Question

Linear displacement at a distance r from the fulcrum for an angular rotation θ (radians) is s = r·θ.
- Work done by a force F applied at that radius during the rotation is
W = F · s = F · r · θ.

Answer 1

Those statements are correct, with a couple of clarifications.

- Arc length: s = r·θ is the exact arc length for a point at distance r from the fulcrum for an angular displacement θ (θ in radians).

- Tangential force: the work formula W = F·s assumes F is the tangential component of the force (the component parallel to the instantaneous direction of motion). If the applied force makes an angle α with the tangent, use F_t = F cosα and W = F_t r θ = F r θ cosα. (Equivalently, if γ is the angle between F and the radial direction, F_t = F sinγ.)

- Torque form: defining torque τ = r × F (magnitude τ = r·F_t), the work can be written as
W = τ·θ.
If τ varies with angle, W = ∫ τ(θ) dθ over the rotation.

- Power: instantaneous power P = dW/dt = τ ω, where ω is angular speed.

- Sign convention: W is positive when the force/torque acts in the direction of the rotation and negative if it opposes the rotation.

Example: r = 0.5 m, purely tangential F = 10 N, θ = 2 rad → W = 10·0.5·2 = 10 J.