## Couples

Two equal and oppositely directed parallel but not collinear forces acting upon a body. The moment of the couple (or torque) is given by the product of one of the forces by the perpendicular distance between them:in the diagram below.

The moment of a forceis only defined with respect to a certain point *P * – the moment of about P - and in general when *P * is changed, the moment changes. However, the moment (torque) of a *couple * is *independent * of the reference point *P *: Any point will give the same moment. In other words, a torque vector, unlike any other moment vector, is a "free vector".

Proof: Suppose there are a set of force vectorsetc. that form a couple, with position vectors (about some origin *P *)etc., respectively. The moment about *P * is

Now we pick a new reference point *P' * that differs from *P * by the vector **r **. The new moment is

We can simplify this as follows:

- The definition of a force couple means that

Therefore,

This proves that the moment is independent of reference point, which is proof that a couple is a free vector.