Last updated: 2026-05-18

Rotational force

Torque — also called the moment of force — is the measure of how effectively a force causes an object to rotate about a pivot. It depends on three quantities: the magnitude of the applied force, the distance from the pivot to the point where the force acts (the moment arm), and the angle between the force and the moment arm. The same force produces more torque when applied farther from the pivot, and only the component of force perpendicular to the moment arm contributes to rotation.

The formula is:

$\tau = F \cdot r \cdot \sin\theta$

where F is the applied force, r is the moment arm length, and θ is the angle between the force vector and the moment arm.

Angle dependence

The factor $\sin\theta$ captures the effective perpendicular component of the force:

$F_\perp = F \cdot \sin\theta$

• At θ = 90°: $\sin 90° = 1$ — all of the force contributes to rotation. Maximum torque.
• At θ = 0° or 180°: $\sin \theta = 0$ — the force is along the arm, causing no rotation.
• At θ = 45°: $\sin 45° \approx 0.707$ — about 70.7% of the force is effective.

This is why pushing a door handle at right angles is more efficient than pushing at a shallow angle.

Units

The SI unit for torque is the newton-metre (N·m). In imperial contexts, torque is given in foot-pounds-force (ft·lbf):

$1 \text{ ft·lbf} \approx 1.356 \text{ N·m}$

Engine torque specs use N·m in Europe and ft·lbf in the United States. Fastener torque values in service manuals may use either, so unit conversion matters in practice.

Worked example

Problem: A mechanic applies 80 N of force to the end of a 0.3 m wrench at 75° to the handle. What is the torque on the bolt?

$\tau = 80 \times 0.3 \times \sin(75°) = 24 \times 0.9659 \approx 23.2 \text{ N·m}$

If the same force were applied perpendicular to the wrench (θ = 90°):

$\tau = 80 \times 0.3 \times 1 = 24 \text{ N·m}$

The 75° angle captures 96.6% of the maximum — nearly as effective as perpendicular.