\/span><span id=\"MathJax-Span-42516\" class=\"mstyle\"><span id=\"MathJax-Span-42517\" class=\"mrow\"><span id=\"MathJax-Span-42518\" class=\"mover\"><span id=\"MathJax-Span-42519\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42520\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42521\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42522\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42523\" class=\"msup\"><span id=\"MathJax-Span-42524\" class=\"mrow\"><span id=\"MathJax-Span-42525\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-42526\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42527\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(4.0i^\u22123.0j^)m\/s2.<\/span><\/span>\u00a0Only two forces act on the mass. If one of the forces is\u00a0<span id=\"MathJax-Element-2011-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42528\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42529\" class=\"mrow\"><span id=\"MathJax-Span-42530\" class=\"semantics\"><span id=\"MathJax-Span-42531\" class=\"mrow\"><span id=\"MathJax-Span-42532\" class=\"mrow\"><span id=\"MathJax-Span-42533\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42534\" class=\"mn\">2.0<\/span><span id=\"MathJax-Span-42535\" class=\"mstyle\"><span id=\"MathJax-Span-42536\" class=\"mrow\"><span id=\"MathJax-Span-42537\" class=\"mover\"><span id=\"MathJax-Span-42538\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42539\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42540\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-42541\" class=\"mn\">1.4<\/span><span id=\"MathJax-Span-42542\" class=\"mstyle\"><span id=\"MathJax-Span-42543\" class=\"mrow\"><span id=\"MathJax-Span-42544\" class=\"mover\"><span id=\"MathJax-Span-42545\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42546\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42547\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42548\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42549\" class=\"mtext\">N,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(2.0i^\u22121.4j^)N,<\/span><\/span>\u00a0what is the magnitude of the other force?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035669542\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165036075410\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035669542-solution\">107<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039345662\">A box is dropped onto a conveyor belt moving at 3.4 m\/s. If the coefficient of friction between the box and the belt is 0.27, how long will it take before the box moves without slipping?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039426530\" class=\"\">\n<section>\n<div id=\"fs-id1165033371416\"><span class=\"os-number\">108<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039273532\">Shown below is a 10.0-kg block being pushed by a horizontal force\u00a0<span id=\"MathJax-Element-2012-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42550\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42551\" class=\"mrow\"><span id=\"MathJax-Span-42552\" class=\"semantics\"><span id=\"MathJax-Span-42553\" class=\"mrow\"><span id=\"MathJax-Span-42554\" class=\"mstyle\"><span id=\"MathJax-Span-42555\" class=\"mrow\"><span id=\"MathJax-Span-42556\" class=\"mover\"><span id=\"MathJax-Span-42557\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42558\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F\u2192<\/span><\/span>\u00a0of magnitude 200.0 N. The coefficient of kinetic friction between the two surfaces is 0.50. Find the acceleration of the block.<\/p>\n<p><span id=\"fs-id1165039091495\"><img decoding=\"async\" id=\"98156\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/e4781e956b67a53c0d88ed7057fa766873a7b064\" alt=\"An illustration of a 10.0 kilogram block being pushed into a slope by a horizontal force F. The slope angles up and to the right at an angle of 30 degrees to the horizontal and the force F points to the right.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035865609\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165038974227\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035865609-solution\">109<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036162461\">As shown below, the mass of block 1 is\u00a0<span id=\"MathJax-Element-2013-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42559\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42560\" class=\"mrow\"><span id=\"MathJax-Span-42561\" class=\"semantics\"><span id=\"MathJax-Span-42562\" class=\"mrow\"><span id=\"MathJax-Span-42563\" class=\"mrow\"><span id=\"MathJax-Span-42564\" class=\"msub\"><span id=\"MathJax-Span-42565\" class=\"mi\">m<\/span><span id=\"MathJax-Span-42566\" class=\"mn\">1<\/span><\/span><span id=\"MathJax-Span-42567\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42568\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42569\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42570\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">m1=4.0kg,<\/span><\/span>\u00a0while the mass of block 2 is\u00a0<span id=\"MathJax-Element-2014-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42571\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42572\" class=\"mrow\"><span id=\"MathJax-Span-42573\" class=\"semantics\"><span id=\"MathJax-Span-42574\" class=\"mrow\"><span id=\"MathJax-Span-42575\" class=\"mrow\"><span id=\"MathJax-Span-42576\" class=\"msub\"><span id=\"MathJax-Span-42577\" class=\"mi\">m<\/span><span id=\"MathJax-Span-42578\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42579\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42580\" class=\"mn\">8.0<\/span><span id=\"MathJax-Span-42581\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42582\" class=\"mtext\">kg<\/span><span id=\"MathJax-Span-42583\" class=\"mtext\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">m2=8.0kg.<\/span><\/span>\u00a0The coefficient of friction between\u00a0<span id=\"MathJax-Element-2015-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42584\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42585\" class=\"mrow\"><span id=\"MathJax-Span-42586\" class=\"semantics\"><span id=\"MathJax-Span-42587\" class=\"mrow\"><span id=\"MathJax-Span-42588\" class=\"mrow\"><span id=\"MathJax-Span-42589\" class=\"msub\"><span id=\"MathJax-Span-42590\" class=\"mi\">m<\/span><span id=\"MathJax-Span-42591\" class=\"mn\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">m1<\/span><\/span>\u00a0and the inclined surface is\u00a0<span id=\"MathJax-Element-2016-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42592\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42593\" class=\"mrow\"><span id=\"MathJax-Span-42594\" class=\"semantics\"><span id=\"MathJax-Span-42595\" class=\"mrow\"><span id=\"MathJax-Span-42596\" class=\"mrow\"><span id=\"MathJax-Span-42597\" class=\"msub\"><span id=\"MathJax-Span-42598\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-42599\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-42600\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42601\" class=\"mn\">0.40<\/span><span id=\"MathJax-Span-42602\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bck=0.40.<\/span><\/span>\u00a0What is the acceleration of the system?<\/p>\n<p><span id=\"fs-id1165035730256\"><img decoding=\"async\" id=\"38422\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/5fc65f8faf919d36f9b0e584c5d6e311986f76a4\" alt=\"Block 1 is on a ramp inclined up and to the right at an angle of 37 degrees above the horizontal. It is connected to a string that passes over a pulley at the top of the ramp, then hangs straight down and connects to  block 2. Block 2 is not in contact with the ramp.