A couple

In mechanics, a couple is a system of forces with a resultant moment but no resultant force.

Its effect is to create rotation without translation, or more generally without any acceleration of the centre of mass.

In rigid body mechanics, force couples are free vectors, meaning their effects on a body are independent of the point of application.

The resultant moment of a couple is called a torque.

This is not to be confused with the term torque as it is used in physics, where it is merely a synonym of moment. Instead, torque is a special case of moment in mechanics.

Torque has special properties that moment does not have, in particular the property of being independent of reference point.

Definition of a couple

A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment.


The simplest kind of couple consists of two equal and opposite forces whose lines of action do not coincide. This is called a "simple couple".

The forces have a turning effect or moment, called a torque, about an axis or turning point which is normal (perpendicular) to the plane of the forces.

The SI unit for the torque of the couple is newton metre. Nm (the same as the unit for a moment).

If the two forces are F and − F, then the magnitude of the torque is given by the following formula:

τ = Fd


τ is the moment of couple

F is the magnitude of the force

d is the perpendicular distance between the two parallel forces.

The magnitude of the torque is equal to F x d, with the direction of the torque given by the unit vector ê, which is perpendicular to the plane containing the two forces and positive being a counter-clockwise couple.

When d is taken as a vector between the points of action of the forces, then the torque is the cross product of d and F

τ = l d x F l