{"id":1454,"date":"2018-02-06T17:09:11","date_gmt":"2018-02-06T17:09:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/?post_type=chapter&p=1454"},"modified":"2018-02-06T17:09:11","modified_gmt":"2018-02-06T17:09:11","slug":"10-chapter-review","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/10-chapter-review\/","title":{"raw":"10 Chapter Review","rendered":"10 Chapter Review"},"content":{"raw":"
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Key Terms<\/span><\/h3>\r\n
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angular acceleration<\/strong><\/dt>\r\n \t
time rate of change of angular velocity<\/dd>\r\n<\/dl>\r\n
\r\n \t
angular position<\/strong><\/dt>\r\n \t
angle a body has rotated through in a fixed coordinate system<\/dd>\r\n<\/dl>\r\n
\r\n \t
angular velocity<\/strong><\/dt>\r\n \t
time rate of change of angular position<\/dd>\r\n<\/dl>\r\n
\r\n \t
instantaneous angular acceleration<\/strong><\/dt>\r\n \t
derivative of angular velocity with respect to time<\/dd>\r\n<\/dl>\r\n
\r\n \t
instantaneous angular velocity<\/strong><\/dt>\r\n \t
derivative of angular position with respect to time<\/dd>\r\n<\/dl>\r\n
\r\n \t
kinematics of rotational motion<\/strong><\/dt>\r\n \t
describes the relationships among rotation angle, angular velocity, angular acceleration, and time<\/dd>\r\n<\/dl>\r\n
\r\n \t
lever arm<\/strong><\/dt>\r\n \t
perpendicular distance from the line that the force vector lies on to a given axis<\/dd>\r\n<\/dl>\r\n
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linear mass density<\/strong><\/dt>\r\n \t
the mass per unit length\u00a0\u03bb<\/span><\/span><\/span><\/span><\/span><\/span>\u03bb<\/span><\/span>\u00a0of a one dimensional object<\/dd>\r\n<\/dl>\r\n
\r\n \t
moment of inertia<\/strong><\/dt>\r\n \t
rotational mass of rigid bodies that relates to how easy or hard it will be to change the angular velocity of the rotating rigid body<\/dd>\r\n<\/dl>\r\n
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Newton\u2019s second law for rotation<\/strong><\/dt>\r\n \t
sum of the torques on a rotating system equals its moment of inertia times its angular acceleration<\/dd>\r\n<\/dl>\r\n
\r\n \t
parallel axis<\/strong><\/dt>\r\n \t
axis of rotation that is parallel to an axis about which the moment of inertia of an object is known<\/dd>\r\n<\/dl>\r\n
\r\n \t
parallel-axis theorem<\/strong><\/dt>\r\n \t
if the moment of inertia is known for a given axis, it can be found for any axis parallel to it<\/dd>\r\n<\/dl>\r\n
\r\n \t
rotational dynamics<\/strong><\/dt>\r\n \t
analysis of rotational motion using the net torque and moment of inertia to find the angular acceleration<\/dd>\r\n<\/dl>\r\n
\r\n \t
rotational kinetic energy<\/strong><\/dt>\r\n \t
kinetic energy due to the rotation of an object; this is part of its total kinetic energy<\/dd>\r\n<\/dl>\r\n
\r\n \t
rotational work<\/strong><\/dt>\r\n \t
work done on a rigid body due to the sum of the torques integrated over the angle through with the body rotates<\/dd>\r\n<\/dl>\r\n
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surface mass density<\/strong><\/dt>\r\n \t
mass per unit area\u00a0\u03c3<\/span><\/span><\/span><\/span><\/span><\/span>\u03c3<\/span><\/span>\u00a0of a two dimensional object<\/dd>\r\n<\/dl>\r\n
\r\n \t
torque<\/strong><\/dt>\r\n \t
cross product of a force and a lever arm to a given axis<\/dd>\r\n<\/dl>\r\n
\r\n \t
total linear acceleration<\/strong><\/dt>\r\n \t
vector sum of the centripetal acceleration vector and the tangential acceleration vector<\/dd>\r\n<\/dl>\r\n
\r\n \t
work-energy theorem for rotation<\/strong><\/dt>\r\n \t