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039114252\" class=\"\">\n<section>\n<div id=\"fs-id1165033370832\"><span class=\"os-number\">110<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035946630\">A student is attempting to move a 30-kg mini-fridge into her dorm room. During a moment of inattention, the mini-fridge slides down a 35 degree incline at constant speed when she applies a force of 25 N acting up and parallel to the incline. What is the coefficient of kinetic friction between the fridge and the surface of the incline?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039077028\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039398265\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039077028-solution\">111<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035640986\">A crate of mass 100.0 kg rests on a rough surface inclined at an angle of\u00a0<span id=\"MathJax-Element-2017-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42603\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42604\" class=\"mrow\"><span id=\"MathJax-Span-42605\" class=\"semantics\"><span id=\"MathJax-Span-42606\" class=\"mrow\"><span id=\"MathJax-Span-42607\" class=\"mrow\"><span id=\"MathJax-Span-42608\" class=\"mn\">37.0<\/span><span id=\"MathJax-Span-42609\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">37.0\u00b0<\/span><\/span>\u00a0with the horizontal. A massless rope to which a force can be applied parallel to the surface is attached to the crate and leads to the top of the incline. In its present state, the crate is just ready to slip and start to move down the plane. The coefficient of friction is\u00a0<span id=\"MathJax-Element-2018-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42610\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42611\" class=\"mrow\"><span id=\"MathJax-Span-42612\" class=\"semantics\"><span id=\"MathJax-Span-42613\" class=\"mrow\"><span id=\"MathJax-Span-42614\" class=\"mrow\"><span id=\"MathJax-Span-42615\" class=\"mn\">80<\/span><span id=\"MathJax-Span-42616\" class=\"mi\">%<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">80%<\/span><\/span>\u00a0of that for the static case. (a) What is the coefficient of static friction? (b) What is the maximum force that can be applied upward along the plane on the rope and not move the block? (c) With a slightly greater applied force, the block will slide up the plane. Once it begins to move, what is its acceleration and what reduced force is necessary to keep it moving upward at constant speed? (d) If the block is given a slight nudge to get it started down the plane, what will be its acceleration in that direction? (e) Once the block begins to slide downward, what upward force on the rope is required to keep the block from accelerating downward?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039470299\" class=\"\">\n<section>\n<div id=\"fs-id1165039297163\"><span class=\"os-number\">112<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039079533\">A car is moving at high speed along a highway when the driver makes an emergency braking. The wheels become locked (stop rolling), and the resulting skid marks are 32.0 meters long. If the coefficient of kinetic friction between tires and road is 0.550, and the acceleration was constant during braking, how fast was the car going when the wheels became locked?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039446833\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035610527\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039446833-solution\">113<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039376043\">A crate having mass 50.0 kg falls horizontally off the back of the flatbed truck, which is traveling at 100 km\/h. Find the value of the coefficient of kinetic friction between the road and crate if the crate slides 50 m on the road in coming to rest. The initial speed of the crate is the same as the truck, 100 km\/h.<\/p>\n<p><span id=\"fs-id1165036158315\"><img decoding=\"async\" id=\"33868\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/49bf81ed6f402b3d4be4e95fab22f4fab9ce3ade\" alt=\"The figure shows a truck moving to the right at 100 kilometers per hour and a 50 kilogram crate on the ground behind the truck.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039111122\" class=\"\">\n<section>\n<div id=\"fs-id1165039074000\"><span class=\"os-number\">114<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039392702\">A 15-kg sled is pulled across a horizontal, snow-covered surface by a force applied to a rope at 30 degrees with the horizontal. The coefficient of kinetic friction between the sled and the snow is 0.20. (a) If the force is 33 N, what is the horizontal acceleration of the sled? (b) What must the force be in order to pull the sled at constant velocity?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039446792\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165036146754\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039446792-solution\">115<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039422312\">A 30.0-g ball at the end of a string is swung in a vertical circle with a radius of 25.0 cm. The rotational velocity is 200.0 cm\/s. Find the tension in the string: (a) at the top of the circle, (b) at the bottom of the circle, and (c) at a distance of 12.5 cm from the center of the circle\u00a0<span id=\"MathJax-Element-2019-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42617\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42618\" class=\"mrow\"><span id=\"MathJax-Span-42619\" class=\"semantics\"><span id=\"MathJax-Span-42620\" class=\"mrow\"><span id=\"MathJax-Span-42621\" class=\"mrow\"><span id=\"MathJax-Span-42622\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42623\" class=\"mrow\"><span id=\"MathJax-Span-42624\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42625\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42626\" class=\"mn\">12.5<\/span><span id=\"MathJax-Span-42627\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42628\" class=\"mtext\">cm<\/span><\/span><span id=\"MathJax-Span-42629\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42630\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(r=12.5cm).<\/span><\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035619639\" class=\"\">\n<section>\n<div id=\"fs-id1165036154120\"><span class=\"os-number\">116<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039003350\">A particle of mass 0.50 kg starts moves through a circular path in the\u00a0<em>xy<\/em>-plane with a position given by\u00a0<span id=\"MathJax-Element-2020-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42631\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42632\" class=\"mrow\"><span id=\"MathJax-Span-42633\" class=\"semantics\"><span id=\"MathJax-Span-42634\" class=\"mrow\"><span id=\"MathJax-Span-42635\" class=\"mrow\"><span id=\"MathJax-Span-42636\" class=\"mstyle\"><span id=\"MathJax-Span-42637\" class=\"mrow\"><span id=\"MathJax-Span-42638\" class=\"mover\"><span