5.7 Conservation of Angular Momentum

Angular Momentum Basics

Angular momentum (L) measures rotational motion: L = Iω, where I is moment of inertia and ω is angular velocity. For a point mass, L = mvr(sinθ). Angular momentum is a vector quantity with direction given by the right-hand rule, and its units are kg·m²/s.

Conservation Law

When net external torque is zero, total angular momentum of a system stays constant: L_initial = L_final. If moment of inertia changes (e.g., a spinning skater pulls in arms), angular velocity must change oppositely to keep Iω constant — this explains ice skaters spinning faster when they retract their arms.

Angular Impulse & Collisions

Angular impulse (torque × time) causes a change in angular momentum: τΔt = ΔL. In rotational collisions (like a disk catching a ball), angular momentum is conserved even though kinetic energy may not be, similar to linear momentum in collisions.

Key terms

Angular momentum (L)

A measure of rotational motion; L = Iω for rigid bodies, or L = mvr sinθ for a point mass. Vector quantity, units kg·m²/s.

Conservation of angular momentum

When net external torque on a system is zero, total angular momentum remains constant: L_initial = L_final.

Angular impulse

The product of torque and time interval (τΔt); equals the change in angular momentum (ΔL).

Why skaters spin faster with arms in

Pulling arms in reduces moment of inertia (I); since L = Iω is conserved, angular velocity (ω) increases.

Angular momentum in collisions

Conserved when no external torque acts on the system, even if kinetic energy is not conserved (e.g., inelastic rotational collisions).

Try an AP-style question

A skater pulls in her arms, halving her rotational inertia. What happens to her angular speed?

1. It stays the same
2. It quadruples
3. It halves
4. It doubles