Angular Momentum

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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:
L_i = L_f
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?

Your Answer:

Solution