id=\"MathJax-Span-42639\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42640\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42641\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42642\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42643\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42644\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42645\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42646\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42647\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42648\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-42649\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42650\" class=\"mn\">3<\/span><span id=\"MathJax-Span-42651\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42652\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42653\" class=\"mstyle\"><span id=\"MathJax-Span-42654\" class=\"mrow\"><span id=\"MathJax-Span-42655\" class=\"mover\"><span id=\"MathJax-Span-42656\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42657\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42658\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42659\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42660\" class=\"mn\">4.0<\/span><span id=\"MathJax-Span-42661\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42662\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-42663\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42664\" class=\"mn\">3<\/span><span id=\"MathJax-Span-42665\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42666\" class=\"mo\">)<\/span><span id=\"MathJax-Span-42667\" class=\"mstyle\"><span id=\"MathJax-Span-42668\" class=\"mrow\"><span id=\"MathJax-Span-42669\" class=\"mover\"><span id=\"MathJax-Span-42670\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42671\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)=(4.0cos3t)i^+(4.0sin3t)j^<\/span><\/span>\u00a0where\u00a0<em>r<\/em>\u00a0is in meters and\u00a0<em>t<\/em>\u00a0is in seconds. (a) Find the velocity and acceleration vectors as functions of time. (b) Show that the acceleration vector always points toward the center of the circle (and thus represents centripetal acceleration). (c) Find the centripetal force vector as a function of time.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039274986\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039371171\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039274986-solution\">117<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035869114\">A stunt cyclist rides on the interior of a cylinder 12 m in radius. The coefficient of static friction between the tires and the wall is 0.68. Find the value of the minimum speed for the cyclist to perform the stunt.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035662464\" class=\"\">\n<section>\n<div id=\"fs-id1165039401204\"><span class=\"os-number\">118<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039046569\">When a body of mass 0.25 kg is attached to a vertical massless spring, it is extended 5.0 cm from its unstretched length of 4.0 cm. The body and spring are placed on a horizontal frictionless surface and rotated about the held end of the spring at 2.0 rev\/s. How far is the spring stretched?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035654187\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165038993163\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035654187-solution\">119<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039297494\">Railroad tracks follow a circular curve of radius 500.0 m and are banked at an angle of\u00a0<span id=\"MathJax-Element-2021-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42672\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42673\" class=\"mrow\"><span id=\"MathJax-Span-42674\" class=\"semantics\"><span id=\"MathJax-Span-42675\" class=\"mrow\"><span id=\"MathJax-Span-42676\" class=\"mrow\"><span id=\"MathJax-Span-42677\" class=\"mn\">5.00<\/span><span id=\"MathJax-Span-42678\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">5.00\u00b0<\/span><\/span>. For trains of what speed are these tracks designed?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039317700\" class=\"\">\n<section>\n<div id=\"fs-id1165039297475\"><span class=\"os-number\">120<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035733788\">A plumb bob hangs from the roof of a railroad car. The car rounds a circular track of radius 300.0 m at a speed of 90.0 km\/h. At what angle relative to the vertical does the plumb bob hang?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036081646\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039300224\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036081646-solution\">121<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035870165\">An airplane flies at 120.0 m\/s and banks at a\u00a0<span id=\"MathJax-Element-2022-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42679\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42680\" class=\"mrow\"><span id=\"MathJax-Span-42681\" class=\"semantics\"><span id=\"MathJax-Span-42682\" class=\"mrow\"><span id=\"MathJax-Span-42683\" class=\"mrow\"><span id=\"MathJax-Span-42684\" class=\"mn\">30<\/span><span id=\"MathJax-Span-42685\" class=\"mtext\">\u00b0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">30\u00b0<\/span><\/span>\u00a0angle. If its mass is\u00a0<span id=\"MathJax-Element-2023-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42686\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42687\" class=\"mrow\"><span id=\"MathJax-Span-42688\" class=\"semantics\"><span id=\"MathJax-Span-42689\" class=\"mrow\"><span id=\"MathJax-Span-42690\" class=\"mrow\"><span id=\"MathJax-Span-42691\" class=\"mn\">2.50<\/span><span id=\"MathJax-Span-42692\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42693\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-42694\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42695\" class=\"msup\"><span id=\"MathJax-Span-42696\" class=\"mrow\"><span id=\"MathJax-Span-42697\" class=\"mn\">10<\/span><\/span><span id=\"MathJax-Span-42698\" class=\"mn\">3<\/span><\/span><span id=\"MathJax-Span-42699\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42700\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.50\u00d7103kg,<\/span><\/span>\u00a0(a) what is the magnitude of the lift force? (b) what is the radius of the turn?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039443997\" class=\"\">\n<section>\n<div id=\"fs-id1165035633482\"><span class=\"os-number\">122<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036147875\">The position of a particle is given by\u00a0<span id=\"MathJax-Element-2024-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42701\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42702\" class=\"mrow\"><span id=\"MathJax-Span-42703\" class=\"semantics\"><span id=\"MathJax-Span-42704\" class=\"mrow\"><span id=\"MathJax-Span-42705\" class=\"mrow\"><span id=\"MathJax-Span-42706\" class=\"mstyle\"><span id=\"MathJax-Span-42707\" class=\"mrow\"><span id=\"MathJax-Span-42708\" class=\"mover\"><span id=\"MathJax-Span-42709\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42710\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42711\" class=\"mrow\"><span id=\"MathJax-Span-42712\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42713\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42714\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42715\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42716\" class=\"mi\">A<\/span><span id=\"MathJax-Span-42717\" class=\"mrow\"><span id=\"MathJax-Span-42718\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42719\" class=\"mrow\"><span id=\"MathJax-Span-42720\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-42721\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42722\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42723\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42724\" class=\"mstyle\"><span