5.7 Angular Momentum and Its Conservation

Defining Angular Momentum

Angular momentum L of a point particle is L = r × p = mvr sinθ, and for a rigid body rotating about an axis, L = Iω. It's a vector quantity, with direction given by the right-hand rule, perpendicular to the plane of rotation. Units are kg·m²/s.

Torque and Rate of Change of L

Just as F_net = dp/dt, rotational motion obeys τ_net = dL/dt. If the net external torque on a system is zero, angular momentum is constant. This is the rotational analog of Newton's second law, linking torque directly to the time rate of change of angular momentum.

Conservation of Angular Momentum

When net external torque = 0, total angular momentum is conserved: L_i = L_f. Since L = Iω, if a system's moment of inertia I changes (e.g., a figure skater pulling in arms), ω must change inversely to keep L constant. This principle explains spinning objects speeding up or slowing down as their mass distribution shifts.

Key terms

Angular momentum (particle)

L = r × p = mvr sinθ, a vector perpendicular to the plane containing r and v.

Angular momentum (rigid body)

L = Iω, where I is the moment of inertia and ω is angular velocity.

Torque–angular momentum relation

τ_net = dL/dt — net external torque equals the rate of change of angular momentum.

Conservation of angular momentum

If τ_net = 0, total angular momentum stays constant: L_i = L_f.

Why skaters spin faster with arms in

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

Try an AP-style question

A 2 kg particle moves at 3 m/s along a line passing 4 m from the origin (perpendicular distance). What is its angular momentum about the origin?

1. 12 kg·m²/s
2. 0
3. 24 kg·m²/s
4. 6 kg·m²/s