In physics and mechanics, torque is the rotational correspondent of linear force. It is also referred to as the moment of force, or simply the moment. 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 axis. For example, when driving a screw, a screwdriver applies torque to the screw, causing it to tend to rotate around its axis. Torque is generally referred to using different vocabulary depending on geographical location and field of study, with torque generally being associated with physics and moment being associated with engineering. This article follows the definition used in US physics in its usage of the word torque. Torque is typically represented mathematically using the lowercase Greek letter tau (𝜏). When being referred to as moment of force, it is commonly denoted by M.
Historical terminology
The term torque (from Latin torquēre, 'to twist') is said to have been suggested by James Thomson and appeared in print in April, 1884. Usage is attested the same year by Silvanus P. Thompson in the first edition of Dynamo-Electric Machinery. Thompson describes his usage of the term as follows:
Just as the Newtonian definition of force is that which produces or tends to produce motion (along a line), so torque may be defined as that which produces or tends to produce torsion (around an axis). It is better to use a term which treats this action as a single definite entity than to use terms like "couple" and "moment", which suggest more complex ideas. The single notion of a twist applied to turn a shaft is better than the more complex notion of applying a linear force (or a pair of forces) with a certain leverage. In mechanical engineering in the UK and the US, torque is generally referred to as moment of force, usually shortened to moment. This terminology can be traced back to at least 1811 in Siméon Denis Poisson's Traité de mécanique. An English translation of Poisson's work appeared in 1842.
Definition and relation to other physical quantities
Torque as a cross product between linear force and the radius about the rotational axis
The torque about an axis can be calculated by multiplying the linear force applied perpendicularly to a lever multiplied by its distance from the lever's fulcrum (the length of the lever arm). Therefore, 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. 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 direction of the torque can be determined by using the right-hand grip rule: if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force, then the thumb points in the direction of the torque. It follows that the torque vector is perpendicular to both the position and force vectors, and defines the plane in which the two vectors lie. The resulting torque vector direction is determined by the right-hand rule. Therefore any force directed parallel to the particle's position vector does not produce a torque. The magnitude of torque applied to a rigid body depends on three quantities: the force applied, the lever arm vector 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:
τ = r × F ⟹ τ = r F ⊥ = r F sin θ {\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} \implies \tau =rF_{\perp }=rF\sin \theta }
where
τ {\displaystyle {\boldsymbol {\tau }}} is the torque vector and τ {\displaystyle \tau } is the magnitude of the torque;
r {\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;
F {\displaystyle \mathbf {F} } is the force vector, F is the magnitude of the force vector, and F⊥ is the amount of force directed perpendicularly to the position of the particle;
× {\displaystyle \times } denotes the cross product, which produces a vector that is perpendicular both to r and to F following the right-hand rule;
θ {\displaystyle \theta } is the angle between the force vector and the lever arm vector. The SI unit for torque is the newton-meter (N⋅m). For more on the units of torque, see § Units.
Relationship with the angular momentum The net torque on a body determines the rate of change of the body's angular momentum,
τ = d L d t {\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}}
where L {\textstyle \mathbf {L} } is the angular momentum vector and t {\textstyle t} is time. For the motion of a point particle,
L = I ω , {\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},}
where I = m r 2 {\textstyle I=mr^{2}} is the moment of inertia and ω {\textstyle {\boldsymbol {\omega }}} is the orbital angular velocity pseudovector. It follows that
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