5.6 Angular Momentum and Angular Impulse

Angular Momentum

Angular momentum (L) measures rotational motion: L = Iω, where I is moment of 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 or a rotating planet both carry angular momentum determined by how their mass is distributed and how fast they spin.

Angular Impulse-Momentum Theorem

Angular impulse equals the change in angular momentum: τΔt = ΔL = Iωf − Iωi. A net torque applied over a time interval changes an object's angular momentum, just as a net force applied over time changes linear momentum. Larger torque or longer duration produces a bigger change in rotational motion.

Conservation of Angular Momentum

When net external torque is zero, angular momentum is conserved: L_initial = L_final. If moment of inertia decreases (mass pulled inward), angular velocity must increase to keep L constant — this is why a spinning skater speeds up when pulling arms in. This principle applies to isolated systems like planets, spinning ice skaters, and divers.

Key terms

Angular momentum (L)

L = Iω; a measure of rotational motion, the rotational analog of linear momentum.

Angular impulse

τΔt — the product of net torque and the time it acts; equals the change in angular momentum.

Angular impulse-momentum theorem

τΔt = ΔL = Iωf − Iωi; net torque over time changes angular momentum.

Conservation of angular momentum

If net external torque = 0, total angular momentum of a system stays constant (L_i = L_f).

Why skaters spin faster pulling arms in

Reducing moment of inertia (I) while L is conserved forces angular velocity (ω) to increase.

Try an AP-style question

An object with I = 2 kg·m² spins at 5 rad/s. What is its angular momentum?

1. 25 kg·m²/s
2. 2.5 kg·m²/s
3. 7 kg·m²/s
4. 10 kg·m²/s