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

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.