Work of a Couple Moment

Consider a rigid body with two equal and opposite forces acting at two points separated by a distance.

This couple produces a moment equal to force times the distance between them. As a result, the rigid body undergoes translation as well as rotation to reach the final position.

During the translation, the work of one force is equal and opposite to the work of another force. So, the net work done by the two forces during translation is zero.

However, during rotation, angular displacement due to one of the forces gives an equivalent linear displacement which results in work.

Expressing the work in terms of angular displacement and using the expression for the magnitude of the moment of the couple gives the final amount of work done, which depends on the couple moment and the angle of rotation.

If the couple moment and the angle of rotation have the same sense, the work done by the couple moment is positive.

In contrast, if they have the opposite sense, the work done is negative.