Torque calculator: moments, angles and balancing

Torque is the turning effect of a force: force times its perpendicular distance from the pivot, τ = Fd, in newton metres (N m). Pushing a spanner with 40 N at 0.3 m from the bolt gives 40 × 0.3 = 12 N m. If the force is at an angle θ to the arm, only part of it turns: τ = Fd sin θ. UK courses usually call torque a moment. This torque calculator is our Moments Calculator: it finds torque, force or distance, handles angled forces, and balances a bar with the principle of moments.

It opens on that spanner. Open the full moments calculator over this page to try your own values.

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Moments Calculator

Balance moments and find the torque about a pivot

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How to use the torque calculator

1. Under Calculate, choose Moment or Balance.
2. In Moment, choose Moment, Force or Distance for τ = Fd, or At an angle for τ = Fd sin θ.
3. In Balance, choose Force or Distance to balance a bar, or Net moment to see which way it turns.
4. Type the values you know, with units from each box.
5. Read the answer and the steps. The diagram draws the bar with each force to scale.

Torque formulas

d is the lever arm: the shortest distance from the pivot to the force's line of action, measured at right angles to it. When the force is at angle θ to the arm, the lever arm is d sin θ. A force pointing straight at the pivot (θ = 0) has no turning effect at all.

The principle of moments

Worked examples you can check in the calculator

Example 1: a spanner

τ = 40 × 0.3 = 12 N m. The calculator adds that at twice the distance, 0.6 m, only 20 N would give the same torque. That is why a longer spanner makes a stiff bolt easier to undo.

Example 2: a force at an angle

The same 40 N acts 0.3 m from the pivot, but at 60° to the spanner. Choose At an angle:

The torque is 10.4 N m, less than the 12 N m a force at 90° would give. The perpendicular distance is 0.3 × sin 60° = 0.26 m.

Example 3: balancing a seesaw

A child weighing 300 N sits 2 m from the pivot. Where must a 400 N child sit to balance it? By the principle of moments, 300 × 2 = 400 × d, so d = 600 ÷ 400 = 1.5 m. Choose Balance, then Distance. Choosing Force instead tells you what weight would balance at a given distance.

Example 4: net torque

On a bar, 400 N acts clockwise 1.5 m from the pivot and 500 N acts anticlockwise 1 m from it. The clockwise moment is 600 N m and the anticlockwise moment is 500 N m, so the net torque is 100 N m clockwise, and the bar turns clockwise. Choose Net moment to check.

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Torque as a vector

In university physics, torque is the cross product τ = r × F, a vector along the axis of rotation, with size rF sin θ. The cross product calculator guide shows how to work it out from components.

Common mistakes

• Using the length along a slanted line. d must be perpendicular to the force, or use d sin θ.
• Using cos instead of sin when θ is the angle between the force and the arm.
• Measuring from the wrong point. Distances are from the pivot, not from the end of the bar.
• Mixing cm and m. Use metres for N m.
• Adding clockwise and anticlockwise moments. They act in opposite directions, so subtract one from the other.

Read next

Physics · MechanicsPrinciple of moments: moment and torque equations explainedThe principle of moments explained: the moment equation M = Fd, balanced seesaws, forces at an angle, beams on two supports, couples and torque, examples.