6.4 Conservation of Angular Momentum

Angular momentum and its conservation

Angular momentum L = Iω (or L = r × p for a point mass) is conserved when the net external torque on a system is zero. Just like linear momentum, angular momentum can be transferred between parts of a system, but the total stays constant if no outside torque acts.

Changing shape changes ω

Since L = Iω stays constant, if a system's moment of inertia changes (e.g., a skater pulling in their arms), angular velocity must change to compensate: I₁ω₁ = I₂ω₂. Decreasing I increases ω, and vice versa — this is why divers spin faster when tucked.

Collisions and angular impulse

In rotational collisions (like a ball sticking to a rotating rod), conserve L about a fixed axis or point, not necessarily kinetic energy. Angular impulse = ∫τ dt = ΔL, the rotational analog of linear impulse-momentum theorem. Always choose a consistent pivot point when computing L = r × p for each object.

Key terms

Angular momentum (L)

L = Iω for rigid bodies, or L = r × p for a point mass; a vector quantity measuring rotational motion.

Conservation of angular momentum

Total angular momentum of a system stays constant if the net external torque acting on it is zero.

I₁ω₁ = I₂ω₂

Equation used when a system's moment of inertia changes with no external torque, so angular velocity adjusts to conserve L.

Angular impulse

∫τ dt = ΔL; the rotational analog of the linear impulse-momentum theorem.

Rotational inelastic collision

A collision where angular momentum is conserved but kinetic energy is not, e.g., a mass sticking to a rotating object.

Try an AP-style question

A 50 kg child runs at 4 m/s tangentially and jumps onto the edge of a merry-go-round (I = 200 kg·m², R = 2 m) at rest. What is the final ω?

1. 2 rad/s
2. 1 rad/s
3. 4 rad/s
4. 0.5 rad/s