momen mômen angular momentum

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Lecture Angular Momentum Tran Thi Ngoc Dung dungttn@gmail.com HCMUT Outline • Angular Momentum • Conservation of Angular Momentum O d  r  F  M  Torque of a force about a fixed point O   F /O  r  F   F / O  plane(OM, F)  The direction of τ is outward F / O  F r sin   F.d O d  r  p  M  Angular momentum about a fixed point O      L /O  r  p  r  mv   L / O  plane(OM , p)  The direction of L is outward L  mvr sin   mv.d Angular momentum and Torque Angular momentumabout a fixed point O      L /O  r  p  r  mv Differentiate by time :       dL /O d r   dp   p r   ( v  mv)   r F       dt dt dt  0   dL /O  dL /   F / O  F /  dt dt The net torque acting on a particle equals the rate of change of the angular momentum of the particle For a system of particles Angular momentum of i th particel about a fixed point O      Li/O  ri  pi  ri  mvi Angular momentum of the system of particles     L / O   Li/O   ri  mvi i i Differenti ate by time :         dL /O  d ri     dpi      pi     ri  vi  mvi )   ri  (Fi,inter  Fi,ext )    ( dt dt  i  i  i  i  dt 0         ri  Fi,inter  0,  ri  Fi,ext   i,ext  net.ext i i i   dL /O  dL /   net ,ext /  net ,ext /  dt dt     dL /  d I  L /  I   dt dt  net ,ext /    d  I  I dt ANGULAR MOMENTUM OF A RIGID OBJECT   d  I  I  net ext /    dt L   I A  dL    net ext /  dt The net external torque acting on a system equals the rate of change of the angular momentum of the system  dL /   net ,ext /  dt   L /  I   Example 10-2 An Atwood' s machine has two blocks of mass m1 and m (m1 > m ), connected by a string of negligible mass that passes over a pulley wit h frictionle ss bearings The pulley is a uniform disk of mass M and radius R The string does not slip on the pulley  dL  Apply Equation  net ,ext dt to the system consisting of both blocks plus pulley to find the angular acceleration of the pulley and the acceleration of the blocks Angular momentum of the system {Block m1 , block m , pulley M) about the axis z through t he center of pulley L  m1vR  m vR  I because that the rope is not sliding on the pulley v  R  a  R dL I    m1aR  m 2aR  I   m1  m  aR dt R   Net torque of external forces : net  m1gR  m 2gR m1  m  m1 moves down, m moves up with the speed v  pulley rotates counter - clockwise  with  outward dL  net,ext dt I    m1  m  aR  m1gR  m 2gR R   m1  m a g I m1  m  R Applying Conservation of Angular Momentum  dL   net,ext dt   if net,ext  L  const If the net external torque acting on a system is zero, the total angular momentum of the system is constant Example A disk is rotating with an initial angular speed i about a frictionle ss shaft through its symmetry ax is as shown Its moment of inertia about this axis is I1 It drops onto another disk of moment of inertia I that is initially at rest on the same shaft Because of surface friction, the two disks eventually attain a common angular speed f Find f External force : weight m1g parallel to the rotation axis    torque net   L  const initial Angular momentum  final angular momentum Li  L f    I1i  (I1  I )f  f  L2 Initial KE : Ti  I1i  2I1 I1  i I1  I L2 Final KE : Tf  (I1  I )f  2(I1  I ) The change in KE : L2  1 L2 I2 T  Tf  Ti       0  I1  I I1  (I1  I )I1 Example A merry - go - round of radius m and moment of inertia 500 kg.m is rotating about a frictionless pivot, making one revolution every s A child of mass 25 kg originally standing at the center walks out to the rim Find the new angular speed of the merry - go - round  External force is the gravitatio nal force mg, parallel to the rotation axis The torque is zero Applying the law of Conservation of Angular momentumLi  Lf (I merry  Ichild,i )i  (I merry  Ichild,f )f f  I merry  Ichild,i I merry  Ichild,f Ichild,i  f  i Ichild,f  mR I merry I merry  mR i 500 2 f   1.05rad / s 500  25  Problem In the figure the incline is frictionless and the string passes through the center of mass of each block The pulley has a moment of inertia I and a radius r (a) Find the net torque acting on the system (the two masses, string, and pulley) about the center of the pulley (b) Write an expression for the total angular momentum of the system about the center of the pulley when the masses are moving with a speed v (c) Find the acceleration of the masses from your results for parts (a) and (b) by setting the net torque equal to the rate of change of the angular momentum of the system a )   m 2g sin R  m1gR b) L  m1vR  m vR  I c) dL  dt I (m1  m  )aR  m 2g sin R  m1gR R m sin   m1 a g I m1  m  R ...  Angular momentum about a fixed point O      L /O  r  p  r  mv   L / O  plane(OM , p)  The direction of L is outward L  mvr sin   mv.d Angular momentum and Torque Angular momentumabout... of change of the angular momentum of the particle For a system of particles Angular momentum of i th particel about a fixed point O      Li/O  ri  pi  ri  mvi Angular momentum of the... attain a common angular speed f Find f External force : weight m1g parallel to the rotation axis    torque net   L  const initial Angular momentum  final angular momentum Li  L