Couple (mechanics)


In physics, a couple is a pair of forces that are equal in magnitude but opposite in their direction of action. A couple produce a pure rotational motion without any translational form.

Simple couple

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 which is normal to the plane of the forces. The SI unit for the torque of the couple is newton metre.
If the two forces are and, then the magnitude of the torque is given by the following formula:
where
The magnitude of the torque is equal to, with the direction of the torque given by the unit vector, which is perpendicular to the plane containing the two forces and positive being a counter-clockwise couple. When is taken as a vector between the points of action of the forces, then the torque is the cross product of and, i.e.

Independence of reference point

The moment of a force is only defined with respect to a certain point and, in general, when is changed, the moment changes. However, the moment of a couple is independent of the reference point : Any point will give the same moment. In other words, a couple, unlike any more general moments, is a "free vector".
The proof of this claim is as follows: Suppose there are a set of force vectors,, etc. that form a couple, with position vectors,,, etc., respectively. The moment about is
Now we pick a new reference point that differs from by the vector. 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.

Forces and couples

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 couple produces an angular acceleration of the rigid body at right angles to the plane of the couple. The force at the center of mass accelerates the body in the direction of the force without change in orientation. The general theorems are:

Applications

Couples are very important in engineering and the physical sciences. A few examples are: