Torque

Torque, or moment of force, is a measure of the influence of a force in producing rotational motion about an axis. For a force F applied at a point with position vector r relative to the axis, the torque is τ = r × F. The magnitude is τ = r F sin θ, where θ is the angle between r and F, and the torque vector points along the rotation axis according to the right-hand rule. Torques are additive, so the net torque about an axis is the sum of the individual torques.

The SI unit of torque is the newton-meter (N·m). This unit is the same as the unit of energy (joule), but torque and energy have different meanings; torque measures rotational effect, while energy measures work done or stored.

Relation to angular motion: for a rigid body rotating about a fixed axis, the net external torque equals the moment of inertia times angular acceleration, ∑τ = Iα. In general, torque equals the time derivative of angular momentum, τ = dL/dt. The direction of torque is determined by the right-hand rule.

Work, power, and dynamics: if the torque is constant and the rotation angle changes by dθ, the incremental work is dW = τ dθ. Instantaneous power delivered by the torque is P = τ ω, where ω is the angular velocity.

Common examples and concepts: applying torque to a door about its hinge, or turning a bolt with a wrench, illustrates how lever arm length and the angle of applied force affect rotational effect. A pure couple consists of two equal and opposite forces whose moment produces rotation without a net force. Sign conventions for torque depend on the chosen rotational direction.