Work Done by a Force (CAIE 9709 Mechanics 4.5) — Mathematics

Introduction

Work is one of the most fundamental ideas in mechanics. In everyday language "work" means effort, but in mathematics it has a precise definition that links forces to motion. The concept underpins the Work–Energy Theorem and the study of power and energy throughout the Mechanics unit of 9709.

A key insight that examiners test repeatedly is that only the component of a force in the direction of motion does work. When a force and a displacement point in different directions, you must resolve before calculating — this is exactly what the formula $W = Fd\cos\theta$ achieves.

Core Concept

When a constant force $F$ acts on an object and the object undergoes a displacement $d$ , the work done by the force is defined as:

W = Fd\cos\theta

where $\theta$ is the angle between the direction of the force and the direction of the displacement.

Interpreting the formula

If the force is parallel to the displacement ( $\theta = 0°$ ), then $\cos\theta = 1$ and $W = Fd$ . All of the force contributes.
If the force is perpendicular to the displacement ( $\theta = 90°$ ), then $\cos\theta = 0$ and $W = 0$ . The force does no work at all — for example, the normal reaction from a horizontal floor does no work on a box sliding along it.
If the force has a component opposing the displacement ( $90° < \theta \leq 180°$ ), then $\cos\theta < 0$ and $W$ is negative — the force removes energy from the object (e.g. friction).

Physical meaning

Work done by a force is the energy transferred by that force. A positive value means energy is given to the object; a negative value means energy is taken from it.

Units

Force in newtons (N), distance in metres (m) → work in joules (J), where $1\text{ J} = 1\text{ N m}$ .

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