W = F·d·cosθ · P = W/t · the work–energy theorem
You will be exhausted, and physics will record zero. Work needs a force AND a distance moved along it, and the arithmetic does not care how tired your arm is. Once you accept that, one short equation connects every push you make to every change of speed that follows.
This page covers work as force times distance along it, the angle that so often makes it zero, power as work per second, and the work–energy theorem that ties them to motion. By the end you'll be able to:
Everything before this chapter is forces; everything after it is energy. The work–energy theorem is the sentence that joins the two halves of mechanics:
Searched as: work done formula, w = fd cos theta, work energy theorem, power formula, why is holding not work, negative work.
Predict the final speed, lock it, then drag the block and watch the area under the force graph turn into kinetic energy. 🟢 real work–energy engine
Guess, then lock — the bench waits until you commit.
Set your force, the distance and the angle — then watch what reaches the block as energy.
Before any arithmetic, decide whether any work is being done at all. Six situations — your answers build the table underneath.
Holding a suitcase still. What work is being done on it?
Ask two things: is there a force, and has it moved anything along itself?
The angle between the force and the motion settles every one of these.
A 2 kg trolley starts at rest. Choose a force so that after 4 m it is moving at the target speed — 3 attempts.
Set the force, then push. W = F·d becomes ½mv².
Each claim breaks the definition somewhere. Tap the broken idea.
Work is force multiplied by the distance moved along it. With no movement there is no distance, so the work on the box is zero however long it is held.
Friction opposes the motion, so the angle between it and the displacement is 180° and cos180° is −1. Its work is negative: it takes energy out.
Work against gravity is mgh and contains no time at all. Running changes the POWER, which is work per second, not the work itself.
Thinking beats memorising. Sort every claim about work into three boxes.
One force, one motion, one decision. The sign of work is where most marks are lost.
Compare the direction of the force with the direction of the movement.
Pick a lifting job, then choose a motor. The energy is fixed by the load and the height; the motor decides only how long it takes.
Work is mgh. Power is that work divided by the time you are allowed.
Read the scene. All three are about the difference between work and power.
Same you, same height, so the same mgh of work. Running delivers those joules in fewer seconds, which is more power — and that is precisely what makes it hard.
The force opposes the motion, so cos180° = −1 and the work is negative. That negative work is exactly the kinetic energy being removed, and the brakes get hot by the same amount.
In a circular orbit gravity is at 90° to the velocity at every instant, so cosθ is zero throughout. That is exactly why the satellite's speed never changes and it needs no fuel.
In plain terms: work is the way energy gets moved from one place to another by a force. If nothing moves along the force, nothing is transferred.
Push at an angle and your force splits into two parts: one along the motion and one across it. Only the first does any work, which is why the formula carries cosθ. At 90° the whole force is across the motion and the work is exactly zero, no matter how large the force is — this is why carrying a bag on the level does no work against gravity.
A force pointing against the motion has θ = 180°, cosθ = −1, and does negative work. Friction always does. So do brakes, and so does gravity when you lift something. Negative work means energy is being taken OUT of the object's motion — which is precisely what slowing down is.
Wnet = ΔKE. Add up the work done by every force, and the total equals the change in ½mv². You never need the acceleration, the time, or even the shape of the path. Messy forces over a messy route still give the final speed in one line, which is why this theorem is worth more than it first appears.
A centripetal force does no work at all, because it is always perpendicular to the motion — which is why orbital speed is constant. Power can also be written P = F·v, so the same engine delivers more power the faster you are already going. And for a changing force, the work is the area under the force–distance graph, which is where a spring's ½kx² comes from.
Exams reward the method. Work it one step at a time — read the thought, predict the line, then reveal it.
Three small labs, each one native to this idea.
Six wrong ideas that survive because the everyday word and the physics word disagree. Tap a card to bust the myth.
Gaspard-Gustave de Coriolis proposed using "work" for force times distance, to give engineers a single number for what a machine actually delivered.
