# Torque

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In physics and mechanics, torque is the rotational analogue of linear force.[1] It is also referred to as the moment of force (also abbreviated to moment). It describes the rate of change of angular momentum that would be imparted to an isolated body.

Quick facts: Torque, Common symbols, SI unit, Other u...
Torque
Relationship between force F, torque τ, linear momentum p, and angular momentum L in a system which has rotation constrained to only one plane (forces and moments due to gravity and friction not considered).
Common symbols
${\displaystyle \tau }$, M
SI unitN⋅m
Other units
pound-force-feet, lbf⋅inch, ozf⋅in
In SI base unitskg⋅m2⋅s−2
Dimension${\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-2}}$
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The concept originated with the studies by Archimedes of the usage of levers, which is reflected in his famous quote: "Give me a lever and a place to stand and I will move the Earth". Just as a linear force is a push or a pull applied to a body, a torque can be thought of as a twist applied to an object with respect to a chosen point. Torque is defined as the product of the magnitude of the perpendicular component of the force and the distance of the line of action of a force from the point around which it is being determined. The law of conservation of energy can also be used to understand torque. The symbol for torque is typically ${\displaystyle {\boldsymbol {\tau }}}$, the lowercase Greek letter tau. When being referred to as moment of force, it is commonly denoted by M.

In three dimensions, the torque is a pseudovector; for point particles, it is given by the cross product of the displacement vector and the force vector. The magnitude of torque applied to a rigid body depends on three quantities: the force applied, the lever arm vector[2] connecting the point about which the torque is being measured to the point of force application, and the angle between the force and lever arm vectors. In symbols:

${\displaystyle \tau =rF\sin \theta ,}$

where

• ${\displaystyle {\boldsymbol {\tau }}}$ is the torque vector and ${\displaystyle \tau }$ is the magnitude of the torque,
• ${\displaystyle \mathbf {r} }$ is the position vector (a vector from the point about which the torque is being measured to the point where the force is applied), and r is the magnitude of the position vector,
• ${\displaystyle \mathbf {F} }$ is the force vector, and F is the magnitude of the force vector,
• ${\displaystyle \times }$ denotes the cross product, which produces a vector that is perpendicular both to r and to F following the right-hand rule,
• ${\displaystyle \theta }$ is the angle between the force vector and the lever arm vector.

The SI unit for torque is the newton-metre (N⋅m). For more on the units of torque, see § Units.

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