Physics
How To Calculate Torque
Torque is the turning effect of a force, and the lever arm matters as much as the force: the same push does far more with a longer spanner.
Quick Answer
torque = force x lever arm
- tau
- Torque, in newton metres
- F
- Force applied, in newtons
- r
- Perpendicular distance from pivot to the line of the force
Multiply the force by the perpendicular distance from the pivot. Pushing a wrench with 150 newtons at 0.4 metres from the bolt gives 60 newton metres, which is about 44 foot pounds. Longer lever arms multiply the same effort.
What Is Torque?
Torque is the turning effect of a force about a pivot. It is force multiplied by the perpendicular distance from the pivot to the line along which the force acts, so pushing a wrench with one hundred and fifty newtons at a distance of 0.4 metres gives sixty newton metres.
The distance is called the lever arm, and it is measured perpendicular to the force. That single word explains most of the confusion: a force pushing straight at the pivot produces no turning at all, because its lever arm is zero.
Torque is what actually loosens a stubborn bolt. Applying the same force further out on the handle increases the torque proportionally, which is why breaker bars are long. Doubling the lever arm doubles the turning effect without any extra effort.
The unit is the newton metre, written N m. Despite looking like a joule, it is not one, because the two quantities are defined differently: energy is force times distance moved, torque is force times distance at right angles to the motion. The units coincide but the concepts do not.
Torque and power are related but distinct. Power equals torque multiplied by angular velocity, so an engine can deliver high torque at low speed or lower torque at high speed for the same power. This is exactly what a gearbox trades between.
The direction matters too. Torque is a vector, and pushing on opposite sides of a pivot turns it opposite ways. A pair of equal and opposite forces applied on either side is called a couple, and it produces pure rotation with no net force.
In static equilibrium the torques must balance. A see saw works because a heavy child close to the pivot can balance a lighter one far away, and the rule is that each weight times its distance must be equal on both sides.
Torque wrenches are set in newton metres or foot pounds because the tightness of a fastener is really a specification of torque. Too little and the joint works loose, too much and the thread strips or the bolt stretches past its elastic limit.
Real lever arms are measured from the pivot to the line of action, not to the point where your hand happens to be. If you push at an angle, the perpendicular distance shrinks, which is why pushing square on a handle is the most effective way to turn it.
Conversion between units is common because vehicle specifications use foot pounds. One newton metre is about 0.7376 foot pounds, and one foot pound is about 1.3558 newton metres. Mixing them up is a classic specification error.
Finally, torque does not require movement. A bolt that will not budge is still experiencing torque, and so is a shelf bracket holding a load. Only when the object actually rotates does the torque do work, and then the work is torque multiplied by the angle turned.
Formula
tau = F x r
Force in newtons times the perpendicular distance from the pivot, in metres. The result is in newton metres.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| tau | Torque | newton metres | Turning effect about the pivot. |
| F | Force | newtons | Force applied, perpendicular to the lever arm. |
| r | Lever arm | metres | Perpendicular distance from pivot to the line of the force. |
P = tau x angular velocity
Power is torque times rotational speed, which is why gearing trades speed against turning force.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Power | watts | Rate of doing rotational work. |
| tau | Torque | newton metres | Turning effect. |
| omega | Angular velocity | rad/s | Rotational speed in radians per second. |
foot pounds = newton metres / 1.3558
Vehicle specifications often quote foot pounds, so the conversion comes up constantly in practice.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| N m | Newton metres | N m | Metric torque unit. |
| lb ft | Foot pounds | lb ft | Imperial torque unit used in vehicle manuals. |
How To Calculate Torque
- 1
Find the pivot
Every torque is measured about a specific point. For a wrench it is the bolt, for a door it is the hinge, and for a see saw it is the fulcrum.
- 2
Measure the perpendicular distance
Distance from the pivot to the line along which the force acts, at right angles to the force. Not the distance to your hand if you are pushing at an angle.
- 3
Convert the force to newtons
A mass in kilograms has to be multiplied by 9.81 to become a weight in newtons before it can be used as a force.
- 4
Multiply force by lever arm
One hundred and fifty newtons at 0.4 metres gives sixty newton metres. Doubling either the force or the distance doubles the torque.
- 5
Convert the unit if a specification demands it
Divide newton metres by 1.3558 for foot pounds, or multiply foot pounds by 1.3558 to go the other way. Never compare two figures without checking which unit each uses.
