A torque does work through an angle
You will be able to: Use constant-torque work and the rotational work–energy theorem with appropriate signs.
Why does turning a wheel through a larger angle transfer more energy?
A motor turns a wheel through 3 radians while exerting a constant 4 N·m torque in the direction of rotation. It transfers 12 joules. A brake turning the same wheel through that angle removes energy because its torque opposes the turn.
A useful starting point: Add translation and rotation without double counting →
Words and symbols before equations
- Torque τ (tau)
- Turning effect of a force about an axis, in N·m.
- Angular displacement Δθ
- Signed angle turned in radians.
- Work W
- Energy transferred by a force acting through motion, in J.
- Net work
- Sum of work by all torques on the rotating body.
What this picture assumes
Fixed-axis rigid wheel, I=2 kg·m², initial K=20 J with positive ω. All allowed states have K≥0. At τ=−5 N·m and θ=4 rad it has just stopped; continuing the torque requires treating the reversal.
Connect the picture to the physics
For a tangential force at radius r, work is F times arc distance rΔθ. Since torque is rF, the same work is τΔθ. This reasoning explains why angle must be in radians. For constant torque, W=τΔθ with signs referred to the same fixed axis.
Positive work increases rotational kinetic energy; negative work decreases it. For a rigid object rotating about a fixed axis, W_net=ΔK_rot=½I(ω_f²−ω_i²). Include resistive torque when determining the net work.
The energy equation gives speed magnitude. It cannot by itself identify whether a wheel reversed direction. A constant braking torque also cannot remove more kinetic energy during a forward turn than the wheel initially has: after stopping, continued torque changes the motion. Check that an assumed interval is physically possible.
A worked example, step by step
A wheel with I=2 kg·m² starts at 2 rad/s. Net torque +3 N·m acts during a positive 4 rad turn. Find its final angular speed.
- W_net=τΔθ=(3)(4)=12 J.
- K_i=½(2)(2²)=4 J.
- K_f=4+12=16 J.
- 16=½(2)ω_f² gives |ω_f|=4 rad/s. The positive torque speeds this positive turn.
Torque and work have related units, but torque is not energy. Work also needs angular displacement.
Can a nonzero torque do zero work?
Compare with an explanation
Yes. If the angular displacement is zero, τΔθ is zero.
Predict. Change one thing. Explain.
Choose a positive or negative net torque, then increase the positive angular displacement. The wheel starts with 20 J. Observe energy gain or loss; the allowed range stops no later than zero energy.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
W=3×3=9 J; K_f=29 J; final angular speed=5.39 rad/s. K remains nonnegative.
Fixed-axis rigid wheel, I=2 kg·m², initial K=20 J with positive ω. All allowed states have K≥0. At τ=−5 N·m and θ=4 rad it has just stopped; continuing the torque requires treating the reversal.
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=1 kg·m² and initial ω=4 rad/s. A constant opposing net torque of magnitude 2 N·m slows its positive turn to rest. (a) Find K_i. (b) State net work to stop. (c) Find stopping angle. (d) Explain the work sign.
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Compare with the answer and four-point rubric
- 1 point: K_i=8 J.
- 1 point: W_net=0−8=−8 J.
- 1 point: −2Δθ=−8, so Δθ=4 rad.
- 1 point: Torque is negative while displacement is positive, so energy leaves the wheel.
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 1Work for a constant torque?
W=τΔθ in radians.
RECALL 2What does net rotational work change?
Rotational kinetic energy for fixed-axis rigid rotation.
RECALL 3Does energy alone give the sign of final ω?
No; it determines speed magnitude.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A torque does work through an angle
- Constant torque: W=τΔθ, using radians.
- Fixed-axis rigid rotation: W_net=ΔK_rot.
Remember: Torque and work have related units, but torque is not energy. Work also needs angular displacement.
Conditions: Fixed-axis rigid wheel, I=2 kg·m², initial K=20 J with positive ω. All allowed states have K≥0. At τ=−5 N·m and θ=4 rad it has just stopped; continuing the torque requires treating the reversal.
Refresh Kid · Unit 6 · Objectives 6.2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.2, objectives 6.2.A. 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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