id=\"MathJax-Span-42725\" class=\"mrow\"><span id=\"MathJax-Span-42726\" class=\"mover\"><span id=\"MathJax-Span-42727\" class=\"mi\">i<\/span><span id=\"MathJax-Span-42728\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42729\" class=\"mo\">+<\/span><span id=\"MathJax-Span-42730\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-42731\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42732\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42733\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42734\" class=\"mstyle\"><span id=\"MathJax-Span-42735\" class=\"mrow\"><span id=\"MathJax-Span-42736\" class=\"mover\"><span id=\"MathJax-Span-42737\" class=\"mi\">j<\/span><span id=\"MathJax-Span-42738\" class=\"mo\">\u02c6<\/span><\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42739\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42740\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r\u2192(t)=A(cos\u03c9ti^+sin\u03c9tj^),<\/span><\/span>\u00a0where\u00a0<span id=\"MathJax-Element-2025-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42741\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42742\" class=\"mrow\"><span id=\"MathJax-Span-42743\" class=\"semantics\"><span id=\"MathJax-Span-42744\" class=\"mrow\"><span id=\"MathJax-Span-42745\" class=\"mi\">\u03c9<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03c9<\/span><\/span>\u00a0is a constant. (a) Show that the particle moves in a circle of radius\u00a0<em>A<\/em>. (b) Calculate\u00a0<span id=\"MathJax-Element-2026-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42746\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42747\" class=\"mrow\"><span id=\"MathJax-Span-42748\" class=\"semantics\"><span id=\"MathJax-Span-42749\" class=\"mrow\"><span id=\"MathJax-Span-42750\" class=\"mrow\"><span id=\"MathJax-Span-42751\" class=\"mrow\"><span id=\"MathJax-Span-42752\" class=\"mrow\"><span id=\"MathJax-Span-42753\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42754\" class=\"mstyle\"><span id=\"MathJax-Span-42755\" class=\"mrow\"><span id=\"MathJax-Span-42756\" class=\"mover\"><span id=\"MathJax-Span-42757\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42758\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42759\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-42760\" class=\"mrow\"><span id=\"MathJax-Span-42761\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42762\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">dr\u2192\/dt<\/span><\/span>\u00a0and then show that the speed of the particle is a constant\u00a0<span id=\"MathJax-Element-2027-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42763\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42764\" class=\"mrow\"><span id=\"MathJax-Span-42765\" class=\"semantics\"><span id=\"MathJax-Span-42766\" class=\"mrow\"><span id=\"MathJax-Span-42767\" class=\"mrow\"><span id=\"MathJax-Span-42768\" class=\"msub\"><span id=\"MathJax-Span-42769\" class=\"mi\">A<\/span><span id=\"MathJax-Span-42770\" class=\"mi\">\u03c9<\/span><\/span><span id=\"MathJax-Span-42771\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">A\u03c9.<\/span><\/span>\u00a0(c) Determine\u00a0<span id=\"MathJax-Element-2028-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42772\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42773\" class=\"mrow\"><span id=\"MathJax-Span-42774\" class=\"semantics\"><span id=\"MathJax-Span-42775\" class=\"mrow\"><span id=\"MathJax-Span-42776\" class=\"mrow\"><span id=\"MathJax-Span-42777\" class=\"mrow\"><span id=\"MathJax-Span-42778\" class=\"mrow\"><span id=\"MathJax-Span-42779\" class=\"msup\"><span id=\"MathJax-Span-42780\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42781\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42782\" class=\"mstyle\"><span id=\"MathJax-Span-42783\" class=\"mrow\"><span id=\"MathJax-Span-42784\" class=\"mover\"><span id=\"MathJax-Span-42785\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42786\" class=\"mo\">\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42787\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-42788\" class=\"mrow\"><span id=\"MathJax-Span-42789\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42790\" class=\"msup\"><span id=\"MathJax-Span-42791\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42792\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">d2r\u2192\/dt2<\/span><\/span>\u00a0and show that\u00a0<em>a<\/em>\u00a0is given by<span id=\"MathJax-Element-2029-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42793\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42794\" class=\"mrow\"><span id=\"MathJax-Span-42795\" class=\"semantics\"><span id=\"MathJax-Span-42796\" class=\"mrow\"><span id=\"MathJax-Span-42797\" class=\"mrow\"><span id=\"MathJax-Span-42798\" class=\"msub\"><span id=\"MathJax-Span-42799\" class=\"mi\">a<\/span><span id=\"MathJax-Span-42800\" class=\"mtext\">c<\/span><\/span><span id=\"MathJax-Span-42801\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42802\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42803\" class=\"msup\"><span id=\"MathJax-Span-42804\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42805\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-42806\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">ac=r\u03c92.<\/span><\/span>\u00a0(d) Calculate the centripetal force on the particle. [<em>Hint<\/em>: For (b) and (c), you will need to use\u00a0<span id=\"MathJax-Element-2030-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42807\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42808\" class=\"mrow\"><span id=\"MathJax-Span-42809\" class=\"semantics\"><span id=\"MathJax-Span-42810\" class=\"mrow\"><span id=\"MathJax-Span-42811\" class=\"mrow\"><span id=\"MathJax-Span-42812\" class=\"mrow\"><span id=\"MathJax-Span-42813\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42814\" class=\"mrow\"><span id=\"MathJax-Span-42815\" class=\"mrow\"><span id=\"MathJax-Span-42816\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42817\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-42818\" class=\"mrow\"><span id=\"MathJax-Span-42819\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42820\" class=\"mi\">t<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42821\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42822\" class=\"mrow\"><span id=\"MathJax-Span-42823\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42824\" class=\"mrow\"><span id=\"MathJax-Span-42825\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-42826\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42827\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42828\" class=\"mi\">t<\/span><\/span><span id=\"MathJax-Span-42829\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42830\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42831\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-42832\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42833\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42834\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-42835\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42836\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42837\" class=\"mi\">t<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(d\/dt)(cos\u03c9t)=\u2212\u03c9sin\u03c9t<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-2031-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42838\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42839\" class=\"mrow\"><span