Torque–time area changes angular momentum
You will be able to: Use signed torque–time areas and angular-momentum graph slopes.
How does a brief push change a wheel’s angular momentum?
A 3 N·m net torque acts on a wheel for 2 seconds. It gives the wheel 6 N·m·s of angular impulse and changes its angular momentum by 6 kg·m²/s. How far it turns is a different question.
A useful starting point: A straight-moving object can have angular momentum →
Words and symbols before equations
- Angular impulse
- Torque accumulated over time, in N·m·s, equivalent to kg·m²/s.
- ΔL
- Final angular momentum minus initial angular momentum.
- Net external torque
- Sum of torques from outside the selected system about the same axis.
- Slope of L versus t
- Change in L divided by time; gives net torque for a straight segment.
What this picture assumes
Initial L=+4 kg·m²/s. Net external torque +3 N·m for 0–2 s, then −2 N·m for 2–5 s. CCW positive. At the torque jump, L remains continuous but its slope changes.
Connect the picture to the physics
For constant net external torque, ΔL=τ_netΔt. When torque varies, sum signed areas under its time graph. Use the average of endpoint torques for a straight segment; do not use the final torque for the entire interval.
The same relationship read backward says the slope of L–t is net external torque. A flat L–t graph has zero net external torque even if L itself is not zero. A negative slope means L is becoming more negative; compare with the current L to decide whether the rotational speed is falling.
For fixed I, ΔL=IΔω and τ_net=Iα. If I changes, torque still governs ΔL, but substituting τ=Iα alone misses the effect of changing mass distribution. Keep the more general angular-momentum balance.
A worked example, step by step
A wheel starts with L=+4 kg·m²/s. Net torque is +3 N·m for 2 s, then −2 N·m for 3 s. Find L after each interval.
- First angular impulse=(3)(2)=+6 kg·m²/s.
- L after 2 s=4+6=10 kg·m²/s.
- Second angular impulse=(−2)(3)=−6 kg·m²/s.
- Final L=4 kg·m²/s. The L–t graph has slopes +3 then −2 N·m; equal opposite impulses restore its starting value.
Use ΔL=L_f−L_i with signs. Equal opposite torque–time areas cancel even when durations differ.
A wheel has constant nonzero L. Must a nonzero torque maintain it?
Compare with an explanation
No. Zero net external torque maintains constant L.
Predict. Change one thing. Explain.
Select elapsed time through both torque intervals. Predict L from accumulated area, then compare with the L–t graph. Explain the change of slope at 2 s.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At t=3 s, angular impulse=4 kg·m²/s and L=8 kg·m²/s. Slope is +3 N·m before 2 s and −2 N·m after 2 s.
Initial L=+4 kg·m²/s. Net external torque +3 N·m for 0–2 s, then −2 N·m for 2–5 s. CCW positive. At the torque jump, L remains continuous but its slope changes.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant physical relationship or contact condition to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA wheel has I=2 kg·m² and initial ω=+1 rad/s. A constant net torque +4 N·m acts for 3 s. (a) Find initial L. (b) Find angular impulse. (c) Find final ω. (d) Describe the slope and endpoints of L–t.
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Compare with the answer and four-point rubric
- 1 point: L_i=2 kg·m²/s.
- 1 point: ΔL=12 kg·m²/s.
- 1 point: L_f=14, so ω_f=14/2=7 rad/s.
- 1 point: Straight line from (0 s,2) to (3 s,14), slope +4 N·m.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What changes L?
Net external angular impulse.
RECALL 2What does the slope of L–t give?
Net external torque.
RECALL 3When can ΔL be replaced by IΔω?
When I is constant about the chosen axis.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Torque–time area changes angular momentum
- ΔL = signed area under τ_net versus t.
- Constant torque: ΔL=τ_netΔt. Slope of L–t=τ_net.
Remember: Use ΔL=L_f−L_i with signs. Equal opposite torque–time areas cancel even when durations differ.
Conditions: Initial L=+4 kg·m²/s. Net external torque +3 N·m for 0–2 s, then −2 N·m for 2–5 s. CCW positive. At the torque jump, L remains continuous but its slope changes.
Refresh Kid · Unit 6 · Objectives 6.3.B, 6.3.C · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.3, objectives 6.3.B, 6.3.C. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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