6.3 Angular Momentum and Angular Impulse

Defining Angular Momentum

Angular momentum for a particle is L = r × p = r × mv, with magnitude L = mvr·sin(θ). For a rigid body rotating about a fixed axis, L = Iω, where I is the moment of inertia and ω is angular velocity. Angular momentum is a vector, directed along the rotation axis using the right-hand rule.

Angular Impulse–Momentum Theorem

Net torque causes a change in angular momentum: τ_net = dL/dt. Integrating over time gives the angular impulse–momentum theorem: ∫τ_net dt = ΔL. This mirrors the linear impulse-momentum theorem, letting you relate a torque applied over a time interval to the resulting change in rotational motion.

Conservation of Angular Momentum

When net external torque is zero, angular momentum is conserved: L_i = L_f. This applies to systems like a spinning skater pulling in their arms (I decreases, ω increases) or a collision involving rotation. Total L of a system is conserved even if internal torques redistribute it between parts.

Key terms

Angular momentum (particle)

L = r × p; magnitude L = mvr·sin(θ), directed by the right-hand rule.

Angular momentum (rigid body)

L = Iω, where I is moment of inertia and ω is angular velocity about a fixed axis.

Angular impulse–momentum theorem

∫τ_net dt = ΔL — net torque applied over time changes angular momentum.

τ_net = dL/dt

Net torque equals the time rate of change of angular momentum, analogous to F = dp/dt.

Conservation of angular momentum

If net external torque is zero, total angular momentum of a system remains constant (L_i = L_f).

Try an AP-style question

A torque τ(t) = 3t² (N·m) acts on a wheel at rest from t = 0 to 2 s. What angular momentum does it gain?

1. 4 kg·m²/s
2. 24 kg·m²/s
3. 8 kg·m²/s
4. 12 kg·m²/s