understand that a couple is a pair of forces that acts to produce rotation only

Turning Effects of Forces

Learning Objective

Students will understand that a couple is a pair of forces that produces rotation without resulting in any translational motion.

Key Concepts

• Torque (or moment of a force) – the turning effect of a force about a point.
• Line of action – the straight line along which a force acts.
• Couple – two equal and opposite forces whose lines of action are parallel but not collinear.
• Resultant torque of a system of forces – the vector sum of individual torques.

Torque Definition

The torque $\tau$ produced by a single force $\mathbf{F}$ about a point $O$ is given by

$$\tau = \mathbf{r} \times \mathbf{F}$$

where $\mathbf{r}$ is the position vector from $O$ to the point of application of the force, and $\times$ denotes the vector cross‑product. The magnitude is

$$\tau = rF\sin\theta$$

with $\theta$ the angle between $\mathbf{r}$ and $\mathbf{F}$. The SI unit is newton‑metre (N·m).

What Is a Couple?

A couple consists of two forces $\mathbf{F}_1$ and $\mathbf{F}_2$ such that:

1. $\mathbf{F}_1 = -\mathbf{F}_2$ (equal magnitude, opposite direction),
2. The forces are parallel, and
3. The lines of action are separated by a perpendicular distance $d$.

The resultant force of a couple is zero, but it produces a non‑zero resultant torque

$$\tau_{\text{couple}} = F d$$

This torque is the same about any point in the plane of the couple; therefore a couple causes pure rotation.

Comparison: Single Force vs. Couple

Aspect	Single Force	Couple
Resultant Force	Non‑zero (produces translation)	Zero (no translation)
Resultant Torque	Depends on point of reference	Same about any point
Effect on Rigid Body	Translation + possible rotation	Pure rotation
Units	N (force) and N·m (torque)	N·m (torque only)

Examples of Couples

• Turning a steering wheel – the driver’s hands apply opposite forces at the rim.
• Opening a door with a wrench – the wrench applies a pair of forces at its ends.
• Torque wrench – calibrated to apply a specific couple to a bolt.

Worked Example

Problem: A wrench of length $0.30\ \text{m}$ is used to loosen a bolt. The mechanic applies a force of $120\ \text{N}$ at one end while an equal opposite force is applied at the other end (the wrench is held stationary at the centre). Calculate the torque produced by the couple.

Solution:

1. Identify the magnitude of each force: $F = 120\ \text{N}$.
2. Determine the perpendicular distance between the lines of action: $d = 0.30\ \text{m}$.
3. Use the couple torque formula: $\tau = F d = 120\ \text{N} \times 0.30\ \text{m} = 36\ \text{N·m}$.

The bolt experiences a clockwise torque of $36\ \text{N·m}$, independent of the point about which the torque is calculated.

Practice Questions

1. Two forces of $50\ \text{N}$ act on a rectangular plate, one upward on the left edge and one downward on the right edge, 0.40 m apart. Determine the magnitude and direction of the torque produced by the couple.
2. A door is 0.80 m wide. A force of $30\ \text{N}$ is applied at the edge of the door, perpendicular to the plane of the door. What torque does this produce about the hinges? Is this a couple?
3. Show that the torque of a couple is the same about any point by calculating the torque about two different points for the couple in the worked example above.

Summary

• A couple consists of two equal, opposite, parallel forces whose lines of action are separated.
• The resultant force of a couple is zero, but it creates a constant torque $\tau = F d$.
• Because the torque of a couple is independent of the reference point, it causes pure rotation of a rigid body.