Couple (mechanics)

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.

, 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, i.e.

[1] In other words, a couple, unlike any more general moments, is a "free vector".

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 Now the distributive property of the cross product implies However, 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.

A force F applied to a rigid body at a distance d from the center of mass has the same effect as the same force applied directly to the center of mass and a couple Cℓ = Fd.

The general theorems are:[3] Couples are very important in engineering and the physical sciences.