Copyright © 2012 Pearson Education Inc. Angular momentum Physics 7C lecture 14 Thursday November 14, 8:00 AM – 9:20 AM Engineering Hall 1200

Copyright © 2012 Pearson Education Inc. Torque as a vector Torque can be expressed as a vector using the vector product. How to find the direction of torque using a right hand rule.

Copyright © 2012 Pearson Education Inc. Rigid body rotation about a moving axis The motion of a rigid body is a combination of translational motion of the center of mass and rotation about the center of mass The kinetic energy of a rotating and translating rigid body is K = 1/2 Mv cm 2 + 1/2 I cm  2.

Copyright © 2012 Pearson Education Inc. Rolling without slipping The condition for rolling without slipping is v cm = R . Figure shows the combined motion of points on a rolling wheel.

Copyright © 2012 Pearson Education Inc. Work and power in rotational motion The total work done on a body by the torque is equal to the change in rotational kinetic energy of the body and the power due to a torque is P =  z  z.

Copyright © 2012 Pearson Education Inc. Angular momentum What’s the analogue of momentum P for angular motion?

Copyright © 2012 Pearson Education Inc. Angular momentum Momentum : p = m v angular momentum: τ = r × p = m r × v

Copyright © 2012 Pearson Education Inc. Angular momentum The angular momentum of a rigid body rotating about a symmetry axis is parallel to the angular velocity and is given by L = I . Right hand rule!  

Copyright © 2012 Pearson Education Inc. Angular momentum For any system of particles  = dL/dt. For a rigid body rotating about the z-axis  z = I  z. It makes sense!

Copyright © 2012 Pearson Education Inc. Conservation of angular momentum When the net external torque acting on a system is zero, the total angular momentum of the system is constant (conserved).

Copyright © 2012 Pearson Education Inc. Conservation of angular momentum

Copyright © 2012 Pearson Education Inc. A spinning figure skater pulls his arms in as he rotates on the ice. As he pulls his arms in, what happens to his angular momentum L and kinetic energy K? A. L and K both increase. B. L stays the same; K increases. C. L increases; K stays the same. D. L and K both stay the same. Q10.11

Copyright © 2012 Pearson Education Inc. A spinning figure skater pulls his arms in as he rotates on the ice. As he pulls his arms in, what happens to his angular momentum L and kinetic energy K? A. L and K both increase. B. L stays the same; K increases. C. L increases; K stays the same. D. L and K both stay the same. A10.11

Copyright © 2012 Pearson Education Inc. Conservation of angular momentum

Copyright © 2012 Pearson Education Inc. A rotational “collision” Find ω. Where is the energy lost?

Copyright © 2012 Pearson Education Inc. Angular momentum in a crime bust A bullet hits a door causing it to swing. Find ω.

Copyright © 2012 Pearson Education Inc. Angular momentum in a crime bust

Copyright © 2012 Pearson Education Inc. Gyroscopes and precession For a gyroscope, the axis of rotation changes direction. The motion of this axis is called precession.

Copyright © 2012 Pearson Education Inc. Non-rotating gyro

Copyright © 2012 Pearson Education Inc. A rotating flywheel For a spinning flywheel, the magnitude of the angular momentum stays the same, but its direction changes continuously.

Copyright © 2012 Pearson Education Inc. A rotating flywheel precession angular frequency: Ω = τ / L.

Copyright © 2012 Pearson Education Inc. A precessing gyroscopic What is the precession direction? CW or CCW?