{"id":88,"date":"2021-07-30T17:11:33","date_gmt":"2021-07-30T17:11:33","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=88"},"modified":"2022-11-01T23:05:19","modified_gmt":"2022-11-01T23:05:19","slug":"summary-of-applications","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-applications\/","title":{"raw":"Summary of Applications","rendered":"Summary of Applications"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<div class=\"PageContent-ny9bj0-0 gLsPXs\" tabindex=\"0\">\r\n<div id=\"main-content\" class=\"MainContent__HideOutline-sc-6yy1if-0 bdVAq sc-bdVaJa ibTDPh\" tabindex=\"-1\" data-dynamic-style=\"false\">\r\n<div id=\"composite-page-27\" class=\"os-eoc os-key-concepts-container\" data-type=\"composite-page\" data-uuid-key=\".key-concepts\">\r\n<div class=\"os-key-concepts\">\r\n<div class=\"os-section-area\"><section id=\"fs-id1170571115987\" class=\"key-concepts\" data-depth=\"1\">\r\n<ul id=\"fs-id1170571115993\" data-bullet-style=\"bullet\">\r\n \t<li>Second-order constant-coefficient differential equations can be used to model spring-mass systems.<\/li>\r\n \t<li>An examination of the forces on a spring-mass system results in a differential equation of the form [latex]mx^{\\prime\\prime}+bx^{\\prime}+kx=f(t)[\/latex], where [latex]m[\/latex]\u00a0represents the mass, [latex]b[\/latex] is the coefficient of the damping force, [latex]k[\/latex]\u00a0is the spring constant, and [latex]f(t)[\/latex]\u00a0represents any net external forces on the system.<\/li>\r\n \t<li>If [latex]b=0[\/latex], there is no damping force acting on the system, and simple harmonic motion results. If\u00a0[latex]b\\ne{0}[\/latex], the behavior of the system depends on whether [latex]b^{2}-4mk&gt;0[\/latex], [latex]b^{2}-4mk=0[\/latex], or\u00a0[latex]b^{2}-4mk&lt;0[\/latex].<\/li>\r\n \t<li>If [latex]b^2-4mk&gt;0[\/latex], the system is overdamped and does not exhibit oscillatory behavior.<\/li>\r\n \t<li>If [latex]b^2-4mk=0[\/latex], the system is critically damped. It does not exhibit oscillatory behavior, but any slight reduction in the damping would result in oscillatory behavior.<\/li>\r\n \t<li>If [latex]b^2-4mk&lt;0[\/latex],\u00a0the system is underdamped. It exhibits oscillatory behavior, but the amplitude of the oscillations decreases over time.<\/li>\r\n \t<li>If [latex]f(t)\\ne{0}[\/latex],\u00a0the solution to the differential equation is the sum of a transient solution and a steady-state solution. The steady-state solution governs the long-term behavior of the system.<\/li>\r\n \t<li>The charge on the capacitor in an [latex]RLC[\/latex]\u00a0series circuit can also be modeled with a second-order constant-coefficient differential equation of the form [latex]L\\frac{d^{2}q}{dt^{2}}+R\\frac{dq}{dt}+\\frac{1}{C}q=E(t)[\/latex], where [latex]L[\/latex] is the inductance,\u00a0[latex]R[\/latex] is the resistance,\u00a0[latex]C[\/latex]\u00a0is the capacitance, and [latex]E(t)[\/latex]\u00a0is the voltage source.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Equation of simple harmonic motion\r\n<\/strong>[latex]x^{\\prime\\prime}+\\omega^{2}x=0[\/latex]<\/li>\r\n \t<li><strong>Solution for simple harmonic motion<\/strong>\r\n[latex]x(t)=c_{1}\\cos{(\\omega{t})}+c_{2}\\sin{(\\omega{t})}[\/latex]<\/li>\r\n \t<li><strong>Alternative form of solution for SHM<\/strong>\r\n[latex]x(t)=A\\sin{(\\omega{t}+\\phi)}[\/latex]<\/li>\r\n \t<li><strong>Forced harmonic motion<\/strong>\r\n[latex]mx^{\\prime\\prime}+bx^{\\prime}+kx=f(t)[\/latex]<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong data-effect=\"bold\">Charge in a<\/strong>\u00a0<strong data-effect=\"bold\"><em data-effect=\"italics\">RLC<\/em><\/strong>\u00a0<strong data-effect=\"bold\">series circuit<\/strong>\r\n[latex]L\\frac{d^{2}q}{dt^{2}}+R\\frac{dq}{dt}+\\frac{1}{C}q=E(t)[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt><em data-effect=\"italics\">RLC<\/em>\u00a0series circuit<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an <\/span><em style=\"font-size: 1em;\" data-effect=\"italics\">RLC<\/em><span style=\"font-size: 1em;\"> series circuit<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>simple harmonic motion<\/dt>\r\n \t<dd>motion described by the equation [latex]x(t)=c_{1}\\cos{(\\omega{t})}+c_{2}\\sin{(\\omega{t})}[\/latex]\u00a0as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>steady-state solution<\/dt>\r\n \t<dd><span style=\"font-size: 1em;\">a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution<\/span><\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<div class=\"PageContent-ny9bj0-0 gLsPXs\" tabindex=\"0\">\n<div id=\"main-content\" class=\"MainContent__HideOutline-sc-6yy1if-0 bdVAq sc-bdVaJa ibTDPh\" tabindex=\"-1\" data-dynamic-style=\"false\">\n<div id=\"composite-page-27\" class=\"os-eoc os-key-concepts-container\" data-type=\"composite-page\" data-uuid-key=\".key-concepts\">\n<div class=\"os-key-concepts\">\n<div class=\"os-section-area\">\n<section id=\"fs-id1170571115987\" class=\"key-concepts\" data-depth=\"1\">\n<ul id=\"fs-id1170571115993\" data-bullet-style=\"bullet\">\n<li>Second-order constant-coefficient differential equations can be used to model spring-mass systems.