id=\"MathJax-Span-42840\" class=\"semantics\"><span id=\"MathJax-Span-42841\" class=\"mrow\"><span id=\"MathJax-Span-42842\" class=\"mrow\"><span id=\"MathJax-Span-42843\" class=\"mrow\"><span id=\"MathJax-Span-42844\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42845\" class=\"mrow\"><span id=\"MathJax-Span-42846\" class=\"mrow\"><span id=\"MathJax-Span-42847\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42848\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-42849\" class=\"mrow\"><span id=\"MathJax-Span-42850\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42851\" class=\"mi\">t<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-42852\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42853\" class=\"mrow\"><span id=\"MathJax-Span-42854\" class=\"mo\">(<\/span><span id=\"MathJax-Span-42855\" class=\"mrow\"><span id=\"MathJax-Span-42856\" class=\"mtext\">sin<\/span><span id=\"MathJax-Span-42857\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42858\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42859\" class=\"mi\">t<\/span><\/span><span id=\"MathJax-Span-42860\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-42861\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42862\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42863\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42864\" class=\"mtext\">cos<\/span><span id=\"MathJax-Span-42865\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42866\" class=\"mi\">\u03c9<\/span><span id=\"MathJax-Span-42867\" class=\"mi\">t<\/span><span id=\"MathJax-Span-42868\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">(d\/dt)(sin\u03c9t)=\u03c9cos\u03c9t.<\/span><\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039504554\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035635058\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039504554-solution\">123<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039073034\">Two blocks connected by a string are pulled across a horizontal surface by a force applied to one of the blocks, as shown below. The coefficient of kinetic friction between the blocks and the surface is 0.25. If each block has an acceleration of\u00a0<span id=\"MathJax-Element-2032-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42869\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42870\" class=\"mrow\"><span id=\"MathJax-Span-42871\" class=\"semantics\"><span id=\"MathJax-Span-42872\" class=\"mrow\"><span id=\"MathJax-Span-42873\" class=\"mrow\"><span id=\"MathJax-Span-42874\" class=\"mn\">2.0<\/span><span id=\"MathJax-Span-42875\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42876\" class=\"msup\"><span id=\"MathJax-Span-42877\" class=\"mrow\"><span id=\"MathJax-Span-42878\" class=\"mtext\">m\/s<\/span><\/span><span id=\"MathJax-Span-42879\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">2.0m\/s2<\/span><\/span>\u00a0to the right, what is the magnitude\u00a0<em>F<\/em>\u00a0of the applied force?<\/p>\n<p><span id=\"fs-id1165035715506\"><img decoding=\"async\" id=\"6962\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/c78d2dc779dd8110971b21760e2a152a36e2e166\" alt=\"Two blocks, 1.0 kilograms on the left and 3.0 kilograms on the right, are connected by a string and are on a horizontal surface. Force F acts on the 3.0 kilogram mass and points up and to the right at a angle of 60 degrees above the horizontal.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039479037\" class=\"\">\n<section>\n<div id=\"fs-id1165035646533\"><span class=\"os-number\">124<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035646536\">As shown below, the coefficient of kinetic friction between the surface and the larger block is 0.20, and the coefficient of kinetic friction between the surface and the smaller block is 0.30. If\u00a0<span id=\"MathJax-Element-2033-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42880\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42881\" class=\"mrow\"><span id=\"MathJax-Span-42882\" class=\"semantics\"><span id=\"MathJax-Span-42883\" class=\"mrow\"><span id=\"MathJax-Span-42884\" class=\"mrow\"><span id=\"MathJax-Span-42885\" class=\"mi\">F<\/span><span id=\"MathJax-Span-42886\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42887\" class=\"mn\">10<\/span><span id=\"MathJax-Span-42888\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42889\" class=\"mtext\">N<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">F=10N<\/span><\/span>\u00a0and\u00a0<span id=\"MathJax-Element-2034-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42890\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42891\" class=\"mrow\"><span id=\"MathJax-Span-42892\" class=\"semantics\"><span id=\"MathJax-Span-42893\" class=\"mrow\"><span id=\"MathJax-Span-42894\" class=\"mrow\"><span id=\"MathJax-Span-42895\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42896\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42897\" class=\"mn\">1.0<\/span><span id=\"MathJax-Span-42898\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42899\" class=\"mtext\">kg<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=1.0kg<\/span><\/span>, what is the tension in the connecting string?<\/p>\n<p><span id=\"fs-id1165039351827\"><img decoding=\"async\" id=\"62415\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/a503dc53b79ac38ef0f48b88d87623571f75b4e8\" alt=\"Two blocks, 2 M on the left and M on the right, are connected by a string and are on a horizontal surface. Force F acts on M and points to the right.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165036007610\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035664399\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165036007610-solution\">125<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035664401\">In the figure, the coefficient of kinetic friction between the surface and the blocks is\u00a0<span id=\"MathJax-Element-2035-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42900\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42901\" class=\"mrow\"><span id=\"MathJax-Span-42902\" class=\"semantics\"><span id=\"MathJax-Span-42903\" class=\"mrow\"><span id=\"MathJax-Span-42904\" class=\"mrow\"><span id=\"MathJax-Span-42905\" class=\"msub\"><span id=\"MathJax-Span-42906\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-42907\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-42908\" class=\"mo\">.<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bck.<\/span><\/span>\u00a0If\u00a0<span id=\"MathJax-Element-2036-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42909\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42910\" class=\"mrow\"><span id=\"MathJax-Span-42911\" class=\"semantics\"><span id=\"MathJax-Span-42912\" class=\"mrow\"><span id=\"MathJax-Span-42913\" class=\"mrow\"><span id=\"MathJax-Span-42914\" class=\"mi\">M<\/span><span id=\"MathJax-Span-42915\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42916\" class=\"mn\">1.0<\/span><span id=\"MathJax-Span-42917\" class=\"mspace\"><\/span><span id=\"MathJax-Span-42918\" class=\"mtext\">kg,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">M=1.0kg,<\/span><\/span>\u00a0find an expression for the magnitude of the acceleration of either block (in terms of\u00a0<em>F<\/em>,\u00a0<span id=\"MathJax-Element-2037-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42919\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42920\" class=\"mrow\"><span id=\"MathJax-Span-42921\" class=\"semantics\"><span id=\"MathJax-Span-42922\" class=\"mrow\"><span id=\"MathJax-Span-42923\" class=\"mrow\"><span id=\"MathJax-Span-42924\" class=\"msub\"><span id=\"MathJax-Span-42925\" class=\"mi\">\u03bc<\/span><span id=\"MathJax-Span-42926\" class=\"mtext\">k<\/span><\/span><span id=\"MathJax-Span-42927\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bck,<\/span><\/span>\u00a0and\u00a0<em>g<\/em>).