the total rotational work done on a rigid body is equal to the change in rotational kinetic energy of the body<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/div>\r\n
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Key Equations<\/span><\/h3>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Angular position<\/td>\r\n\u03b8<\/span>=<\/span>s<\/span>r<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b8=sr<\/span><\/span><\/td>\r\n<\/tr>\r\n
Angular velocity<\/td>\r\n\u03c9<\/span>=<\/span>lim<\/span><\/span>\u0394<\/span>t<\/span>\u2192<\/span>0<\/span><\/span><\/span>\u0394<\/span>\u03b8<\/span><\/span>\u0394<\/span>t<\/span><\/span><\/span>=<\/span>d<\/span>\u03b8<\/span><\/span>d<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9=lim\u0394t\u21920\u0394\u03b8\u0394t=d\u03b8dt<\/span><\/span><\/td>\r\n<\/tr>\r\n
Tangential speed<\/td>\r\nv<\/span>t<\/span><\/span>=<\/span>r<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span>vt=r\u03c9<\/span><\/span><\/td>\r\n<\/tr>\r\n
Angular acceleration<\/td>\r\n\u03b1<\/span>=<\/span>lim<\/span><\/span>\u0394<\/span>t<\/span>\u2192<\/span>0<\/span><\/span><\/span>\u0394<\/span>\u03c9<\/span><\/span>\u0394<\/span>t<\/span><\/span><\/span>=<\/span>d<\/span>\u03c9<\/span><\/span>d<\/span>t<\/span><\/span><\/span>=<\/span>d<\/span>2<\/span><\/span>\u03b8<\/span><\/span>d<\/span>t<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1=lim\u0394t\u21920\u0394\u03c9\u0394t=d\u03c9dt=d2\u03b8dt2<\/span><\/span><\/td>\r\n<\/tr>\r\n
Tangential acceleration<\/td>\r\na<\/span>t<\/span><\/span>=<\/span>r<\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span>at=r\u03b1<\/span><\/span><\/td>\r\n<\/tr>\r\n
Average angular velocity<\/td>\r\n\u03c9<\/span>\u2013<\/span><\/span>=<\/span>\u03c9<\/span>0<\/span><\/span>+<\/span>\u03c9<\/span>f<\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c9\u2013=\u03c90+\u03c9f2<\/span><\/span><\/td>\r\n<\/tr>\r\n
Angular displacement<\/td>\r\n\u03b8<\/span>f<\/span><\/span>=<\/span>\u03b8<\/span>0<\/span><\/span>+<\/span>\u03c9<\/span>\u2013<\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span>\u03b8f=\u03b80+\u03c9\u2013t<\/span><\/span><\/td>\r\n<\/tr>\r\n
Angular velocity from constant angular acceleration<\/td>\r\n\u03c9<\/span>f<\/span><\/span>=<\/span>\u03c9<\/span>0<\/span><\/span>+<\/span>\u03b1<\/span>t<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9f=\u03c90+\u03b1t<\/span><\/span><\/td>\r\n<\/tr>\r\n
Angular velocity from displacement and\r\n
<\/div>\r\nconstant angular acceleration<\/td>\r\n
\u03b8<\/span>f<\/span><\/span>=<\/span>\u03b8<\/span>0<\/span><\/span>+<\/span>\u03c9<\/span>0<\/span><\/span>t<\/span>+<\/span>1<\/span>2<\/span><\/span>\u03b1<\/span>t<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b8f=\u03b80+\u03c90t+12\u03b1t2<\/span><\/span><\/td>\r\n<\/tr>\r\n
Change in angular velocity<\/td>\r\n\u03c9<\/span>2<\/span>f<\/span><\/span>=<\/span>\u03c9<\/span>2<\/span>0<\/span><\/span>+<\/span>2<\/span>\u03b1<\/span>(<\/span>\u0394<\/span>\u03b8<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>\u03c9f2=\u03c902+2\u03b1(\u0394\u03b8)<\/span><\/span><\/td>\r\n<\/tr>\r\n
Total acceleration<\/td>\r\na<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>a<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>c<\/span><\/span><\/span>+<\/span>a<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>a\u2192=a\u2192c+a\u2192t<\/span><\/span><\/td>\r\n<\/tr>\r\n
Rotational kinetic energy<\/td>\r\nK<\/span>=<\/span>1<\/span>2<\/span><\/span>(<\/span>\u2211<\/span>j<\/span><\/span>m<\/span>j<\/span><\/span>r<\/span>2<\/span>j<\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>K=12(\u2211jmjrj2)\u03c92<\/span><\/span><\/td>\r\n<\/tr>\r\n
