, torque is the rotational equivalent of linear force
. It is also referred to as the moment, moment of force, rotational force or turning effect, depending on the field of study. The concept originated with the studies by Archimedes
of the usage of lever
s. Just as a linear force is a push or a pull, a torque can be thought of as a twist to an object around a specific axis. Another definition of torque is the product of the magnitude of the force and the perpendicular distance of the line of action
of a force from the axis of rotation
. The symbol for torque is typically
, the lowercase Greek letter
''. When being referred to as moment
of force, it is commonly denoted by .
In three dimensions, the torque is a pseudovector
; for point particles
, it is given by the cross product
of the position vector (distance vector
) and the force vector. The magnitude of torque of 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:
is the torque vector and
is the magnitude of the torque,
is the position vector (a vector from the point about which the torque is being measured to the point where the force is applied),
is the force vector,
denotes the cross product
, which produces a vector that is perpendicular
to both and following the right-hand rule
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
, the brother of Lord Kelvin
, introduced the term ''torque'' into English scientific literature in 1884. However, torque is referred to using different vocabulary depending on geographical location and field of study. This article follows the definition used in US physics in its usage of the word ''torque''.
[''Physics for Engineering'' by Hendricks, Subramony, and Van Blerk, Chinappi page 148]
/ref> In the UK and in US mechanical engineering, torque is referred to as ''moment of force'', usually shortened to ''moment''.
These terms are interchangeable in US physics and UK physics terminology, unlike in US mechanical engineering, where the term ''torque'' is used for the closely related "resultant moment of a couple".
Torque and moment in the US mechanical engineering terminology
In US mechanical engineering, ''torque'' is defined mathematically as the rate of change of angular momentum of an object (in physics it is called "net torque"). The definition of torque states that one or both of the angular velocity or the moment of inertia of an object are changing. ''Moment'' is the general term used for the tendency of one or more applied forces to rotate an object about an axis, but not necessarily to change the angular momentum of the object (the concept which is called ''torque'' in physics).
Kane, T.R. Kane and D.A. Levinson (1985). ''Dynamics, Theory and Applications'' pp. 90–99]
For example, a rotational force applied to a shaft causing acceleration, such as a drill bit accelerating from rest, results in a moment called a ''torque''. By contrast, a lateral force on a beam produces a moment (called a bending moment), but since the angular momentum of the beam is not changing, this bending moment is not called a ''torque''. Similarly with any force couple on an object that has no change to its angular momentum, such moment is also not called a ''torque''.
Definition and relation to angular momentum
A force applied perpendicularly 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 a force of one newton applied six metres from the fulcrum. 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.
More generally, the torque on a point particle (which has the position r in some reference frame) can be defined as the cross product:
where r is the particle's position vector relative to the fulcrum, and F is the force acting on the particle. The magnitude ''τ'' of the torque is given by
where ''r'' is the distance from the axis of rotation to the particle, ''F'' is the magnitude of the force applied, and ''θ'' is the angle between the position and force vectors. Alternatively,
where ''F''⊥ is the amount of force directed perpendicularly to the position of the particle. Any force directed parallel to the particle's position vector does not produce a torque.
It follows from the properties of the cross product that the ''torque vector'' is perpendicular to both the ''position'' and ''force'' vectors. Conversely, the ''torque vector'' defines the plane in which the ''position'' and ''force'' vectors lie. The resulting ''torque vector'' direction is determined by the right-hand rule.
The net torque on a body determines the rate of change of the body's angular momentum,
where L is the angular momentum vector and ''t'' is time.
For the motion of a point particle,
where is the moment of inertia and ω is the orbital angular velocity pseudovector. It follows that
where α is the angular acceleration of the particle, and ''p''|| is the radial component of its linear momentum. This equation is the rotational analogue of Newton's Second Law for point particles, and is valid for any type of trajectory. Note that although force and acceleration are always parallel and directly proportional, the torque τ need not be parallel or directly proportional to the angular acceleration α. This arises from the fact that although mass is always conserved, the moment of inertia in general is not.
Proof of the equivalence of definitions
The definition of angular momentum for a single point particle is:
where p is the particle's linear momentum and r is the position vector from the origin. The time-derivative of this is:
This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definition of force (whether or not mass is constant) and the definition of velocity
The cross product of momentum with its associated velocity is zero because velocity and momentum are parallel, so the second term vanishes.
