τ = rF·sinθ — Torque (Moment of Force)

Torque — the rotational equivalent of force — with the lever-arm formula and a worked wrench example.

\tau = rF\sin\theta

Torque equals lever arm times force times the sine of the angle between them.

What it means

Torque measures twisting effectiveness: force times lever arm, times the sine of the angle between them. Maximum twist comes from pushing perpendicular at the far end — long wrenches and right angles are why bolts come loose.

Variables

Symbol	Name	Unit	Meaning
$τ$	Torque	N·m	Twisting effect about the pivot
$r$	Lever arm	m	Distance from pivot to force application
$F$	Force	N	Applied push or pull
$θ$	Angle	degrees or rad	Angle between the lever arm and force

Assumptions

Force applied at a single point
r measured from the pivot to the force's line of application
θ is the angle between r and F

Worked example

Problem

You push a 0.30 m wrench with 50 N perpendicular to its handle. What torque do you apply to the bolt?

Given:

r = 0.30 m (wrench length)
F = 50 N
θ = 90° (push perpendicular)

Write the formula: $\tau = rF\sin\theta$
Substitute: $\tau = 0.30 \times 50 \times \sin 90°$
Calculate: $\tau = 0.30 \times 50 \times 1 = 15\ \mathrm{N·m}$

15 N·m

15 N·m of torque twists the bolt. Pushing at an angle wastes effort — only the perpendicular component (F·sinθ) turns the wrench, which is why you push at 90°.