Moment of inertia<\/td>\r\nI<\/span>=<\/span>\u2211<\/span>j<\/span><\/span>m<\/span>j<\/span><\/span>r<\/span>2<\/span>j<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>I=\u2211jmjrj2<\/span><\/span><\/td>\r\n<\/tr>\r\n
Rotational kinetic energy in terms of the\r\n
<\/div>\r\nmoment of inertia of a rigid body<\/td>\r\n
K<\/span>=<\/span>1<\/span>2<\/span><\/span>I<\/span>\u03c9<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>K=12I\u03c92<\/span><\/span><\/td>\r\n<\/tr>\r\n
Moment of inertia of a continuous object<\/td>\r\nI<\/span>=<\/span>\u222b<\/span>r<\/span>2<\/span><\/span>d<\/span>m<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>I=\u222br2dm<\/span><\/span><\/td>\r\n<\/tr>\r\n
Parallel-axis theorem<\/td>\r\nI<\/span>parallel-axis<\/span><\/span><\/span>=<\/span>I<\/span>initial<\/span><\/span><\/span>+<\/span>m<\/span>d<\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Iparallel-axis=Iinitial+md2<\/span><\/span><\/td>\r\n<\/tr>\r\n
Moment of inertia of a compound object<\/td>\r\nI<\/span>total<\/span><\/span><\/span>=<\/span>\u2211<\/span>i<\/span><\/span>I<\/span>i<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>Itotal=\u2211iIi<\/span><\/span><\/td>\r\n<\/tr>\r\n
Torque vector<\/td>\r\n\u03c4<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>r<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>F<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4\u2192=r\u2192\u00d7F\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Magnitude of torque<\/td>\r\n\u2223\u2223<\/span>\u03c4<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u2223\u2223<\/span><\/span>=<\/span>r<\/span>\u22a5<\/span><\/span>F<\/span><\/span><\/span><\/span><\/span><\/span>|\u03c4\u2192|=r\u22a5F<\/span><\/span><\/td>\r\n<\/tr>\r\n
Total torque<\/td>\r\n\u03c4<\/span>net<\/span><\/span><\/span>=<\/span>\u2211<\/span>i<\/span><\/span>\u2223\u2223<\/span>\u03c4<\/span>i<\/span><\/span><\/span>\u2223\u2223<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03c4net=\u2211i|\u03c4i|<\/span><\/span><\/td>\r\n<\/tr>\r\n
Newton\u2019s second law for rotation<\/td>\r\n\u2211<\/span>i<\/span><\/span>\u03c4<\/span>i<\/span><\/span><\/span><\/span><\/span>=<\/span>I<\/span>\u03b1<\/span><\/span><\/span><\/span><\/span><\/span>\u2211i\u03c4i=I\u03b1<\/span><\/span><\/td>\r\n<\/tr>\r\n
Incremental work done by a torque<\/td>\r\nd<\/span>W<\/span>=<\/span>(<\/span>\u2211<\/span>i<\/span><\/span>\u03c4<\/span>i<\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>d<\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span>dW=(\u2211i\u03c4i)d\u03b8<\/span><\/span><\/td>\r\n<\/tr>\r\n
Work-energy theorem<\/td>\r\nW<\/span>A<\/span>B<\/span><\/span><\/span>=<\/span>K<\/span>B<\/span><\/span>\u2212<\/span>K<\/span>A<\/span><\/span><\/span><\/span><\/span><\/span><\/span>WAB=KB\u2212KA<\/span><\/span><\/td>\r\n<\/tr>\r\n
Rotational work done by net force<\/td>\r\nW<\/span>A<\/span>B<\/span><\/span><\/span>=<\/span>\u222b<\/span>\u03b8<\/span>A<\/span><\/span><\/span>\u03b8<\/span>B<\/span><\/span><\/span><\/span>(<\/span>\u2211<\/span>i<\/span><\/span>\u03c4<\/span>i<\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>d<\/span>\u03b8<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>WAB=\u222b\u03b8A\u03b8B(\u2211i\u03c4i)d\u03b8<\/span><\/span><\/td>\r\n<\/tr>\r\n
Rotational power<\/td>\r\nP<\/span>=<\/span>\u03c4<\/span>\u03c9<\/span><\/span><\/span><\/span><\/span><\/span>P=\u03c4\u03c9<\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><\/div>\r\n
<\/div>\r\n<\/div>\r\n \r\n
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Summary<\/h3>\r\n<\/div>\r\n
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10.1<\/span>\u00a0<\/span>Rotational Variables<\/span><\/h4>\r\n