<\/p>\n<p><span id=\"fs-id1165035748378\"><img decoding=\"async\" id=\"53635\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/58bc9a491f5e85eeb1c061f73130320572ec1165\" alt=\"Two blocks, M on the left and 3 M on the right, are connected by a string and are on a horizontal surface. The following forces are indicated: f sub k 2 acting on M and pointing to the right, f sub k 1 acting on 3 M and pointing to the right, F acting on 3 M and pointing to the left, N sub 2 acting on M and pointing up, N sub 1 acting on 3 M and pointing up, M g acting on M and pointing down, , 3 M g acting on 3 M and pointing down.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035662213\" class=\"\">\n<section>\n<div id=\"fs-id1165039209290\"><span class=\"os-number\">126<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039209292\">Two blocks are stacked as shown below, and rest on a frictionless surface. There is friction between the two blocks (coefficient of friction\u00a0<span id=\"MathJax-Element-2038-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42928\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42929\" class=\"mrow\"><span id=\"MathJax-Span-42930\" class=\"semantics\"><span id=\"MathJax-Span-42931\" class=\"mrow\"><span id=\"MathJax-Span-42932\" class=\"mi\">\u03bc<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03bc<\/span><\/span>). An external force is applied to the top block at an angle\u00a0<span id=\"MathJax-Element-2039-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42933\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42934\" class=\"mrow\"><span id=\"MathJax-Span-42935\" class=\"semantics\"><span id=\"MathJax-Span-42936\" class=\"mrow\"><span id=\"MathJax-Span-42937\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0with the horizontal. What is the maximum force\u00a0<em>F<\/em>\u00a0that can be applied for the two blocks to move together?<\/p>\n<p><span id=\"fs-id1165035758924\"><img decoding=\"async\" id=\"48921\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/713b7cb5d8a0235c6ef30aef5f07c19e68d9106d\" alt=\"Rectangular block M sub 2 is on a horizontal surface. Rectangular block M sub 1 is on top of block M sub 2. A force F pushes on block M sub 1. Force F is directed down and to the right, at a angle theta to the horizontal.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039496071\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039443360\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039496071-solution\">127<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039433646\">A box rests on the (horizontal) back of a truck. The coefficient of static friction between the box and the surface on which it rests is 0.24. What maximum distance can the truck travel (starting from rest and moving horizontally with constant acceleration) in 3.0 s without having the box slide?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039232615\" class=\"\">\n<section>\n<div id=\"fs-id1165039257182\"><span class=\"os-number\">128<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039070262\">A double-incline plane is shown below. The coefficient of friction on the left surface is 0.30, and on the right surface 0.16. Calculate the acceleration of the system.<\/p>\n<\/div>\n<div><span id=\"fs-id1165035627255\"><img decoding=\"async\" id=\"32363\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/bc9f831a81c4a58a2915254a9632088e78bfa1b0\" alt=\"Two carts connected by a string passing over a pulley are on either side of a double inclined plane. The string passes over a pulley attached to the top of the double incline. On the left, the incline makes an angle of 37 degrees with the horizontal and the cart on that side has mass 10 kilograms. On the right, the incline makes an angle of 53 degrees with the horizontal and the cart on that side has mass 15 kilograms.\" \/><\/span><\/div>\n<div><\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"os-review-challenge-container\">\n<h3><span class=\"os-text\">Challenge Problems<\/span><\/h3>\n<section id=\"fs-id1165039325516\" class=\"review-challenge\">\n<div id=\"fs-id1165039026859\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035723079\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039026859-solution\">129<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165036007334\">In a later chapter, you will find that the weight of a particle varies with altitude such that\u00a0<span id=\"MathJax-Element-2040-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42938\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42939\" class=\"mrow\"><span id=\"MathJax-Span-42940\" class=\"semantics\"><span id=\"MathJax-Span-42941\" class=\"mrow\"><span id=\"MathJax-Span-42942\" class=\"mrow\"><span id=\"MathJax-Span-42943\" class=\"mi\">w<\/span><span id=\"MathJax-Span-42944\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42945\" class=\"mfrac\"><span id=\"MathJax-Span-42946\" class=\"mrow\"><span id=\"MathJax-Span-42947\" class=\"mi\">m<\/span><span id=\"MathJax-Span-42948\" class=\"mi\">g<\/span><span id=\"MathJax-Span-42949\" class=\"msub\"><span id=\"MathJax-Span-42950\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42951\" class=\"mn\">0<\/span><\/span><span id=\"MathJax-Span-42952\" class=\"msup\"><span id=\"MathJax-Span-42953\" class=\"mrow\"><\/span><span id=\"MathJax-Span-42954\" class=\"mn\">2<\/span><\/span><\/span><span id=\"MathJax-Span-42955\" class=\"mrow\"><span id=\"MathJax-Span-42956\" class=\"msup\"><span id=\"MathJax-Span-42957\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42958\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">w=mgr02r2<\/span><\/span>\u00a0where\u00a0<span id=\"MathJax-Element-2041-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42959\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42960\" class=\"mrow\"><span id=\"MathJax-Span-42961\" class=\"semantics\"><span id=\"MathJax-Span-42962\" class=\"mrow\"><span id=\"MathJax-Span-42963\" class=\"mrow\"><span id=\"MathJax-Span-42964\" class=\"msub\"><span id=\"MathJax-Span-42965\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42966\" class=\"mn\">0<\/span><\/span><span id=\"MathJax-Span-42967\" class=\"msup\"><span id=\"MathJax-Span-42968\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">r0<\/span><\/span>\u00a0is the radius of Earth and\u00a0<em>r<\/em>\u00a0is the distance from Earth\u2019s center. If the particle is fired vertically with velocity\u00a0<span id=\"MathJax-Element-2042-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42969\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42970\" class=\"mrow\"><span id=\"MathJax-Span-42971\" class=\"semantics\"><span id=\"MathJax-Span-42972\" class=\"mrow\"><span id=\"MathJax-Span-42973\" class=\"mrow\"><span id=\"MathJax-Span-42974\" class=\"msub\"><span id=\"MathJax-Span-42975\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42976\" class=\"mn\">0<\/span><\/span><span id=\"MathJax-Span-42977\" class=\"msup\"><span id=\"MathJax-Span-42978\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v0<\/span><\/span>\u00a0from Earth\u2019s surface, determine its velocity as a function of position\u00a0<em>r<\/em>. (<em>Hint:<\/em>\u00a0use\u00a0<span id=\"MathJax-Element-2043-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42979\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42980\" class=\"mrow\"><span id=\"MathJax-Span-42981\" class=\"semantics\"><span id=\"MathJax-Span-42982\" class=\"mrow\"><span id=\"MathJax-Span-42983\" class=\"mrow\"><span id=\"MathJax-Span-42984\" class=\"msup\"><span id=\"MathJax-Span-42985\" class=\"mi\">a<\/span><\/span><span id=\"MathJax-Span-42986\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42987\" class=\"mi\">r<\/span><span id=\"MathJax-Span-42988\" class=\"mo\">=<\/span><span id=\"MathJax-Span-42989\" class=\"msup\"><span id=\"MathJax-Span-42990\" class=\"mi\">v<\/span><\/span><span id=\"MathJax-Span-42991\" class=\"mi\">d<\/span><span id=\"MathJax-Span-42992\" class=\"mi\">v<\/span><span id=\"MathJax-Span-42993\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">adr=vdv,<\/span><\/span>\u00a0the rearrangement mentioned in the text.)