define and apply the torque of a couple

Turning Effects of Forces

1. Definition of Torque (Moment of a Force)

The torque (or moment) of a force about a point O is a measure of its tendency to cause rotation about that point.

Mathematically, torque is defined by the vector product

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

where

• \(\mathbf{r}\) is the position vector from the point O to the point of application of the force.
• \(\mathbf{F}\) is the applied force.
• \(\times\) denotes the cross‑product.

The magnitude of the torque is

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

with \(\theta\) the angle between \(\mathbf{r}\) and \(\mathbf{F}\). The direction of \(\boldsymbol{\tau}\) is given by the right‑hand rule.

2. Torque of a Couple

A couple consists of two equal and opposite forces whose lines of action do not coincide. Because the forces are equal and opposite, the resultant force is zero, but they produce a net turning effect.

The torque of a couple is the same about any point and is given by

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

where

• \(F\) is the magnitude of either force in the couple.
• \(d\) is the perpendicular distance between the two lines of action.

The direction of the couple’s torque follows the right‑hand rule applied to the rotation it would cause.

3. Key Characteristics of a Couple

Feature	Single Force	Couple
Resultant force	Non‑zero (produces translation)	Zero (forces cancel)
Resultant torque	Depends on point of reference	Same about any point
Effect	Translation + possible rotation	Pure rotation
Formula for magnitude	\(\tau = rF\sin\theta\)	\(\tau = Fd\)

4. Applying the Torque of a Couple

When solving problems involving a couple, follow these steps:

1. Identify the two forces and confirm they are equal in magnitude and opposite in direction.
2. Determine the perpendicular distance \(d\) between their lines of action.
3. Calculate the magnitude of the couple’s torque using \(\tau = Fd\).
4. Apply the right‑hand rule to establish the sense (clockwise or anticlockwise) of the torque.
5. Use equilibrium conditions if required:
   • For rotational equilibrium, the sum of torques about any point must be zero: \(\sum \tau = 0\).
   • Since a couple’s torque is independent of the reference point, it can be added directly to other torques.

5. Example Problem

Problem: A wrench of length 0.30 m is used to loosen a bolt. A force of 120 N is applied at the end of the wrench, perpendicular to the wrench. What is the torque produced by this force? If a second force of 120 N is applied in the opposite direction at a point 0.10 m from the bolt, what is the net torque of the resulting couple?

Solution:

1. Torque due to the first force: $$\tau_1 = (0.30\ \text{m})(120\ \text{N}) = 36\ \text{N·m}$$
2. The second force creates a torque in the opposite sense: $$\tau_2 = (0.10\ \text{m})(120\ \text{N}) = 12\ \text{N·m}$$
3. The two forces form a couple. The net torque of the couple is the difference because they act in opposite senses: $$\tau_{\text{couple}} = \tau_1 - \tau_2 = 36\ \text{N·m} - 12\ \text{N·m} = 24\ \text{N·m}$$
4. The direction is anticlockwise (as defined by the right‑hand rule for the larger torque).

6. Summary

• Torque measures the turning effect of a force: \(\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}\).
• A couple consists of two equal, opposite forces whose lines of action are separated by a distance \(d\).
• The torque of a couple is \(\tau_{\text{couple}} = Fd\) and is the same about any point.
• Couples produce pure rotation without translation.
• In problem solving, identify the forces, measure the perpendicular separation, compute \(Fd\), and apply the right‑hand rule for direction.