Angular Momentum
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.
Angular momentum is the rotational version of momentum. For a particle, L = r × p, with magnitude mvr⊥. For a rigid body spinning about a fixed axis, L = Iω. A net torque changes it: τnet = dL/dt.
1. Particles and rigid bodies
2. Things to notice
Angular momentum always depends on the chosen point (or axis). State it: “about O.”
For a particle moving at constant velocity, r⊥ never changes, so L about any point is constant: no force, no torque.
Direction: right-hand rule from r to p. In 2-D problems, counterclockwise is +, clockwise is −.
Angular impulse ∫τ dt is the area under a τ–t graph, just like linear impulse.
3. Example, then try it
A 2.0 kg ball moves at 5.0 m/s along a line passing 3.0 m from point O. Find its angular momentum about O.
L = mvr⊥ = (2.0)(5.0)(3.0) = 30 kg·m²/s.
A wheel (I = 2.0 kg·m²) starts from rest. A torque τ(t) = 6t − t² (N·m) acts on it from t = 0 to 3.0 s. Find the change in angular momentum, the final angular velocity, and the average torque.
Angular impulse: ΔL = ∫03(6t − t²) dt = [3t² − t³/3] = 27 − 9 = 18 kg·m²/s.
Final ω: L = Iω → ω = 18 ÷ 2.0 = 9.0 rad/s.
Average torque: ΔL/Δt = 18 ÷ 3.0 = 6.0 N·m.
Answer: 18 kg·m²/s, 9.0 rad/s, 6.0 N·m average.
4. Common mistakes
Ball A (1 kg) moves at 4 m/s along a line passing 2 m from point O. Ball B (2 kg) moves at 2 m/s along a line passing 1 m from O. Which has more angular momentum about O?
L = mvr⊥: A has (1)(4)(2) = 8 kg·m²/s; B has (2)(2)(1) = 4 kg·m²/s.
Ready to practice?
4 questions on Angular Momentum, each with a hint and an explanation for every wrong answer.