Couples and torques

In a nutshell

A couple is a pair of equal forces acting in the opposite direction to produce rotational motion. The moment of a couple is called a torque.

Equations

Variable definitions

Couples

A couple is described as a pair of equal forces which act in the opposite direction. The forces are applied parallel to each other which produce a rotational motion about the centre point, the pivot. The couple produces no translational movement.

Torque of a couple

The moment of a couple is described mathematically as:

​$$moment = \biggl( F \times \frac{d}{2} \biggr) + \biggl(F \times \frac{d}{2} \biggr) = Fd$$

Where these relationships are shown in the image below:

It is defined as the magnitude of one of the forces multiplied by the perpendicular distance between them. 

The moment of a couple is known as a torque. The torque of a couple is:

​$$torque \ of \ a \ couple = one \ of \ the \ forces \times perpendicular \ distance \ between \ forces = Fd$$

Exam tip: This equation is very similar to moment equation, but the difference is the force is only one of the forces and the distance is the distance between the forces not the distance to the pivot.

Example:

A pair of bicycle pedals are an example of a couple. A downward force of $$30 \ N$$ is applied to one of the pedals. The pedals are aligned horizontally and are $$50 \ cm$$ apart. Calculate the torque of the couple.

​

Firstly state the variables from the question:

$$F = 30 \ N\newline d = 0.5 \ m$$

Next state the equation needed:

$$moment =Fd$$

Finally substitute in and solve:

​$$moment = 30 \times 0.5 = 15 \ Nm$$​

The torque on the pedals is $$\underline{15 \ Nm}$$.​