6.3 Angular Momentum and Angular Impulse

Defining Angular Momentum

Angular momentum (L) measures rotational motion: L = Iω, where I is rotational inertia and ω is angular velocity. It's the rotational analog of linear momentum (p = mv). Units are kg·m²/s. A spinning figure skater, a rotating planet, and a rolling wheel all possess angular momentum determined by how mass is distributed and how fast the object spins.

Angular Impulse Changes Angular Momentum

Angular impulse equals torque applied over a time interval: J = τΔt = ΔL. This is the rotational version of the impulse-momentum theorem. A net external torque acting over time changes an object's angular momentum. If torque varies, angular impulse equals the area under a torque-vs-time graph.

Conservation of Angular Momentum

When net external torque is zero, angular momentum is conserved: L_initial = L_final. This explains why a skater spins faster when pulling arms in (I decreases, so ω increases to keep Iω constant) and governs collisions between rotating objects, like a person jumping onto a rotating platform.

Key terms

Angular momentum (L)

L = Iω; the rotational analog of linear momentum, measured in kg·m²/s.

Angular impulse

J = τΔt = ΔL; torque applied over time changes angular momentum.

Conservation of angular momentum

When net external torque = 0, total angular momentum stays constant (L_i = L_f).

Why does a spinning skater speed up when pulling arms in?

Decreasing rotational inertia (I) with L conserved causes angular velocity (ω) to increase.

Area under a torque vs. time graph

Represents angular impulse, which equals the change in angular momentum (ΔL).

Try an AP-style question

A solid disk (I = ½MR²) of mass 4 kg and radius 0.5 m spins at 10 rad/s. What is its angular momentum?

1. 5 kg·m²/s
2. 20 kg·m²/s
3. 2.5 kg·m²/s
4. 10 kg·m²/s