<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039315132\" class=\"\">\n<section>\n<div id=\"fs-id1165039315134\"><span class=\"os-number\">130<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035926222\">A large centrifuge, like the one shown below, is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries. (a) At what angular velocity is the centripetal acceleration 10<em>g<\/em>\u00a0if the rider is 15.0 m from the center of rotation? (b) The rider\u2019s cage hangs on a pivot at the end of the arm, allowing it to swing outward during rotation as shown in the bottom accompanying figure. At what angle\u00a0<span id=\"MathJax-Element-2044-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42994\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-42995\" class=\"mrow\"><span id=\"MathJax-Span-42996\" class=\"semantics\"><span id=\"MathJax-Span-42997\" class=\"mrow\"><span id=\"MathJax-Span-42998\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0below the horizontal will the cage hang when the centripetal acceleration is 10<em>g<\/em>? (<em>Hint:<\/em>\u00a0The arm supplies centripetal force and supports the weight of the cage. Draw a free-body diagram of the forces to see what the angle\u00a0<span id=\"MathJax-Element-2045-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-42999\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43000\" class=\"mrow\"><span id=\"MathJax-Span-43001\" class=\"semantics\"><span id=\"MathJax-Span-43002\" class=\"mrow\"><span id=\"MathJax-Span-43003\" class=\"mi\">\u03b8<\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">\u03b8<\/span><\/span>\u00a0should be.)<\/p>\n<p><span id=\"fs-id1165039487271\"><img decoding=\"async\" id=\"64783\" class=\"aligncenter\" src=\"https:\/\/cnx.org\/resources\/94622e0aba8bdb935d2dde3c8cc055dec14c39bb\" alt=\"(a) A photograph of a high g training centrifuge. The astronaut sits in a cage at the end of a long arm that rotates in a horizontal plane. (b) An illustration of a top view of the centrifuge along with an illustration of the forces. The free body diagram shows the weight, w, pointing vertically down and the force F sub arm pointing up and to the left. The forces are then shown rearranged to form a right triangle. F sub arm is the hypotenuse of the triangle pointing up and left, w is the vertical side pointing down, and F sub c is the base pointing to the left. The F sub c arrow is then shown separately with the notation that vector F sub c equals F sub net.\" \/><\/span><\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035646117\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035772540\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035646117-solution\">131<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035772542\">A car of mass 1000.0 kg is traveling along a level road at 100.0 km\/h when its brakes are applied. Calculate the stopping distance if the coefficient of kinetic friction of the tires is 0.500. Neglect air resistance. (<em>Hint:<\/em>\u00a0since the distance traveled is of interest rather than the time,\u00a0<em>x<\/em>\u00a0is the desired independent variable and not\u00a0<em>t<\/em>. Use the Chain Rule to change the variable:\u00a0<span id=\"MathJax-Element-2046-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43004\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43005\" class=\"mrow\"><span id=\"MathJax-Span-43006\" class=\"semantics\"><span id=\"MathJax-Span-43007\" class=\"mrow\"><span id=\"MathJax-Span-43008\" class=\"mrow\"><span id=\"MathJax-Span-43009\" class=\"mfrac\"><span id=\"MathJax-Span-43010\" class=\"mrow\"><span id=\"MathJax-Span-43011\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43012\" class=\"mi\">v<\/span><\/span><span id=\"MathJax-Span-43013\" class=\"mrow\"><span id=\"MathJax-Span-43014\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43015\" class=\"mi\">t<\/span><\/span><\/span><span id=\"MathJax-Span-43016\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43017\" class=\"mfrac\"><span id=\"MathJax-Span-43018\" class=\"mrow\"><span id=\"MathJax-Span-43019\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43020\" class=\"mi\">v<\/span><\/span><span id=\"MathJax-Span-43021\" class=\"mrow\"><span id=\"MathJax-Span-43022\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43023\" class=\"mi\">x<\/span><\/span><\/span><span id=\"MathJax-Span-43024\" class=\"mspace\"><\/span><span id=\"MathJax-Span-43025\" class=\"mfrac\"><span id=\"MathJax-Span-43026\" class=\"mrow\"><span id=\"MathJax-Span-43027\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43028\" class=\"mi\">x<\/span><\/span><span id=\"MathJax-Span-43029\" class=\"mrow\"><span id=\"MathJax-Span-43030\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43031\" class=\"mi\">t<\/span><\/span><\/span><span id=\"MathJax-Span-43032\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43033\" class=\"mi\">v<\/span><span id=\"MathJax-Span-43034\" class=\"mfrac\"><span id=\"MathJax-Span-43035\" class=\"mrow\"><span id=\"MathJax-Span-43036\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43037\" class=\"mi\">v<\/span><\/span><span id=\"MathJax-Span-43038\" class=\"mrow\"><span id=\"MathJax-Span-43039\" class=\"mi\">d<\/span><span id=\"MathJax-Span-43040\" class=\"mi\">x<\/span><\/span><\/span><span id=\"MathJax-Span-43041\" class=\"mo\">.<\/span><span id=\"MathJax-Span-43042\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">dvdt=dvdxdxdt=vdvdx.)<\/span><\/span><\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035679412\" class=\"\">\n<section>\n<div id=\"fs-id1165039494317\"><span class=\"os-number\">132<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039331601\">An airplane flying at 200.0 m\/s makes a turn that takes 4.0 min. What bank angle is required? What is the percentage increase in the perceived weight of the passengers?