A torque (τ) in physics, also called a moment (of force), is a pseudo-vector that measures the tendency of a force to rotate an object about some axis (center). The magnitude of a torque is defined as the product of a force and the length of the lever arm (radius). Just as a force is a push or a pull, a torque can be thought of as a twist.

The SI unit for torque is the newton meter (N m). In Imperial and U.S. customary units, it is measured in foot pounds (ft·lbf) (also known as 'pound feet') and for smaller measurement of torque: inch pounds (in·lbf) or even inch ounces (in·ozf) . The symbol for torque is τ, the Greek letter tau. .

History

The concept of torque, also called moment or couple, originated with the studies of Archimedes on levers. The rotational analogues of force, mass, and acceleration are torque, moment of inertia, and angular acceleration, respectively.

Explanation

The force applied to a lever multiplied by its distance from the lever's fulcrum, the length of the lever arm, is its torque. A force of three newtons applied two meters from the fulcrum, for example, exerts the same torque as one newton applied six meters from the fulcrum. This assumes the force is in a direction at right angles to the straight lever. The direction of the torque can be determined by using the right hand grip rule: curl the fingers of your right hand the direction of rotation and stick your thumb out so it is aligned with the axis of rotation. Your thumb points in the direction of the torque vector.

Mathematically, the torque on a particle (which has the position r in some reference frame) can be defined as the cross product:

$\backslash mathbf\{\backslash tau\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{F\}$

where

r is the particle's position vector relative to the fulcrum

F is the force acting on the particle.

The torque on a body determines the rate of change of its angular momentum,

$\backslash mathbf\{\backslash tau\}=\backslash frac\{\backslash mathrm\{d\}\backslash mathbf\{L\}\}\{\backslash mathrm\{d\}t\}$

where

L is the angular momentum vector

t stands for time.

As can be seen from either of these relationships, torque is a vector, which points along the axis of the rotation it would tend to cause.

Proof of the equivalence of definitions

The definition of angular momentum for a single particle is:

$\backslash mathbf\{L\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{p\}$

where "×" indicates the vector cross product. The time-derivative of this is:

$\backslash frac\{d\backslash mathbf\{L\}\}\{dt\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash frac\{d\backslash mathbf\{p\}\}\{dt\}\; +\; \backslash frac\{d\backslash mathbf\{r\}\}\{dt\}\; \backslash times\; \backslash mathbf\{p\}$