### Angular momentum

```Physics 7C lecture 14
Angular momentum
Thursday November 14, 8:00 AM – 9:20 AM
Engineering Hall 1200
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.
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 Mvcm2 + 1/2 Icm2.
Rolling without slipping
• The condition for rolling without slipping is vcm = R.
• Figure 10.13 shows the combined motion of points on a
rolling wheel.
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.
Angular momentum
• What’s the analogue of momentum P for angular motion?
Angular momentum
• Momentum : p = m v
• angular momentum: τ = r × p = m r × v
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!
Angular momentum
• For any system of particles  = dL/dt.
• For a rigid body rotating about the z-axis z = Iz.
• It makes sense!
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).
Conservation of angular momentum
Q10.11
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
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.
Conservation of angular momentum
A rotational “collision”
• Find ω. Where is
the energy lost?
Angular momentum in a crime bust
• A bullet hits a door causing it to swing. Find ω.
Angular momentum in a crime bust
Gyroscopes and precession
• For a gyroscope, the axis of
rotation changes direction.
The motion of this axis is
called precession.
Non-rotating gyro
A rotating flywheel
• For a spinning flywheel, the magnitude of the angular
momentum stays the same, but its direction changes
continuously.