<\/li>\n<li>An examination of the forces on a spring-mass system results in a differential equation of the form [latex]mx^{\\prime\\prime}+bx^{\\prime}+kx=f(t)[\/latex], where [latex]m[\/latex]\u00a0represents the mass, [latex]b[\/latex] is the coefficient of the damping force, [latex]k[\/latex]\u00a0is the spring constant, and [latex]f(t)[\/latex]\u00a0represents any net external forces on the system.<\/li>\n<li>If [latex]b=0[\/latex], there is no damping force acting on the system, and simple harmonic motion results. If\u00a0[latex]b\\ne{0}[\/latex], the behavior of the system depends on whether [latex]b^{2}-4mk>0[\/latex], [latex]b^{2}-4mk=0[\/latex], or\u00a0[latex]b^{2}-4mk<0[\/latex].<\/li>\n<li>If [latex]b^2-4mk>0[\/latex], the system is overdamped and does not exhibit oscillatory behavior.<\/li>\n<li>If [latex]b^2-4mk=0[\/latex], the system is critically damped. It does not exhibit oscillatory behavior, but any slight reduction in the damping would result in oscillatory behavior.<\/li>\n<li>If [latex]b^2-4mk<0[\/latex],\u00a0the system is underdamped. It exhibits oscillatory behavior, but the amplitude of the oscillations decreases over time.<\/li>\n<li>If [latex]f(t)\\ne{0}[\/latex],\u00a0the solution to the differential equation is the sum of a transient solution and a steady-state solution. The steady-state solution governs the long-term behavior of the system.<\/li>\n<li>The charge on the capacitor in an [latex]RLC[\/latex]\u00a0series circuit can also be modeled with a second-order constant-coefficient differential equation of the form [latex]L\\frac{d^{2}q}{dt^{2}}+R\\frac{dq}{dt}+\\frac{1}{C}q=E(t)[\/latex], where [latex]L[\/latex] is the inductance,\u00a0[latex]R[\/latex] is the resistance,\u00a0[latex]C[\/latex]\u00a0is the capacitance, and [latex]E(t)[\/latex]\u00a0is the voltage source.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Equation of simple harmonic motion<br \/>\n<\/strong>[latex]x^{\\prime\\prime}+\\omega^{2}x=0[\/latex]<\/li>\n<li><strong>Solution for simple harmonic motion<\/strong><br \/>\n[latex]x(t)=c_{1}\\cos{(\\omega{t})}+c_{2}\\sin{(\\omega{t})}[\/latex]<\/li>\n<li><strong>Alternative form of solution for SHM<\/strong><br \/>\n[latex]x(t)=A\\sin{(\\omega{t}+\\phi)}[\/latex]<\/li>\n<li><strong>Forced harmonic motion<\/strong><br \/>\n[latex]mx^{\\prime\\prime}+bx^{\\prime}+kx=f(t)[\/latex]<\/li>\n<\/ul>\n<ul id=\"fs-id1169736654976\">\n<li><strong data-effect=\"bold\">Charge in a<\/strong>\u00a0<strong data-effect=\"bold\"><em data-effect=\"italics\">RLC<\/em><\/strong>\u00a0<strong data-effect=\"bold\">series circuit<\/strong><br \/>\n[latex]L\\frac{d^{2}q}{dt^{2}}+R\\frac{dq}{dt}+\\frac{1}{C}q=E(t)[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt><em data-effect=\"italics\">RLC<\/em>\u00a0series circuit<\/dt>\n<dd><span style=\"font-size: 1em;\">a complete electrical path consisting of a resistor, an inductor, and a capacitor; a second-order, constant-coefficient differential equation can be used to model the charge on the capacitor in an <\/span><em style=\"font-size: 1em;\" data-effect=\"italics\">RLC<\/em><span style=\"font-size: 1em;\"> series circuit<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>simple harmonic motion<\/dt>\n<dd>motion described by the equation [latex]x(t)=c_{1}\\cos{(\\omega{t})}+c_{2}\\sin{(\\omega{t})}[\/latex]\u00a0as exhibited by an undamped spring-mass system in which the mass continues to oscillate indefinitely<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>steady-state solution<\/dt>\n<dd><span style=\"font-size: 1em;\">a solution to a nonhomogeneous differential equation related to the forcing function; in the long term, the solution approaches the steady-state solution<\/span><\/dd>\n<\/dl>\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-88\">\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>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/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":17533,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-88","chapter","type-chapter","status-publish","hentry"],"part":25,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/88","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/88\/revisions"}],"predecessor-version":[{"id":3800,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/88\/revisions\/3800"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/25"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/88\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=88"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=88"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=88"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=88"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}