What torque is
Torque is the rotational equivalent of force: it measures how strongly a force tends to twist an object about an axis. Engineers often call it the moment of the force, and the two words mean the same thing. Its unit is the newton-metre, formed the same way as the joule but describing something entirely different — a turning effect, not an energy — which is why torque is written N·m and never J.
Three quantities set it. The size of the force is the obvious one. The distance from the axis is the reason a long breaker bar loosens a nut that a short spanner will not. And the angle matters because only the component of force perpendicular to the arm actually turns anything; a pull directed straight at the axis just tries to move the whole assembly sideways.
Combine those into τ = Fr sin θ and you have the whole subject. The product r sin θ is the moment arm: the perpendicular distance from the axis to the force's line of action. Sketch that perpendicular and you will never make a sign or geometry error again, because the moment arm is always shorter than the physical arm unless the force is exactly perpendicular.
Torque is what turns a screw, corners a car through the steering rack, holds a beam up at its supports, and defines what an engine can pull. It is also what limits a fastener: bolts are tightened to a torque figure because torque is what a person with a wrench can control, even though the quantity that actually matters is the tension in the bolt.
Why the sine is there, and why torque is not power
Torque is properly a vector cross product, τ = r × F, whose magnitude is Fr sin θ and whose direction is along the axis of rotation. The sine is what the cross product does: it keeps the perpendicular component and discards the parallel one. At 90° the sine is 1 and all the force turns the shaft; at 0° or 180° the sine is zero and none of it does, because the force's line of action passes straight through the axis.
Notice the symmetry: sin 30° and sin 150° are both 0.5, so a force pulling 30° off the arm and one pushing 30° the other way produce the same magnitude of torque. The calculator's angle sweep shows the whole curve, and it is the same shape whatever your force and radius.
The single most consequential thing to understand about torque is that it is not power. Power is torque multiplied by angular velocity, P = τω, so the same torque delivers ten times the power at ten times the speed. A stalled motor produces its full torque and zero power — every watt going in becomes heat. This is why gearing works: a reduction gearbox multiplies torque and divides speed in the same ratio, leaving power unchanged apart from losses. You never get anything for free; you trade speed for turning effort.
Torque also has its own version of Newton's second law: τ = Iα, where I is the moment of inertia about the axis and α is the angular acceleration in rad/s². It is the direct rotational analogue of F = ma, with every trap intact: the torque must be the net torque, and the moment of inertia must be taken about the axis actually being rotated.
Worked example: loosening a wheel nut
A wheel nut is specified at 110 N·m. You have a 380 mm wheel brace. How hard do you have to pull, and what changes if you cannot pull square to it?
- Perpendicular pull. F = τ ÷ (r sin θ) = 110 ÷ (0.380 × 1) = 289.5 N, about 29.5 kgf. That is a firm two-handed pull but well within reach.
- Pull at 60° instead. sin 60° = 0.86603, so the effective arm falls to 0.380 × 0.86603 = 0.32909 m and the force needed rises to 110 ÷ 0.32909 = 334.3 N — 15% more for the same result.
- Pull at 30°. sin 30° = 0.5, effective arm 0.190 m, force needed 110 ÷ 0.190 = 578.9 N. Half the effective arm, double the force. The geometry of how you stand matters as much as the length of the tool.
- Extend the bar. Slip a 600 mm tube over the brace, giving 0.98 m total, and the perpendicular force falls to 110 ÷ 0.98 = 112.2 N. That is the whole reason cheater bars exist — and also why they overtighten so easily, since the same comfortable pull now delivers 0.98 ÷ 0.380 = 2.6 times the torque.
- Check the units. 110 N·m × 0.7375621 = 81.1 lbf·ft, the figure a US-market service manual would quote.
The last step is worth doing every time. Mistaking lbf·ft for N·m under-tightens a joint by 26%; mistaking N·m for lbf·ft over-tightens it by 36%, which is enough to yield a small fastener.
Reading the result, and what torque does not tell you
Check the units before anything else. Newton-metres and pound-feet differ by a factor of 1.356, and pound-inches by a further factor of twelve, so a torque figure without a unit is worthless. Small fasteners and instruments are usually specified in pound-inches or newton-centimetres, vehicle fasteners in newton-metres or pound-feet, and industrial drives in newton-metres or kilonewton-metres.
Then check the effective arm. The calculator reports r sin θ separately for exactly this reason: it is the number that actually does the work, and if it is much smaller than your physical arm, your geometry is costing you.
