Work Done

Push a shopping trolley across a car park, drag a sledge up a snowy hill, haul a bucket of water up from a well — in each case you feel yourself using something up. You get tired, you breathe harder, you want a rest. In physics we have a precise word for what you have spent: you have done work.

Work isn't just "effort", though. Standing still holding a heavy suitcase makes your arm ache, but in the physicist's sense you are doing no work on the case at all — because it isn't going anywhere. Work is done only when a force actually moves an object through a distance. No movement, no work. That one sentence is the whole idea of this page, and the rest is learning to count it in joules.

The equation: force times distance

To do work you need two things at once: a force, and a distance moved in the direction of that force. Multiply them together and you have the work done:

The work done by a force is the force multiplied by the distance moved along the force:

W = F\,s

W is the work done, measured in joules (J).
F is the force, measured in newtons (N).
s is the distance moved in the direction of the force, measured in metres (m).

A bigger push, or a longer journey, means more work. The units fit together neatly: because W = F\,s, one joule is one newton multiplied by one metre. That is exactly what a joule means:

1\ \text{J} = 1\ \text{N} \times 1\ \text{m} = 1\ \text{N·m}.

So doing 1 J of work is pushing with 1 N for 1 m — roughly the work of lifting a small apple from the floor to a table. Everyday jobs quickly run into thousands of joules (kilojoules, kJ).

Work done is energy transferred

Here is the deep reason work matters. When you do work on something, you don't make the energy vanish — you move it from one store to another. Doing work is simply energy being transferred by a force. In fact this is so exact that we can write it as a rule:

\text{work done} = \text{energy transferred.}

That's why work and energy share the very same unit, the joule. When you lift a box, the work your muscles do is transferred into the box's gravitational store (it gains the ability to fall back down). When an engine does work driving a car forward, chemical energy from the fuel is transferred into the car's movement (kinetic) store. Count the joules of work, and you have counted the joules of energy moved — they are two names for the same number.

Worked examples

Example 1 — finding the work W. A gardener pushes a wheelbarrow with a steady force of 150 N along a path 12 m long. How much work is done?

W = F\,s = 150\ \text{N} \times 12\ \text{m} = 1800\ \text{J}.

That's 1800 J (or 1.8 kJ) transferred from the gardener into moving the barrow.

Example 2 — finding the force F. A crane does 6000 J of work lifting a crate straight up by 3 m. What lifting force did it use? Rearrange W = F\,s into F = \dfrac{W}{s}:

F = \frac{W}{s} = \frac{6000\ \text{J}}{3\ \text{m}} = 2000\ \text{N}.

Example 3 — finding the distance s. A tractor engine provides a driving force of 800 N and does 20\,000 J of work pulling a plough. How far did the plough move? Rearrange to s = \dfrac{W}{F}:

s = \frac{W}{F} = \frac{20\,000\ \text{J}}{800\ \text{N}} = 25\ \text{m}.

The same little triangle of quantities — W, F, s — solves all three: cover the one you want and read off the other two.

Try it: drag the box and watch the joules

Below is a box on the ground. One slider sets the force pushing it (the orange arrow), the other sets how far it is dragged along the ground. As you move them, the box slides across and the running total at the top shows W = F\,s being worked out live — and, right beneath it, the very same number of joules of energy transferred. Notice that doubling either the force or the distance doubles the work, and that with the distance at zero the work is zero however hard you push.

Only motion along the force counts

The s in W = F\,s is not just "how far the object went" — it's how far it went in the direction the force points. This matters enormously when the force and the movement aren't lined up.

Carry a heavy school bag across a flat playground at a steady walk. The bag's weight pulls straight down, but the bag travels sideways. There is no sideways movement in the direction of the weight, so the work done against gravity is 0 J — even though your arm is aching. To do work against the bag's weight you'd have to lift it, moving it in the same direction as (opposite to) the downward pull.

Aching arms are not work. Holding a weight still, or carrying it along the flat, does no work against its weight — because it isn't moving in the direction of that force. Your muscles get tired for biological reasons, but W = F\,s counts only movement along the force.
No movement means no work, no matter how hard you push. Shove a brick wall with all your might and it doesn't budge: s = 0, so W = F \times 0 = 0 J. Force alone is never enough — something has to move.
Watch your units. The distance must be in metres and the force in newtons to get joules. Centimetres or grams will give a wrong answer — convert first.

Work against friction warms things up

When you drag something across a rough surface, part of your work goes into fighting friction. That energy isn't lost — but it doesn't end up as useful movement either. Instead it is dissipated: transferred to the thermal (heat) store of the surfaces, so they warm up. Rub your two cold hands together on a winter morning and they get toasty within seconds; that heat is exactly the work you did against friction turning into warmth.

It's why a bicycle brake pad gets hot after a long downhill, why a drill bit is warm after cutting, and why sliding down a rope can burn your palms. Whenever a force does work against friction, expect heat.

A shooting star is a speck of rock — often no bigger than a grain of sand — tearing into the atmosphere at tens of kilometres per second. As it rips through the air, the enormous force of air resistance does a colossal amount of work against the meteor's motion. All that work is dissipated as heat — enough to make the rock (and the air around it) glow white-hot and vaporise in a streak of light. The very same physics that gently warms your rubbed hands, scaled up, is what lights up the night sky. Spacecraft returning to Earth face the same problem, which is why they wear thick heat shields to survive the work done against the air.