Work and the Joule

Work in physics means something much narrower than it does in ordinary speech: it is what a force does when it actually moves something, and it is measured in joules. By the end of this lesson you should be able to calculate the work done by a force acting through a distance, run that calculation backwards to find either the force or the distance, and say why a force that moves nothing at all — however hard it is to hold — does no work whatsoever.

Force acting through a distance

Two quantities from the mechanics module are all you need. A force is a push or a pull, measured in newtons, and one newton is roughly the weight of a small apple. A distance is how far something moves, measured in metres. Put them together: when a force acts on an object and the object moves along the line the force is pushing, something has been transferred from whatever supplied the force to the object — and work is the measure of how much. Push a crate 4 metres and you have done twice the work of pushing it 2 metres, because you kept pushing for twice as far; push it with twice the force over the same 4 metres and you have again done twice the work, because every metre cost you twice as much. Work is therefore the force multiplied by the distance, and neither factor can be dropped.

The definition carries one qualification that the arithmetic hides, and it is the qualification the whole lesson turns on: the distance must be measured in the direction of the force. Not the total distance travelled, and not the distance in some other direction — the distance moved along the line the force points.

That is the whole relationship, and the two rearrangements you will need come straight out of it. Divide both sides of work = force × distance by the distance: on the right the distances cancel and leave the force alone, so $\text{force} = \frac{\text{work}}{\text{distance}}$. Divide the original by the force instead and the forces cancel, giving $\text{distance} = \frac{\text{work}}{\text{force}}$. Three arrangements, one fact — and you should derive them rather than memorise them, because the same procedure works on every formula in this module.

The joule

The unit of work is the joule, named after James Joule and written with a capital J because it commemorates a person. One joule is the work done when a force of one newton moves something one metre along the direction of the push. So lifting a small apple a metre off the table is about a joule of work, which tells you immediately that the joule is a small unit for everyday purposes: a person climbing a flight of stairs does a few thousand joules. Where the numbers get large, the kilojoule is used instead, and the prefix kilo- always means a thousand, so one kilojoule is 1000 joules.

One more thing about the joule matters more than anything else in this lesson, because the rest of the module depends on it. Doing work on something transfers energy to it, and the amount transferred is exactly the work done — so energy is measured in joules as well. Every energy quantity in this module, and every one anywhere later in this course, is a number of joules; there is no separate unit for movement energy, stored energy or heat. That single shared unit is what makes it possible to say that one form of energy has turned into another, which is the subject of the next three lessons.

No movement means no work

Hold a heavy suitcase out at arm's length and keep perfectly still. Your arm aches within seconds, you are plainly making an effort, and the physics says you are doing no work on the suitcase at all — because the distance it has moved is zero, and a force multiplied by zero is zero. Push against a brick wall as hard as you like and the answer is the same. This is the first place ordinary language and physics part company, and the parting is deliberate rather than pedantic: work is defined as it is because it measures energy transferred, and nothing has been transferred to an object that has not budged.

There is an honest complication worth stating rather than hiding. You do burn energy holding that suitcase, because your muscle fibres are repeatedly contracting and relaxing to hold the tension, and each of those tiny movements costs energy that ends up as heat inside you. But that is work done inside your body, not work done on the suitcase, and it is the suitcase the question is about. Whenever an item describes a large force and no movement, the work done on the object is zero.

A force at right angles does no work either

Now walk across the room with the same suitcase held at a steady height. It moves several metres, so there is a distance this time, but the force you are applying to it points straight up while the movement is horizontal. The distance moved in the direction of the force is still zero, because the suitcase never rose, so the upward force still does no work. The rule generalises: a force acting at right angles to the movement does no work at all, which is why the ground pushing up on a sliding block does none, and why holding something up while walking with it is tiring but not, in this sense, work.

Between those two extremes sits the angled case in the figure. A rope pulling a sledge upwards and forwards at once can be thought of as two pushes applied together: one along the ground, which is the part doing the moving, and one straight up, which is not. Only the along-the-ground part enters the work calculation, so an angled pull does less work than the same-sized pull applied horizontally — you are spending part of your effort in a direction the sledge is not going. Working out exactly what fraction lies along the motion is a piece of geometry that depends on the angle, and it is beyond what the knowledge tests ask: their items keep the force either along the movement or at right angles to it. What you must be able to do is recognise which of those two cases you are looking at.

Why it matters in aviation

The largest single demand an engine meets in normal operation is not acceleration but the climb, and a climb is work done against weight. Because work is force multiplied by distance, raising an aircraft through a given height costs a fixed number of joules however quickly it is done: the force is the weight and the distance is the vertical gain, so a heavier aircraft simply has more newtons to lift through the same metres. That is why the fuel used getting up to a cruising level rises markedly with take-off weight, and why the same aircraft flown light reaches the same level for noticeably less.

It also explains why a cruising altitude is chosen rather than simply maximised. The thinner air higher up makes the cruise itself cheaper per mile, but every extra thousand feet has to be bought with work first, and on a short sector there may not be enough cruising left to repay what the climb cost. Deciding a level is largely this piece of arithmetic: energy spent buying height set against energy saved once you have it.

What the test asks

The distractor to expect treats effort, or force on its own, as work. An item describes somebody straining to hold a mass steady, or leaning on an immovable object, and offers a large number of joules as the answer — often the force itself, with the units quietly changed from newtons to joules. Nothing moved, so the work is zero, and an answer that never multiplied by a distance cannot be right. The same trap appears in reverse when a question gives you a distance travelled in a direction the force does not point, most often somebody carrying a load along a corridor: the load moved, but not along the line of the upward force, so again the work is zero. The third and mildest trap is arithmetic dressed as physics — a question that quotes a distance in centimetres, which must become metres before it enters the formula, since a joule is defined with a newton and a metre and nothing else.