Angular Impulse
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.
A torque acting over time changes angular momentum: ΔL = τΔt. This is the rotational version of impulse (Δp = FΔt). On a torque–time graph, ΔL is the area under the graph.
1. Torque over time
2. Same idea as Unit 4
Linear: a force for a time changes momentum. Rotational: a torque for a time changes angular momentum.
Use the torque, not the force. A force applied near the axis delivers less angular impulse than the same force at the rim.
A brake that pushes against a spinning wheel gives a negative torque, so L decreases. The bigger the torque or the longer it acts, the bigger the change.
3. Example, then try it
A motor applies a 0.80 N·m torque for 5.0 s to a fan (I = 0.20 kg·m²) that starts at rest. How fast is it spinning afterward?
ΔL = τΔt = (0.80)(5.0) = 4.0 kg·m²/s.
ω = L ÷ I = 4.0 ÷ 0.20 = 20 rad/s.
A wheel (I = 2.0 kg·m²) spins at 15 rad/s. A brake applies a 5.0 N·m torque against the spin. How long does it take to stop? How fast is it spinning if the brake acts for only 3.0 s?
Starting angular momentum: L = Iω = (2.0)(15) = 30 kg·m²/s.
To stop, ΔL = −30: Δt = 30 ÷ 5.0 = 6.0 s.
After 3.0 s: ΔL = (−5.0)(3.0) = −15, so L = 30 − 15 = 15 kg·m²/s.
Angular speed: ω = 15 ÷ 2.0 = 7.5 rad/s.
Answer: 6.0 s to stop; after 3.0 s it spins at 7.5 rad/s.
4. Common mistakes
The same torque acts for the same time on a small wheel and on a large, heavy wheel. How do their changes in angular momentum compare?
ΔL = τΔt depends only on the torque and the time, so ΔL is the same. The small wheel’s ω changes more, because it has less rotational inertia.
Ready to practice?
4 questions on Angular Impulse, each with a hint and an explanation for every wrong answer.