Examples
Example 1: A wrench pushed with 150 newtons at 0.4 metres
- Force
- 150 N
- Lever arm
- 0.4 m
| Step | Calculation | Result |
|---|---|---|
| Torque in foot pounds | 60 / 1.3558 | 44.25 |
| Torque in kilogram force metres | 60 / 9.80665 | 6.12 |
| Torque in newton metres | 150 x 0.4 | 60 |
Result: Pushing 150 newtons on a 0.4 metre wrench applies 60 newton metres of torque, which is about 44.25 foot pounds.
Example 2: The same force on a longer breaker bar
- Force
- 150 N
- Lever arm
- 1.2 m
| Step | Calculation | Result |
|---|---|---|
| Torque in foot pounds | 180 / 1.3558 | 132.76 |
| Torque in kilogram force metres | 180 / 9.80665 | 18.35 |
| Torque in newton metres | 150 x 1.2 | 180 |
Result: A bar three times as long turns the same 150 newton push into 180 newton metres, triple the torque for no extra effort.
Example 3: A wheel nut tightened with 80 newtons at 1.5 metres
- Force
- 80 N
- Lever arm
- 1.5 m
| Step | Calculation | Result |
|---|---|---|
| Torque in foot pounds | 120 / 1.3558 | 88.51 |
| Torque in kilogram force metres | 120 / 9.80665 | 12.24 |
| Torque in newton metres | 80 x 1.5 | 120 |
Result: Eighty newtons at 1.5 metres gives 120 newton metres of torque, comfortably past the 110 newton metres most car wheel nuts require.
Calculator
Torque
60
- Torque in foot pounds
- 44.2537
- Torque in inch pounds
- 531.0447
- Torque in kilogram force metres
- 6.1183
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the Torque calculator page.
Common Mistakes
Using the distance along the handle instead of perpendicular
The lever arm is the perpendicular distance from the pivot to the line of the force. Pushing at an angle reduces it, which is why a push that is not square on feels so much weaker.
Confusing torque with power
Torque is a turning force and power is how fast work is done. An engine can make large torque at low speed and small torque at high speed while producing the same power.
Treating newton metres as joules
The units look identical and are dimensionally the same, but torque and energy are different concepts. Torque times the angle turned gives work, not torque alone.
Mixing foot pounds with newton metres
Specifications from different sources use different units, and the two differ by about 1.36. Comparing them directly leads to fasteners that are dangerously over or under tightened.
Forgetting to convert mass into force
Standing on a wrench applies your weight, not your mass. A seventy kilogram person applies about 687 newtons, and using seventy instead understates the torque by a factor of ten.
Ignoring the direction of the force
Torque depends on which way the force pushes relative to the pivot. Equal forces on opposite sides cancel, and a force aimed at the pivot produces no rotation at all.
Assuming a longer spanner always helps
Beyond the elastic limit of the fastener, more torque simply stretches or shears the bolt. Torque specifications exist precisely because more is not better.
FAQ
What is the formula for torque?
Torque equals force multiplied by the perpendicular distance from the pivot to the line of the force. Force in newtons times distance in metres gives newton metres.
What is the difference between torque and power?
Torque is a turning force, measured in newton metres. Power is the rate of doing work, measured in watts, and equals torque multiplied by rotational speed.
Why does a longer wrench make bolts easier to loosen?
Because torque is force times lever arm. Moving your hand further from the bolt increases the distance, so the same push produces proportionally more turning effect.
Is a newton metre the same as a joule?
The units have the same dimensions but mean different things. A joule is energy, a newton metre is torque, and you cannot use one where the other is required.
How do you convert newton metres to foot pounds?
Divide by 1.3558. Sixty newton metres is about 44.25 foot pounds. Going the other way, multiply foot pounds by 1.3558.
Does torque require motion?
No. A stuck bolt under load experiences torque without moving. Work is only done when the object actually rotates, and then it is torque times the angle turned.
References
- [1]Wikipedia, Torque — https://en.wikipedia.org/wiki/Torque
- [2]Wikipedia, Lever — https://en.wikipedia.org/wiki/Lever
- [3]Wikipedia, Couple (mechanics) — https://en.wikipedia.org/wiki/Couple_(mechanics)
- [4]Wikipedia, Torque wrench — https://en.wikipedia.org/wiki/Torque_wrench