Work Done by a Constant Force

What work means in physics

In everyday language, "work" means any physical or mental effort. In physics, work has a precise definition that is quite different.

Work is done on a body only when a force causes displacement in the direction of the force.

If you push a wall with all your strength and it does not move — no work is done, however exhausted you feel. If you carry a heavy bag horizontally — the force (upward, supporting the bag) is perpendicular to the displacement (horizontal) — no work is done by the carrying force.

The formula

For a constant force $\vec{F}$ acting on a body that undergoes displacement $\vec{d}$:

$W = \vec{F} \cdot \vec{d} = Fd\cos\theta$

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

Why $\cos\theta$

Only the component of force along the displacement does work. The perpendicular component has no effect on the motion along the displacement direction.

The component of $\vec{F}$ along $\vec{d}$ is $F\cos\theta$.

Work = (component of force along displacement) × displacement = $F\cos\theta \times d = Fd\cos\theta$.

Sign of work

Positive work: force aids the motion (engine driving a car)

Zero work: force perpendicular to motion (normal force on a moving body, centripetal force on circular motion)

Negative work: force opposes motion (friction on a sliding body, brakes on a car)

Units

$[W] = [F][d] = \text{N} \cdot \text{m} = \text{J (Joule)}$

One Joule is the work done when a force of 1 N moves a body through 1 m in the direction of the force.

Examples

Pulling a suitcase: Force $F = 50$ N at angle $\theta = 30°$ to horizontal, displacement $d = 10$ m:

$W = 50 \times 10 \times \cos30° = 500 \times \frac{\sqrt{3}}{2} \approx 433 \text{ J}$

Lifting a box: Force = $mg$ upward, displacement = $h$ upward, $\theta = 0°$:

$W = mgh\cos0° = mgh$

Friction: Friction force $f$ backward, displacement $d$ forward, $\theta = 180°$:

$W_{friction} = fd\cos180° = -fd$

Friction always does negative work on a sliding body.

Work is a scalar

Although force and displacement are vectors, their dot product — work — is a scalar. Work has no direction, only magnitude and sign.