6.2 Torque and Work

Torque does rotational work

Just as force does work along a displacement, torque does work through an angular displacement: W = ∫τ dθ. If torque is constant, W = τΔθ. This work changes a rigid body's rotational kinetic energy, mirroring the linear work-energy theorem.

Rotational work-energy theorem

The net work done by torques equals the change in rotational kinetic energy: W_net = ΔKE_rot = ½Iω_f² − ½Iω_i², where I is moment of inertia. This parallels W = ΔKE for translation, with torque replacing force and angular displacement replacing linear displacement.

Power in rotation

Rotational power is the rate of doing rotational work: P = τω (constant torque) or P = dW/dt = τ(dθ/dt). This is analogous to P = Fv in translational motion, showing torque and angular velocity play roles similar to force and velocity.

Key terms

Work done by constant torque

W = τΔθ, the rotational analog of W = FΔx.

Work done by variable torque

W = ∫τ dθ, the area under a torque vs. angular displacement graph.

Rotational work-energy theorem

W_net = ΔKE_rot = ½Iω_f² − ½Iω_i².

Rotational power

P = τω for constant torque; generally P = dW/dt.

Analogy: torque & work vs force & work

Torque (τ) and angular displacement (θ) play the same roles as force (F) and linear displacement (x) in translational work.

Try an AP-style question

A torque τ(θ) = 2θ (N·m) acts from θ = 0 to θ = 3 rad. How much work does it do?

1. 3 J
2. 6 J
3. 9 J
4. 18 J