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This article is about the physical concept. For another meaning see Torque (jewellery)

The concept of torque in physics, also called moment or couple, originated with the work of Archimedes on levers. Informally, torque can be thought of as "rotational force". The weight that rests on a lever, multiplied by its distance from the lever's fulcrum, is the torque. For example, a weight of three newtons resting two metres from the fulcrum exerts the same torque as one newton resting six metres from the fulcrum. This assumes the force is in a direction at right angles to a straight lever. More generally, one may define torque as the cross product:

where r is the vector from the axis of rotation to the point on which the force is acting, and F is the vector of force. Torque is important in the design of machines such as engines.

Torque has dimensions of distance × force; the same as energy. However, the units of torque are usually stated as "newton-metres" or "foot-pounds" rather than joules. Of course this 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 and θ is the angle moved, in radians.

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. Note that 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 which is 0.5 m long, the torque will be 5 N·m, assuming that the person pulls the spanner in the direction best suited to turning bolts.

If the force is at an angle θ from the perpendicular then, from the definition of cross product, the magnitude of the torque arising is:

For an object to be at static equilibrium, not only must the sum of the forces be zero, but also the sum of the torques. For a two-dimensional situation, the sum of the forces requirement is two equations, and the torque a third equation. That is, to solve equilibrium problems in two-dimensions, we use three equations.

Torque is the time-derivative of angular momentum, just as force is the time derivative of linear momentum. For multiple torques acting simultaneously:

where L is angular momentum. See also proof of angular momentum.

Torque on a rigid body can be written in terms of its moment of inertia and its angular velocity :

so if is constant,

where α is angular acceleration, a quantity usually measured in rad/s2.

The measurement of torque is important in automotive engineering, being concerned with the transmission of power from the drive train to the wheels of a vehicle. A torque wrench is used where the tightness of screws and bolts is crucial. Torque is also the easiest way to explain mechanical advantage in just about every simple machine except the pulley.