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165039416120\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165035975002\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165039416120-solution\">133<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035975004\">A skydiver is at an altitude of 1520 m. After 10.0 seconds of free fall, he opens his parachute and finds that the air resistance,\u00a0<span id=\"MathJax-Element-2047-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43043\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43044\" class=\"mrow\"><span id=\"MathJax-Span-43045\" class=\"semantics\"><span id=\"MathJax-Span-43046\" class=\"mrow\"><span id=\"MathJax-Span-43047\" class=\"mrow\"><span id=\"MathJax-Span-43048\" class=\"msub\"><span id=\"MathJax-Span-43049\" class=\"mi\">F<\/span><span id=\"MathJax-Span-43050\" class=\"mtext\">D<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">FD<\/span><\/span>, is given by the formula\u00a0<span id=\"MathJax-Element-2048-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43051\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43052\" class=\"mrow\"><span id=\"MathJax-Span-43053\" class=\"semantics\"><span id=\"MathJax-Span-43054\" class=\"mrow\"><span id=\"MathJax-Span-43055\" class=\"mrow\"><span id=\"MathJax-Span-43056\" class=\"msub\"><span id=\"MathJax-Span-43057\" class=\"mi\">F<\/span><span id=\"MathJax-Span-43058\" class=\"mtext\">D<\/span><\/span><span id=\"MathJax-Span-43059\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43060\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-43061\" class=\"mi\">b<\/span><span id=\"MathJax-Span-43062\" class=\"mi\">v<\/span><span id=\"MathJax-Span-43063\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">FD=\u2212bv,<\/span><\/span>\u00a0where\u00a0<em>b<\/em>\u00a0is a constant and\u00a0<em>v<\/em>\u00a0is the velocity. If\u00a0<span id=\"MathJax-Element-2049-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43064\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43065\" class=\"mrow\"><span id=\"MathJax-Span-43066\" class=\"semantics\"><span id=\"MathJax-Span-43067\" class=\"mrow\"><span id=\"MathJax-Span-43068\" class=\"mrow\"><span id=\"MathJax-Span-43069\" class=\"mi\">b<\/span><span id=\"MathJax-Span-43070\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43071\" class=\"mn\">0.750<\/span><span id=\"MathJax-Span-43072\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">b=0.750,<\/span><\/span>\u00a0and the mass of the skydiver is 82.0 kg, first set up differential equations for the velocity and the position, and then find: (a) the speed of the skydiver when the parachute opens, (b) the distance fallen before the parachute opens, (c) the terminal velocity after the parachute opens (find the limiting velocity), and (d) the time the skydiver is in the air after the parachute opens.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035694730\" class=\"\">\n<section>\n<div id=\"fs-id1165035654245\"><span class=\"os-number\">134<\/span><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165035654247\">In a television commercial, a small, spherical bead of mass 4.00 g is released from rest at\u00a0<span id=\"MathJax-Element-2050-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43073\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43074\" class=\"mrow\"><span id=\"MathJax-Span-43075\" class=\"semantics\"><span id=\"MathJax-Span-43076\" class=\"mrow\"><span id=\"MathJax-Span-43077\" class=\"mrow\"><span id=\"MathJax-Span-43078\" class=\"mi\">t<\/span><span id=\"MathJax-Span-43079\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43080\" class=\"mn\">0<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">t=0<\/span><\/span>\u00a0in a bottle of liquid shampoo. The terminal speed is observed to be 2.00 cm\/s. Find (a) the value of the constant\u00a0<em>b<\/em>\u00a0in the equation\u00a0<span id=\"MathJax-Element-2051-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: left;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><span id=\"MathJax-Span-43081\" class=\"math\" role=\"math\"><span id=\"MathJax-Span-43082\" class=\"mrow\"><span id=\"MathJax-Span-43083\" class=\"semantics\"><span id=\"MathJax-Span-43084\" class=\"mrow\"><span id=\"MathJax-Span-43085\" class=\"mrow\"><span id=\"MathJax-Span-43086\" class=\"mi\">v<\/span><span id=\"MathJax-Span-43087\" class=\"mo\">=<\/span><span id=\"MathJax-Span-43088\" class=\"mfrac\"><span id=\"MathJax-Span-43089\" class=\"mrow\"><span id=\"MathJax-Span-43090\" class=\"mi\">m<\/span><span id=\"MathJax-Span-43091\" class=\"mi\">g<\/span><\/span><span id=\"MathJax-Span-43092\" class=\"mi\">b<\/span><\/span><span id=\"MathJax-Span-43093\" class=\"mo\">(<\/span><span id=\"MathJax-Span-43094\" class=\"mn\">1<\/span><span id=\"MathJax-Span-43095\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-43096\" class=\"msup\"><span id=\"MathJax-Span-43097\" class=\"mi\">e<\/span><span id=\"MathJax-Span-43098\" class=\"mrow\"><span id=\"MathJax-Span-43099\" class=\"mtext\">\u2212<\/span><span id=\"MathJax-Span-43100\" class=\"mrow\"><span id=\"MathJax-Span-43101\" class=\"mrow\"><span id=\"MathJax-Span-43102\" class=\"mi\">b<\/span><span id=\"MathJax-Span-43103\" class=\"mi\">t<\/span><\/span><span id=\"MathJax-Span-43104\" class=\"mtext\">\/<\/span><span id=\"MathJax-Span-43105\" class=\"mi\">m<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-43106\" class=\"mo\">)<\/span><span id=\"MathJax-Span-43107\" class=\"mo\">,<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"MJX_Assistive_MathML\" role=\"presentation\">v=mgb(1\u2212e\u2212bt\/m),<\/span><\/span>\u00a0and (b) the value of the resistive force when the bead reaches terminal speed.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165035685642\" class=\"os-hasSolution\">\n<section>\n<div id=\"fs-id1165039511826\"><a class=\"os-number\" href=\"https:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@9.1:e6aa980a-5228-5ee3-9de8-cdb28cc6d537#fs-id1165035685642-solution\">135<\/a><span class=\"os-divider\">.<\/span><\/p>\n<p id=\"fs-id1165039511828\">A boater and motor boat are at rest on a lake. Together, they have mass 200.0 kg. If the thrust of the motor is a constant force of 40.0 N in the direction of motion, and if the resistive force of the water is numerically equivalent to 2 times the speed\u00a0<em>v<\/em>of the boat, set up and solve the differential equation to find: (a) the velocity of the boat at time\u00a0<em>t<\/em>; (b) the limiting velocity (the velocity after a long time has passed).<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1422\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>OpenStax University Physics. <strong>Authored by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/1Q9uMg_a@10.16:Gofkr9Oy@15\">https:\/\/cnx.org\/contents\/1Q9uMg_a@10.16:Gofkr9Oy@15<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@10.16<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":44985,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax University Physics\",\"author\":\"OpenStax CNX\",\"organization\":\"\",\"url\":\"https:\/\/cnx.org\/contents\/1Q9uMg_a@10.16:Gofkr9Oy@15\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/d50f6e32-0fda-46ef-a362-9bd36ca7c97d@10.16\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1422","chapter","type-chapter","status-publish","hentry"],"part":560,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters\/1422","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/users\/44985"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters\/1422\/revisions"}],"predecessor-version":[{"id":1423,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters\/1422\/revisions\/1423"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/parts\/560"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapters\/1422\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/media?parent=1422"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/pressbooks\/v2\/chapter-type?post=1422"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/contributor?post=1422"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/wp-json\/wp\/v2\/license?post=1422"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}