Torque — Loosening a Bolt with a Wrench (Worked Example)

Easy

Problem

A mechanic pulls perpendicular to a 0.35 m wrench with a force of 180 N to loosen a stubborn bolt. What torque does she apply?

Given

r = 0.35 m
F = 180 N
θ = 90° (perpendicular)

Find

The torque τ about the bolt.

Principle

Torque is the rotational equivalent of force: the lever arm times the applied force (times sinθ for the angle between them).

Formula

\tau = rF\sin\theta

τ = rF sinθ

Solution

Write the torque formula: $\tau = rF\sin\theta$
The pull is perpendicular, so $\sin 90° = 1$ : $\tau = rF$
Substitute: $\tau = 0.35 \times 180$
Multiply: $\tau = 63\ \mathrm{N\cdot m}$

63 N·m counterclockwise (loosening direction)

Unit check

m × N = N·m, the unit of torque — dimensionally consistent.

What this means physically

63 N·m is a healthy torque for a car lug bolt (typical spec ~110 N·m, so she may need a longer wrench or both hands). Doubling the wrench length would double the torque for the same effort.

Common mistake

Using the full wrench length when the force isn't applied at the end, or forgetting sinθ when pulling at an angle. Only the perpendicular component of force (and the distance from the pivot) counts.