Calculator of Alexandria

Torque calculator

Torque from force and lever arm at any angle, in N·m and lbf·ft, with τ = rF sin θ derived and an explanation of why longer wrenches help.

From the pivot (the bolt’s centre) to where the force is applied.

90° when you push square to the wrench.

Torqueτ\tau
Torque (lbf·ft)
Force square to the arm (N)F⊥F_\perp

Formula

τ=rFsin⁡θ\tau = rF\sin\theta
How it's derived

The formula

τ=rFsin⁡θ\tau = r F \sin\theta

τ\tau is the torque, rr the distance from the pivot to where the force acts (the lever arm), FF the force and θ\theta the angle between the arm and the force.

Where it comes from

Torque measures how strongly a force turns something about a pivot. Two things make a turn stronger: pushing harder, and pushing farther from the pivot. That’s why torque is force times distance.

Only the part of the force at right angles to the arm turns it. Split FF into a part along the arm, Fcos⁡θF\cos\theta, which pushes or pulls straight into the pivot and does nothing, and a part square to the arm:

F⊥=Fsin⁡θF_\perp = F\sin\theta

Torque is that perpendicular part times the arm length:

τ=r F⊥=rFsin⁡θ\tau = r\,F_\perp = r F \sin\theta

There’s a second way to see it. Extend the line of the force, and the perpendicular distance from the pivot to that line is rsin⁡θr\sin\theta. Torque is the full force times that “moment arm”. Both views give the same number.

Why does a longer arm help? Turn a wrench through a small angle and your hand travels farther on a long wrench. The work in equals the work out, so more distance at the hand means more force at the bolt.

Worked example

Pushing with 150 N at the end of a 30 cm wrench, square to the handle:

τ=0.30×150×sin⁡90∘=45 N⋅m\tau = 0.30 \times 150 \times \sin 90^\circ = 45\ \text{N·m}

That’s 33.2 lbf·ft. Push at 60° to the handle instead, and sin⁡60∘=0.866\sin 60^\circ = 0.866, so the torque drops to 39 N·m.

Watch out for

  • N·m isn’t joules here. Torque and energy share units, but they’re different things. Torque is written in N·m, never J.
  • Torque specs are for clean threads. A bolt’s torque spec assumes a certain thread condition. Oil or anti-seize can make the same torque stretch the bolt much more.
  • Measure the arm to where your hand pushes, not to the end of the handle.

Sources

Last checked