By definition, torque τ = r × F. Therefore, torque on a particle is ''equal'' to the
first derivative of its angular momentum with respect to time.
If multiple forces are applied, Newton's second law instead reads , and it follows that
This is a general proof for point particles.
The proof can be generalized to a system of point particles by applying the above proof to each of the point particles and then summing over all the point particles. Similarly, the proof can be generalized to a continuous mass by applying the above proof to each point within the mass, and then integrating over the entire mass.
Torque has the dimension of force times distance, symbolically . Although those fundamental dimensions are the same as that for energy or work, official SI literature suggests using the unit ''newton metre'' (N⋅m) and never the joule.
[From th] The unit ''newton metre'' is properly denoted N⋅m.
official SI website
example, the quantity torque is the cross product of a position vector and a force vector.
The SI unit is newton metre. Even though torque has the same dimension as energy (SI unit
joule), the joule is never used for expressing torque."
The traditional Imperial and U.S. customary units for torque are the pound foot (lbf-ft), or for small values the pound inch (lbf-in). Confusingly, in US practice torque is most commonly referred to as the foot-pound (denoted as either lb-ft or ft-lb) and the inch-pound (denoted as in-lb). [ Demonstration that, as in most US industrial settings, the torque ranges are given in ft-lb rather than lbf-ft.] Practitioners depend on context and the hyphen in the abbreviation to know that these refer to torque and not to energy or moment of mass (as the symbolism ft-lb would properly imply).
Special cases and other facts
Moment arm formula
A very useful special case, often given as the definition of torque in fields other than physics, is as follows:
The construction of the "moment arm" is shown in the figure to the right, along with the vectors r and F mentioned above. The problem with this definition is that it does not give the direction of the torque but only the magnitude, and hence it is difficult to use in three-dimensional cases. If the force is perpendicular to the displacement vector r, the moment arm will be equal to the distance to the centre, and torque will be a maximum for the given force. The equation for the magnitude of a torque, arising from a perpendicular force:
For example, if a person places a force of 10 N at the terminal end of a wrench that is 0.5 m long (or a force of 10 N exactly 0.5 m from the twist point of a wrench of any length), the torque will be 5 N⋅m – assuming that the person moves the wrench by applying force in the plane of movement and perpendicular to the wrench.
For an object to be in static equilibrium, not only must the sum of the forces be zero, but also the sum of the torques (moments) about any point. For a two-dimensional situation with horizontal and vertical forces, the sum of the forces requirement is two equations: Σ''H'' = 0 and Σ''V'' = 0, and the torque a third equation: Σ''τ'' = 0. That is, to solve statically determinate equilibrium problems in two-dimensions, three equations are used.
Net force versus torque
When the net force on the system is zero, the torque measured from any point in space is the same. For example, the torque on a current-carrying loop in a uniform magnetic field is the same regardless of your point of reference. If the net force is not zero, and is the torque measured from , then the torque measured from is …
Torque forms part of the basic specification of an engine: the power output of an engine is expressed as its torque multiplied by its rotational speed of the axis. Internal-combustion engines produce useful torque only over a limited range of rotational speeds (typically from around 1,000–6,000 rpm for a small car). One can measure the varying torque output over that range with a dynamometer, and show it as a torque curve.
Steam engines and electric motors tend to produce maximum torque close to zero rpm, with the torque diminishing as rotational speed rises (due to increasing friction and other constraints). Reciprocating steam-engines and electric motors can start heavy loads from zero rpm without a clutch.
Relationship between torque, power, and energy
If a force is allowed to act through a distance, it is doing mechanical work. Similarly, if torque is allowed to act through a rotational distance, it is doing work. Mathematically, for rotation about a fixed axis through the center of mass, the work ''W'' can be expressed as
where ''τ'' is torque, and ''θ''1 and ''θ''2 represent (respectively) the initial and final angular positions of the body.
The work done by a variable force acting over a finite linear displacement is given by integrating the force with respect to an elemental linear displacement
However, the infinitesimal linear displacement is related to a corresponding angular displacement and the radius vector as
Substitution in the above expression for work gives
The expression is a scalar triple product given by