Torque from perpendicular force:
formula and worked example

What torque measures

Torque is the rotational counterpart of force. A force makes things move in a line; a torque makes things turn. Opening a door, tightening a wheel nut and pedalling a bicycle all rely on torque, and the same push produces a very different turning effect depending on where it is applied.

That dependence on position is why door handles sit far from the hinges. A modest push near the hinge barely moves the door, while the same push at the handle swings it easily.

Formula, inputs and the right-angle condition

• Force: the push or pull in newtons, measured at the point where it is applied.
• Lever arm: the perpendicular distance from the axis of rotation to the line of action of the force, in metres.

The product uses the perpendicular distance. If the force is applied at an angle θ to the lever, only the perpendicular component counts, and torque equals force × distance × sin θ. At 90 degrees, sin θ is one, and the simple product applies.

Torque is quoted in newton-metres. Although this is the same combination as joules, torque is not energy, so we write N·m and keep J for work.

An 80 N push on a half-metre lever

1. 1Make sure the force is at right angles to the lever, so no angle correction is needed.
2. 2Take the lever arm as 0.5 m, measured from the axis to the point of push.
3. 3Multiply: 80 × 0.5 = 40.
4. 4Express the unit: 40 N·m.

As a check, the same 40 N·m could be produced by a 40 N force on a 1 m lever or a 160 N force on a 0.25 m lever. The torque stays the same while the force changes to match the lever.

Making use of a torque figure

Compare the result with the torque required by the task. Fasteners have specified tightening torques, and a wrench turned with less than that will leave them loose. Applying much more can strip threads or snap the bolt.

• Net torque zero: the object does not start rotating, or keeps turning at a steady rate.
• Net torque positive: angular speed increases in the chosen direction.
• Opposing torques: add with signs, so a seesaw balances when the two products are equal.

Torque also links to power. A shaft turning at an angular speed ω delivers torque × ω watts. Torque alone does not tell you how much power the machine has without the speed.

How force and lever arm trade off

A 10% rise in force, 88 N instead of 80, gives 44 N·m, exactly 10% more. Torque is linear in both force and lever arm.

This symmetry explains why a breaker bar works: adding a pipe to double the handle length doubles the torque for the same effort. In design, a longer lever buys either more turning effect or the same effect from a smaller person.

Slips that corrupt torque numbers

• Measuring distance along an angled handle instead of the perpendicular distance from the line of the force.
• Using centimetres for the lever arm in a formula meant for metres, giving a result 100 times too big.
• Ignoring the angle when the push is not at 90 degrees.
• Confusing torque with work or energy because the units look alike.

Practical scenario: a tight bolt

A bolt needs 40 N·m. With a 0.25 m spanner, you need 160 N, a hard pull. Swap to a 0.5 m spanner and 80 N is enough. A 1 m breaker bar brings it down to 40 N, approachable for most people.

The catch is accuracy. A long lever makes it easy to overshoot a tightening specification, so for fasteners with a specified value use a torque wrench and set the target number.