Torque , also called moment or moment of force (see the terminology below), is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist.
In more basic terms, torque measures how hard something is rotated. For example, imagine a wrench or spanner trying to twist a nut or bolt. The amount of "twist" (torque) depends on how long the wrench is, how hard you push on it, and how well you are pushing it in the correct direction.
The terminology for this concept is not straightforward: In physics, it is usually called "torque", and in mechanical engineering, it is called "moment". However, in mechanical engineering, the term "torque" means something different , described below. In this article, the word "torque" is always used in the physics sense, synonymous with "moment" in engineering.
The symbol for torque is typically τ , the Greek letter tau . When it is called moment, it is commonly denoted M .
The magnitude of torque depends on three quantities: First, the force applied; second, the length of the lever arm connecting the axis to the point of force application; and third, the angle between the two. In symbols:
where
The length of the lever arm is particularly important; choosing this length appropriately lies behind the operation of levers, pulleys, gears, and most other simple machines involving a mechanical advantage.
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). For more on the units of torque, see below.
Terminology
See also: Couple (mechanics)In mechanical engineering (unlike physics), the terms "torque" and "moment" are not interchangeable. "Moment" is the general term for the tendency of one or more applied forces to rotate an object about an axis (the concept which in physics is called torque). "Torque" is a special case of this: If the applied force vectors add to zero (i.e., their "resultant" is zero), then the forces are called a "couple" and their moment is called a "torque".
For example, a rotational force down a shaft, such as a turning screw-driver, forms a couple, so the resulting moment is called a "torque". By contrast, a lateral force on a beam produces a moment (called a bending moment), but since the net force is nonzero, this bending moment is not called a "torque".
This article follows physics terminology by calling all moments by the term "torque", whether or not they are associated with a couple.
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.
Definition and relation to angular momentum
A force applied at a right angle 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 meters 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 curl in the direction of rotation and the thumb points along the axis of rotation, then the thumb also points in the direction of the torque.
More generally, the torque on a 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. It points along the axis of rotation, and its direction is determined by the right-hand rule.
The 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. If multiple torques are acting on the body, it is instead the net torque which determines the rate of change of the angular momentum:
For rotation about a fixed axis,
where I is the moment of inertia and ω is the angular velocity. It follows that
where α is the angular acceleration of the body, measured in rad s −2 .
Proof of the equivalence of definitions
The definition of angular momentum for a single particle is:
where "×" indicates the vector cross product and p is the particle's linear momentum. 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 definitions of velocity v = d r / dt , acceleration a = d v / dt and linear momentum p = m v ,
The cross product of any vector with itself is zero, so the second term vanishes. Hence with the definition of force F = m a (Newton's 2nd law),
Then by definition, torque τ = r × F .
If multiple forces are applied, Newton's second law instead reads F net = m a , and it follows that
The proof relies on the assumption that mass is constant; this is valid only in non-relativistic systems in which no mass is being ejected.
Units
Torque has dimensions of force times distance. Official SI literature suggests using the unit newton meter (N m) or joule per radian. The unit newton meter is properly denoted "N m" or "N·m", but not other combinations (this avoids ambiguity—for example, mN is the symbol for millinewtons, nm is the symbol for nanometers, etc.)
The joule, which is the SI unit for energy or work, is dimensionally equivalent to a N m, but this unit is not used for torque. Energy and torque are entirely different concepts, so the practice of using different unit names for them helps avoid mistakes and misunderstandings. The dimensional equivalence of these units, of course, is not simply a coincidence: a torque of 1 N m applied through a full revolution will require an energy of exactly 2π joules. Mathematically,
where E is the energy, τ is magnitude of the torque, and θ is the angle moved (in radians). This equation motivates the alternate unit name of "joules per radian".
Other non-SI units of torque include "pound-force-feet" or "foot-pounds-force" or "inch-pounds-force" or "ounce-force-inches" or "meter-kilograms-force" or "kilogrammeter" (kgm). For all these units, the word "force" is often left out, for example abbreviating "pound-force-foot" to simply "pound-foot". (In this case, it would be implicit that the "pound" is pound-force and not pound-mass.)
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 below, 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 on a spanner (wrench) which is 0.5 m long, the torque will be 5 N m, assuming that the person pulls the spanner by applying force perpendicular to the spanner.
Static equilibrium
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