Torque


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, driving a screw uses torque to force it into an object, which is applied by the screwdriver rotating around its axis to the drives on the head.
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. When being referred to as moment of force, it is commonly denoted by.

Historical terminology

The term torque 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:
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.
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:
where
  • is the torque vector and is the magnitude of the torque;
  • is the position vector, and r is the magnitude of the position vector;
  • 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;
  • denotes the cross product, which produces a vector that is perpendicular both to and to 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-meter. For more on the units of torque, see .

Relationship with the angular momentum

The net torque on a body determines the rate of change of the body's angular momentum,
where is the angular momentum vector and is time. For the motion of a point particle,
where is the moment of inertia and is the orbital angular velocity pseudovector. It follows that
using the derivative of a vector is
This equation is the rotational analogue of Newton's second law for point particles, and is valid for any type of trajectory. In some simple cases like a rotating disc, where only the moment of inertia on rotating axis is, the rotational Newton's second law can bewhere.

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. But because the rate of change of linear momentum is force and the rate of change of position is velocity,
The cross product of momentum with its associated velocity is zero because velocity and momentum are parallel, so the second term vanishes. Therefore, torque on a particle is equal to the first derivative of its angular momentum with respect to time. If multiple forces are applied, according Newton's second law it follows that
This is a general proof for point particles, but it 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.

Derivatives of torque

In physics, rotatum is the derivative of torque with respect to time
where τ is torque.
This word is derived from the Latin word rotātus meaning 'to rotate'. The term rotatum is not universally recognized but is commonly used. There is not a universally accepted lexicon to indicate the successive derivatives of rotatum, even if sometimes various proposals have been made.
Using the cross product definition of torque, an alternative expression for rotatum is:

Because the rate of change of force is yank and the rate of change of position is velocity, the expression can be further simplified to:

Relationship with power and energy

The law of conservation of energy can also be used to understand torque. If a force is allowed to act through a distance, it is doing mechanical work. Similarly, if torque is allowed to act through an angular displacement, 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 θ1 and θ2 represent the initial and final angular positions of the body.
It follows from the work–energy principle that W also represents the change in the rotational kinetic energy Er of the body, given by
where I is the moment of inertia of the body and ω is its angular speed.
Power is the work per unit time, given by
where P is power, τ is torque, ω is the angular velocity, and represents the scalar product.
Algebraically, the equation may be rearranged to compute torque for a given angular speed and power output. The power injected by the torque depends only on the instantaneous angular speed – not on whether the angular speed increases, decreases, or remains constant while the torque is being applied.

Proof

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 inside the integral is a scalar triple product, but as per the definition of torque, and since the parameter of integration has been changed from linear displacement to angular displacement, the equation becomes
If the torque and the angular displacement are in the same direction, then the scalar product reduces to a product of magnitudes; i.e., giving

Principle of moments

The principle of moments, also known as Varignon's theorem states that the resultant torques due to several forces applied to about a point is equal to the sum of the contributing torques:
From this it follows that the torques resulting from N number of forces acting around a pivot on an object are balanced when

Units

Official SI literature indicates the newton-meter as the standard unit for torque, properly denoted using N⋅m; although this is dimensionally equivalent to the joule, which is not used for torque. In the case of torque, the unit is assigned to a vector, whereas for energy, it is assigned to a scalar. This means that the dimensional equivalence of the newton-metre and the joule may be applied in the former but not in the latter case. This problem is addressed in orientational analysis, which treats the radian as a base unit rather than as a dimensionless unit.
Torque has the dimension of force times distance, symbolically, and those fundamental dimensions are the same as that for energy or work.
The traditional imperial units for torque are the pound foot, or, for small values, the pound inch. In the US, torque is most commonly referred to as the foot-pound and the inch-pound. 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.

Conversion to other units

A conversion factor may be necessary when using different units of power or torque. For example, if rotational speed is used in place of angular speed, we must multiply by 2 radians per revolution. In the following formulas, P is power, τ is torque, and ν is rotational speed.
Showing units:
Dividing by 60 seconds per minute gives us the following.
where rotational speed is in revolutions per minute.
Some people use horsepower for power, foot-pounds for torque and rpm for rotational speed. This results in the formula changing to:
The constant below changes with the definition of the horsepower; for example, using metric horsepower, it becomes approximately 32,550.
The use of other units would require a different custom conversion factor.