Angular Momentum
A rotating rigid body has angular momentum analogous
to a translating body with linear momentum. Similar to linear momentum, angular momentum is
also a conserved quantity when there is zero net external force on the system.
Definition of Angular Momentum:
Consider a single particle of mass m, position vector r subject to a force F. The linear
momentum is p = mv and L, the angular momentum about the origin O is given by
L = r x p = r x mv
dL/dt = r x mdv/dt = r x F = t
Just like the relationship between the linear momentum and force, we obtain the same
result for angular momentum and torque. Now we define angular momentum for rotational
dynamics.
L = Iw
t = Ia
Angular momentum is the product of moment of inertia and the angular velocity. Again, we
see that L is simply the time derivative of torque in rotational dynamics.
It was said earlier that angular momentum is a conserved quantity. Where do we see this
in real life? A good example would be an ice-skater who often spins and controls the
angular velocity by using his/her arms. When the arms are pulled in, the moment of inertia
I is decreased and thus the angular velocity increases. And the opposite, when the arms are
pulled out, the moment of inertia increases and the angular velocity decreases.
Conservation of Angular Momentum:
Li = Lf
The angular momentum of initial state of system is equal to its final state.
Sample Problem
A cue has w = 2rev/s and hits an object ball which
was initially at rest then slows down to w = 0.5rev/s.
What is the angular velocity of the object ball?