Torque with a Lever Arm (τ = rF sin θ)

Torque is the turning effect of a force about a pivot, and $\tau = rF\sin\theta$ says it depends on three things: how hard you push, how far from the pivot you push, and the direction you push in. Only the component of force perpendicular to the lever actually turns anything, which is the whole job of the $\sin\theta$. Push straight along a wrench handle, $\theta = 0$, and the bolt does not care at all. Push at right angles, $\theta = 90°$, and every newton counts.

A 200 N push applied perpendicular at the end of a 300 mm wrench delivers $\tau = 0.30 \times 200 \times \sin 90° = 60$ N·m. Slip a 600 mm breaker bar on instead and the same 200 N gives 120 N·m — the reason breaker bars exist, and the reason a wheel nut torqued to 120 N·m can be undone by a person who could not possibly generate that force directly.

The idea is old. Archimedes set out the law of the lever in the third century BC and is supposed to have said that with a place to stand he could move the Earth, which is this formula pushed to its limit: any torque is available if $r$ is large enough. The rotational world is built on it — the moment of a force in statics, the bending moment in a beam, and $\tau = I\alpha$ in dynamics are all the same quantity in different contexts.

The measurement error is measuring $r$ to the wrong point. The lever arm runs from the axis of rotation to the point where the force is applied, and both ends get mistaken. On a torque wrench with an extension or a crow's-foot adapter fitted, the effective length is no longer the wrench's marked length, and the reading on the scale no longer equals the torque at the fastener — that correction catches experienced people. On a bolted joint, the pivot is the bolt axis, not the edge of the bracket. The second mistake is a symbol swap: work is $W = Fd\cos\theta$ and torque is $\tau = rF\sin\theta$, and the two angles are measured the same way but enter through different trigonometric functions, because work wants the component along the displacement and torque wants the component across the lever. Getting them the wrong way round turns a maximum into a zero. One more, for when you solve for $\theta$: arcsine returns only the principal branch up to 90°, and the supplementary angle $180° - \theta$ produces exactly the same torque, so check which geometry your setup actually has.