6.2 Torque and Work

Work done by a torque

When a torque acts on a rotating object through an angular displacement, it does work: W = τΔθ, where θ is in radians. This is the rotational analog of W = Fd. If the torque is constant and in the same direction as the rotation, it adds energy to the system's rotational motion.

Rotational work-energy theorem

Just like linear work changes translational KE, rotational work changes rotational kinetic energy: W_net = ΔKE_rot = ½Iω_f² − ½Iω_i². A net torque that speeds up rotation does positive work; one that opposes rotation (like friction) does negative work, removing energy.

Power in rotational systems

Rotational power is the rate of doing rotational work: P = τω (torque times angular velocity), analogous to P = Fv. This tells you how quickly energy is being transferred to or from a rotating system, such as a motor spinning a wheel.

Key terms

Work done by torque (formula)

W = τΔθ, where Δθ is angular displacement in radians.

Rotational work-energy theorem

W_net = ΔKE_rot = ½Iω_f² − ½Iω_i²; net rotational work changes rotational kinetic energy.

Rotational power

P = τω, the rate at which a torque does work on a rotating system.

Sign of work from opposing torque

Negative — a torque acting opposite to the direction of rotation removes rotational kinetic energy.

Why must angle be in radians for W = τΔθ?

Radians are dimensionless, keeping units consistent so torque (N·m) × angle gives energy in joules.

Try an AP-style question

A constant 10 N·m torque turns a wheel one full revolution. How much work does it do?

1. 20 J
2. 10π J
3. 20π J
4. 10 J