Read rotational work from an angle-dependent torque

1. Frame the question

A tangential force acting through a small arc transfers work FₜΔs. With Δs=rΔθ and τ=rFₜ, the same transfer is τΔθ. For a constant signed torque about a fixed axis, W=τΔθ, where radians are used. Torque and angular displacement with opposite signs transfer negative work.

2. Build the mechanism

When torque changes with angle, add small torque-angle areas. For the linear profile in this lesson, the exact signed area is a trapezoid: W=½(τstart+τend)Δθ. Areas above and below zero can cancel. A positive final energy does not generally prove the path was reachable; a body might stop earlier. Here the stipulated initial energy and bounded torques keep energy positive throughout the selected path.

Net work changes rotational kinetic energy for this fixed-axis constant-inertia model. If you calculate work from only one torque, other interactions may also transfer energy. An ideal stationary pivot does no translational work at its fixed point, but bearing friction would supply a torque and must be included in the net accounting.

Graph labels determine the meaning of area. Torque versus angle yields work in joules; torque versus time yields angular impulse in N m s. The two areas answer different questions. A torque-angle profile plus initial energy can determine final speed without giving elapsed time, so do not report a duration from the graph width in radians.

3. Work an example

The net torque changes linearly from +4 to −2 N m over 2 rad. Work is ½×(4−2)×2=2 J. With initial rotational energy 20 J and I=2 kg m², final energy is 22 J and positive angular velocity is √22≈4.690 rad/s. The negative late torque removes some energy but does not reverse the rotor in this supported case.

4. Predict and investigate

Predict first: Can a graph include both positive and negative torque while still adding net energy?

Compare start/end torques (4,−2), (2,−2) and (−4,−4). Compute each signed area and energy change. Sketch the cumulative work, identify where it reaches an extreme, and check that kinetic energy never reaches zero before the endpoint. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

A constant net torque −3 N m acts while a body turns +2 rad. Find work.

Your numerical result (J) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

What additional information would be required to infer angular impulse from a torque-angle graph?

Retain your first response, system and axis choices, separate scalar energy and signed angular-momentum ledgers, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic captures are not a performed physical experiment.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 6 topic mapping: 6.2. Checked 30 October 2026. Planar signed analysis without required three-dimensional angular-momentum direction. Slipping evolution is qualitative, in accordance with the framework boundary; no rolling-resistance model. Framework cross-reference, not AP authorization.