Do not read a torque figure as a measure of how much work a machine can do. That is power. A motor rated at 200 N·m at 1,500 rpm delivers 31.4 kW; the same 200 N·m at 150 rpm delivers 3.14 kW. Comparing engines or motors on torque alone, without stating the speed, compares nothing.
Finally, on fasteners: torque is a proxy for what you actually want, which is the clamping tension in the bolt. Most of the torque you apply is spent overcoming friction under the head and in the threads, and only a small fraction becomes tension. The usual relation is T = K · D · Fpreload, with D the nominal thread diameter and K a nut factor around 0.2 for plain steel and lower when lubricated — figures that engineering handbooks give as rules of thumb, not as design data. Because K varies with lubrication, plating, surface finish and reuse, torque control alone is an imprecise way to set preload, which is why critical joints use angle control, bolt stretch measurement or direct tension indicators instead.
Torque unit conversions
| Unit | In newton-metres | 1 N·m equals | Typically used for |
|---|---|---|---|
| Newton-metre (N·m) | 1 | 1 | The SI unit — everything |
| Pound-foot (lbf·ft) | 1.3558179 | 0.7375621 | US vehicle and structural work |
| Pound-inch (lbf·in) | 0.1129848 | 8.8507458 | Small fasteners, instruments |
| Ounce-inch (oz·in) | 0.0070616 | 141.6119 | Small motors, watchmaking |
| Kilogram-force metre (kgf·m) | 9.80665 | 0.1019716 | Older European and Japanese manuals |
| Kilogram-force centimetre (kgf·cm) | 0.0980665 | 10.19716 | Small assemblies, hobby servos |
| Kilonewton-metre (kN·m) | 1,000 | 0.001 | Structural moments, large drives |
A torque of 100 N·m is 73.76 lbf·ft, 885.07 lbf·in or 10.197 kgf·m. Always carry the unit — the numbers differ by more than a factor of eight between the two most commonly confused pairs.
Torque specifications come from the joint, not from a calculator
This page tells you the relationship between force, lever arm and torque. It cannot tell you what torque a particular fastener should be tightened to. That figure depends on the bolt grade, diameter, thread pitch, the material being clamped, the lubrication state and the joint's function, and it must come from the manufacturer's specification or the governing standard for the assembly. Applying a torque figure from a similar-looking joint is how threads get stripped and bolts get yielded.
Common mistakes
- Confusing pound-feet with pound-inches. A factor of twelve, and the single most expensive unit error in the fastener world.
- Writing torque in joules. Both are newton-metres dimensionally, but torque is a turning effect and energy is not. The SI convention keeps them apart deliberately.
- Using the physical arm instead of the moment arm. If the force is not perpendicular, the effective arm is r·sin θ, and it can be far shorter than the tool you are holding.
- Comparing motors on torque alone. Torque without a speed says nothing about capability. Power is torque times angular velocity, and it is the quantity that survives gearing.
- Forgetting that a gearbox trades one for the other. A 10:1 reduction multiplies torque by roughly ten and divides speed by ten. Power out never exceeds power in.
- Treating torque as a direct measure of bolt tension. Most of the applied torque goes into friction. Use angle control, stretch measurement or tension indicators where preload really matters.
- Adding torques without regard to sense. Clockwise and anticlockwise moments have opposite signs. The net torque is the signed sum, which is exactly what τ = Iα requires.
Torque in the wider picture of rotation
Torque is one corner of a complete rotational mechanics that mirrors the linear one exactly. Force becomes torque, mass becomes moment of inertia, acceleration becomes angular acceleration, and Newton's second law becomes τ = Iα. Momentum becomes angular momentum L = Iω, with its own conservation law, paralleling the linear momentum calculator.
Energy transfers the same way. Work done by a torque is τθ with the angle in radians, and rotational kinetic energy is ½Iω². Those two are why a flywheel spun up by a given torque through a given angle ends at a predictable speed, and why the energy stored scales with the square of that speed.
In statics, torque appears as the moment of a force about a support, and the condition for equilibrium is that both the net force and the net moment vanish. That pair of conditions is the entire basis of beam analysis, of lever and pulley calculations, and of any free-body problem where a body is not free to translate. And in circular motion, a torque applied to a rotating body changes its speed, while the centripetal force merely holds it on its path without changing its speed at all — the clearest illustration of why forces along and perpendicular to the motion do such different things.