AP® Physics C: Mechanics · Unit 6 · Topic 6.2
Work Done by a Torque
Work done by a torque equals the torque times the angle it turns through, or the integral of torque over angle when torque changes. It is the rotational twin of W = F times d.
Turning a stiff bolt takes real effort because you are pushing a force through a distance, just the rotational version of it. Every point on the wrench moves along an arc as the bolt turns, and your hand's force acts along that arc the whole way. Multiply the twisting effort by how far it rotates and you get energy transferred into the bolt system, same as pushing a crate across a floor transfers energy via force times distance.
Linear work is the integral of F dx. Rotational work is the integral of tau d(theta), torque integrated over angle instead of force over distance. If torque stays constant, the integral collapses to simple multiplication: W = tau times theta, just like W = F times d when force is constant. If torque varies with angle, you have to integrate tau(theta) across the angular range, the same way you would integrate a changing force over position. The units work out too: torque in newton meters times angle in radians gives joules, consistent because radians are dimensionless.
This topic sets up the rotational work-energy theorem: the net work done by torques equals the change in rotational kinetic energy, (1/2) I omega2. That connects directly to angular momentum and rolling motion later in the unit, where energy and momentum bookkeeping both rely on knowing how much work a torque actually delivers. Grasping the integral here also makes the area-under-the-curve picture on a torque-angle graph intuitive, since that area is exactly the work done.
Key terms
- Torque
- The rotational analog of force, equal to force times the lever arm, measured in newton meters.
- Rotational work
- Energy transferred by a torque acting through an angle, W = integral of tau d(theta).
- Radian
- A dimensionless unit of angle, so torque times radians still gives joules.
- Torque versus angle graph
- A plot whose area under the curve equals the work done by the torque.
Watch it explained
- How it worksHow does the integral of torque over angle mirror the linear work integral?
- Worked exampleFind the work done by a steady 15 N m torque turning a wheel through 4 radians.
- Worked exampleFind the work done by a torque of 6 theta newton meters as the angle goes from 0 to 2 radians.
- Going furtherWhat does the area under a torque versus angle graph physically represent?
Check yourself
End of topic.