Torque Calculator — turning force (τ = rF sin θ)

How it works

Torque is the rotational equivalent of force: it measures how effectively a force twists an object about an axis. OpenStax University Physics gives it as force times lever arm:

τ = r × F × sin θ

where r is the distance from the pivot to where the force is applied, F is the force, and θ is the angle between the lever arm and the force. Distance is in metres and force in newtons, so torque comes out in newton-metres (N·m).

Two things set the torque. First, the lever arm — the farther from the pivot you apply the force, the more torque you get, which is why a longer wrench loosens a stubborn bolt more easily. Second, the direction — only the component of force perpendicular to the arm turns it. Pushing at right angles (θ = 90°, sin θ = 1) gives the full r × F; pushing straight along the arm (θ = 0°) gives zero torque no matter how hard you push.

Worked example

A 50 N force on the end of a 0.3 m wrench, applied at 90° to the handle:

τ = 0.3 × 50 × sin 90° = 0.3 × 50 × 1 = 15 N·m

Apply that same force at 30° instead and only half of it turns the bolt: 0.3 × 50 × sin 30° = 7.5 N·m.

Frequently asked questions

What is torque?

Torque is the measure of the turning or twisting effect of a force about a pivot or axis. A larger force, a longer lever arm, or a more perpendicular push all increase the torque.

What is the formula for torque?

τ = r × F × sin θ, where r is the lever arm (distance from the pivot), F is the applied force, and θ is the angle between them. With metres and newtons, torque is in newton-metres (N·m).

Why does a longer wrench make it easier to loosen a bolt?

Because torque grows with the lever arm. Doubling the distance from the pivot doubles the torque for the same force, so a longer wrench delivers more twist with the same effort.

Why is the angle in the formula?

Only the part of the force perpendicular to the lever arm creates torque. The sine of the angle captures that: sin 90° = 1 gives maximum torque, while a force pointing along the arm (sin 0° = 0) produces none.

Are newton-metres the same as joules?

They have the same base units, but they describe different things. A joule is a newton-metre of energy (force along a distance); torque is a newton-metre of turning effect (force across a lever arm). By convention torque is written in N·m and energy in joules to keep them distinct.

How can I increase the torque?

Three ways: apply a larger force, use a longer lever arm (push farther from the pivot), or make the force more perpendicular to the arm. A breaker bar loosens a bolt a short wrench cannot because its extra length multiplies the same force into more torque.