4.3.3 Couple

A couple consists of two equal but opposite forces whose lines of action are parallel but non-collinear.

In the example shown below, a couple is formed by two parallel forces F applied at both ends of a rod of length L.

Notice that both forces are going to cause the rod to rotate clockwise about the CM (center of mass) of the rod. So the total moment of the couple (also called the torque) is given by

\displaystyle \begin{aligned}\tau &=F\times \frac{L}{2}+F\times \frac{L}{2}\\&=F\times L\end{aligned}

If the forces are not perpendicular to the beam, we just have to work with the components perpendicular to the rod, leading us to

\displaystyle \tau =F\sin \theta \times L

Alternatively, we can obtain the perpendicular distance between the two forces, leading us to

\displaystyle \tau =F\times L\sin \theta

Notice that the net force of a couple is always zero. This is why some people think of couples as pure moments, since they produce only rotational effect but no translational effect. This is also why the total moment of a couple evaluates to be the same magnitude about any pivot point.

Concept Test

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