Chapter 11 Torque and Angular Momentum I Torque II Angular momentum Definition III Newton s second law in angular form IV Angular momentum System of particles Rigid body Conservation I Torque Vector quantity r F Direction right hand rule Magnitude r F sin r F r sin F r F Torque is calculated with respect to about a point Changing the point can change the torque s magnitude and direction 1 II Angular momentum l r p m r v Vector quantity Magnitude Units kg m2 s l r p sin r m v sin r m v r p r sin p r p r m v Direction right hand rule l positive counterclockwise l negative clockwise Direction of l is always perpendicular to plane formed by r and p III Newton s second law in angular form Linear Angular dp Fnet dt net dl dt Single particle The vector sum of all torques acting on a particle is equal to the time rate of change of the angular momentum of that particle Proof dl dv dr l m r v m r v m r a v v m r a dt dt dt dl r ma r Fnet r F net dt V Angular momentum System of particles n L l1 l2 l3 ln li i 1 2 dL n dli n dL net i net dt i 1 dt i 1 dt Includes internal torques due to forces between particles within system and external torques due to forces on the particles from bodies outside system Forces inside system third law force pairs torqueint sum 0 The only torques that can change the angular momentum of a system are the external torques acting on a system The net external torque acting on a system of particles is equal to the time rate of change of the system s total angular momentum L Rigid body rotating about a fixed axis with constant angular speed Magnitude li ri pi sin 90 ri mi vi vi ri li ri mi ri mi ri 2 Direction li perpendicular to ri and pi n n n 2 Lz liz mi ri mi ri 2 I i 1 i 1 i 1 Lz I dLz d dL I I z ext dt dt dt L I Rotational inertia of a rigid body about a fixed axis 3 Conservation of angular momentum dL Newton s second law net dt If no net external torque acts on the system isolated system Law of conservation of angular momentum Li L f dL 0 L cte dt isolated system Net angular momentum at time ti Net angular momentum at later time tf If the net external torque acting on a system is zero the angular momentum of the system remains constant no matter what changes take place within the system If the component of the net external torque on a system along a certain axis is zero the component of the angular momentum of the system along that axis cannot change no matter what changes take place within the system This conservation law holds not only within the frame of Newton s mechanics but also for relativistic particles speeds close to light and subatomic particles I i i I f f Ii f i f refer to rotational inertia and angular speed before and after the redistribution of mass about the rotational axis 4 Examples Spinning volunteer If Ii mass closer to rotation axis Torque ext 0 Ii i If f f i Springboard diver Center of mass follows parabolic path When in air no net external torque about COM Diver s angular momentum L constant throughout dive magnitude and direction L is perpendicular to the plane of the figure inward Beginning of dive She pulls arms legs closer Intention I is reduced increases End of dive layout position Purpose I increases slow rotation rate less water splash 5 Rotation Translation F Force Linear momentum p dP F dt Conservation law Angular momentum Closed isolated system L li l r p System of particles i L I Newton s second law P cte r F Angular momentum Linear P pi MvCOM i momentum system of particles rigid body Newton s second law Torque Rigid body fixed axis L component along that axis net dL dt Conservation law L cte Closed isolated system IV Precession of a gyroscope Gyroscope wheel fixed to shaft and free to spin about shaft s axis Non spinning gyroscope dL dt If one end of shaft is placed on a support and released Gyroscope falls by rotating downward about the tip of the support The torque causing the downward rotation fall changes angular momentum of gyroscope Torque caused by gravitational force acting on COM Mgr sin 90 Mgr 6 Rapidly spinning gyroscope If released with shaft s angle slightly upward first rotates downward then spins horizontally about vertical axis z precession due to non zero initial angular momentum Simplification i L due to rapid spin L due to precession ii shaft horizontal when precession starts L I I rotational moment of gyroscope about shaft angular speed of wheel about shaft Vector L along shaft parallel to r Torque perpendicular to L can only change the Direction of L not its magnitude dL dt dL dt Mgrdt d dL Mgrdt L I Rapidly spinning gyroscope dL dt dL dt Mgrdt d dL Mgrdt L I Precession rate d Mgr dt I 7
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