Couple (mechanics)

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Simple couple[edit]




\tau = F d \,


\tau is the torque
F is the magnitude of one of the forces
d is the perpendicular distance between the forces, sometimes called the arm of the couple

The magnitude of the torque is always equal to F d, with the direction of the torque given by the unit vector \hat{e}, which is perpendicular to the plane containing the two forces. When d is taken as a vector between the points of action of the forces, then the couple is the cross product of d and F. I.e.,

 \mathbf{\tau} = \mathbf{d} \times \mathbf{F} .

Independence of reference point[edit]



The proof of this claim is as follows: Suppose there are a set of force vectors F1, F2, etc. that form a couple, with position vectors (about some origin P) r1, r2, etc., respectively. The moment about P is

M = \mathbf{r}_1\times \mathbf{F}_1 + \mathbf{r}_2\times \mathbf{F}_2 + \cdots


M' = (\mathbf{r}_1+\mathbf{r})\times \mathbf{F}_1 + (\mathbf{r}_2+\mathbf{r})\times \mathbf{F}_2 + \cdots

Now the distributive property of the cross product implies

M' = \left(\mathbf{r}_1\times \mathbf{F}_1 + \mathbf{r}_2\times \mathbf{F}_2 + \cdots\right) + \mathbf{r}\times \left(\mathbf{F}_1 + \mathbf{F}_2 + \cdots \right).

However, the definition of a force couple means that

\mathbf{F}_1 + \mathbf{F}_2 + \cdots = 0.




Forces and couples[edit]

Force and couple.PNG





  • The forces exerted by one's hand on a screw-driver
  • The forces exerted by the tip of a screw-driver on the head of a screw
  • Drag forces acting on a spinning propeller
  • Forces on an electric dipole in a uniform electric field.
  • The reaction control system on a spacecraft.



See also[edit]

