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# Torque

The concept of torque in physics 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 Nm applied through a full revolution will require an energy of exactly 2π J — mathematically, E = τ θ, 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:

\τ = moment arm × force

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, 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. This gives rise to the approximation

τ = distance to centre × force

For example, if a person places a force of 9.8 N (1 kg) on a spanner which is 0.5 m long, the torque will be approximately 4.9 Nm, assuming that the person pulls the spanner in the direction best suited to turning bolts.

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