6.4 Conservation of Angular Momentum

Angular momentum defined

Angular momentum (L) measures rotational motion: L = Iω, where I is rotational inertia and ω is angular velocity. For a point mass, L = mvr sin(θ). Like linear momentum, angular momentum is a vector, and its direction follows the right-hand rule along the rotation axis.

Conservation law

If the net external torque on a system is zero, total angular momentum is conserved: L_initial = L_final. This applies even when a system's shape changes, causing I to change — since L = Iω stays constant, ω must change inversely with I (a skater spinning faster when pulling arms in).

Applying conservation to systems

When objects in a system interact (collisions, mergers, changes in radius), use I₁ω₁ = I₂ω₂ if no external torque acts. Internal forces/torques redistribute angular momentum within the system but don't change the total. Check whether torque is truly zero — friction or external pushes violate conservation.

Key terms

Angular momentum (L)

A measure of rotational motion; L = Iω for rigid bodies, or L = mvr sin(θ) for a point mass.

Conservation of angular momentum

Total angular momentum stays constant when net external torque on a system is zero: L_initial = L_final.

Why skaters spin faster with arms in

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

I₁ω₁ = I₂ω₂

Equation used to relate angular momentum before and after a system's rotational inertia changes, with no external torque.

Direction of angular momentum

A vector quantity; its direction is found using the right-hand rule along the axis of rotation.

Try an AP-style question

A child walks from the edge of a freely spinning merry-go-round toward the center. What happens to its angular speed?

1. It stops
2. It increases
3. It stays the same
4. It decreases