What Is Torque?

F is the force, r the distance from the axis to where the force acts, and θ the angle between them. At 90°, sin θ = 1 and the formula becomes τ = F × r.

Pull with 80 N at the end of a 0.25 m wrench, square to the handle:

τ = 80 × 0.25 = 20 N·m, or 14.75 lb·ft

Pull at 60° to the handle and only part of the force turns the bolt: τ = 80 × 0.25 × 0.866 = 17.3 N·m. The torque calculator handles SI and US units. In US units, 60 lb on a 1.5 ft bar gives 90 lb·ft.

N·m has the same dimensions as the joule, but the two are different quantities. The SI Brochure says the joule is never used for torque. Torque acts about an axis. Energy is a scalar. They connect through angle: work = torque × angle in radians. So 20 N·m turned through one full revolution does 2π × 20 = 125.7 J of work.

Force is a push or a pull. Torque is what a force does about an axis. You cannot turn one into the other without a distance. "Moment" is the same physical idea. By habit, engineers say torque for twisting about a shaft's own axis, and bending moment for a load that bends a beam. A pair of equal and opposite forces, a couple, gives pure torque with no net push on the part.

Whether something rotates does not decide how much torque it carries. The balance of torques does.

A shaft at steady speed can carry a large torque with no acceleration at all. Only the net torque changes speed, and Newton's second law for rotation gives its size:

τ_net = I × α

A flywheel with I = 2 kg·m² going from 0 to 1,500 rpm in 5 s has α = 1,500 × 2π ÷ 60 ÷ 5 = 31.4 rad/s². It needs 2 × 31.4 = 62.8 N·m on top of any load and friction. That extra torque is why start-up often sets the size of a drive.

Power is torque multiplied by rotational speed:

A 5.5 kW motor at 1,450 rpm gives T = 9,550 × 5.5 ÷ 1,450 = 36.2 N·m. A 7.5 hp motor at 1,750 rpm gives 22.5 lb·ft. The power calculator runs this the other way, from torque and speed to power.

Neither number alone describes an engine. For the same power, an engine at low rpm delivers more torque and one at high rpm delivers less. That trade is what a gearbox exploits.

A gear pair lowers speed and raises torque by the same ratio, less friction. Take a 20-tooth pinion driving an 80-tooth gear, with the motor above and 97% efficiency:

The ideal output is 144.8 N·m. Power does not increase. The gear ratio calculator gives both figures.

A shaft carrying torque is in shear, with stress highest at its surface. For a solid round shaft, the peak shear stress is 16 T ÷ (π d³), so doubling the diameter cuts the stress by eight. That is covered with examples in What Is Shear Stress?. On a large scale the numbers are huge: NREL's 5 MW reference wind turbine carries about 4.18 million N·m in its low-speed shaft at 12.1 rpm, because power is so large and speed so low.

Bolt torque is a means to an end. The goal is clamp force, called preload. The usual estimate is:

K is the nut factor, d the nominal diameter and F the preload. For an M16 class 8.8 bolt at K = 0.2, the screw torque calculator gives 218.5 N·m for a preload of 68.3 kN, which is 75% of proof load.

K is an estimate, and it changes with the surface. Apply the same 218.5 N·m to an oiled thread at K = 0.12 and the preload jumps to 113.8 kN, or 125% of proof load. The bolt may be permanently stretched or damaged, even though the wrench read correctly. Use the torque value for the actual lubrication, as a chart will show for dry and oiled threads, and treat the result as an estimate.