His book on the calculation of machine action let a water wheel, a horse and a steam engine be compared by the same measure for the first time.
James Joule showed that mechanical work and heat are the same currency, which is why the unit of both now carries his name.
The unit of power was named for James Watt, whose engines had been sold in horsepower — a marketing figure he invented to compare against the animals he was replacing.
Everyday speech still uses "work" for effort and "power" for importance, which is why this chapter spends most of its time on definitions.
The energy per lift is mgh whatever motor you fit. The motor's rating decides only how many seconds each lift takes.
Identical work either way. Running compresses the same joules into fewer seconds, which is the whole of why it is harder.
P = F·v means the same thrust delivers more power the faster you already are, which is why acceleration eases off near top speed.
One sentence each: what is doing work, and on what.
Drag the slider. Six jobs, from a few joules to a few hundred gigajoules.
Eleven orders of magnitude, and the same three letters underneath every one.
Each one is a build and a measurement — the measuring is what turns a demo into a project.
Build: a flight of stairs, a tape measure and a stopwatch.
Measure: your mass, the total height, and the time. Compare walking with running — same joules, different watts.
Build: a trolley pulled the same distance by the same spring balance, at several rope angles.
Measure: the final speed each time. The flatter the rope, the faster it ends up.
Build: a trolley pulled by a hanging weight over a known distance, with a timing gate.
Measure: the speed for several distances, then plot v² against distance. It should be a straight line.
Build: a spring stretched in steps, with the force recorded at each extension.
Measure: plot F against x and find the area under it. Compare with ½kx².
Build: a trolley given a known push, allowed to run until it stops.
Measure: its initial kinetic energy and the distance it travelled; divide to get the friction force, then check it against μmg.
Build: a load raised directly, then through a two-pulley system.
Measure: force and distance in both cases. The force halves, the rope length doubles, and the work barely changes.
Straight answers, each one complete on its own:
Work is force multiplied by the distance moved along that force. A held object does not move, so there is no distance and the work done on it is zero. Your muscles do burn energy, because they hold tension by contracting and relaxing constantly, but that energy goes into your body rather than into the object.
It picks out the part of the force that lies along the motion, because only that part transfers energy. At 0° the whole force counts, at 60° half of it does, and at 90° none of it does. The angle is measured between the force and the displacement, not between the force and the ground.
When the force opposes the motion, the angle is 180° and cosθ is −1. Friction, brakes and gravity on an object moving upwards all do negative work, which means they are removing energy from the object's motion rather than adding to it. That removed energy has to go somewhere, usually as heat.
It says the total work done by all the forces equals the change in kinetic energy. That lets you find a final speed without knowing the acceleration, the time, or even the shape of the path — which is often far quicker than using F = ma and an equation of motion, and sometimes the only practical route.
Work is the total energy transferred, measured in joules. Power is how fast it is transferred, measured in watts, which are joules per second. Climbing the same stairs slowly or quickly involves the same work; only the power differs, and it is the power that makes the fast climb feel hard.
No. It is always perpendicular to the motion, so cosθ is zero at every instant and the work is zero over any path. This is exactly why an object in uniform circular motion keeps a constant speed, and why a satellite in a circular orbit needs no fuel to stay there.
Because they are the same quantity in different forms. Joule's experiments showed that a fixed amount of mechanical work always produced the same amount of heat, which established energy as a single currency. That is why the joule is used for work, kinetic energy, potential energy and heat alike.
Then the work is the area under the force–distance graph rather than a simple product. For a constant force that area is a rectangle, F × d. For a spring, where the force grows as you stretch it, the area is a triangle and gives ½kx², which is exactly the elastic energy stored.
Spot these before the exam does.
Seven question formats, the way LLOS.ai serves them — multiple choice, multiple-correct, fill-in-the-blank, match, sequence, read-think-connect, and write-your-own. Every question has layered hints: a quick nudge, the reasoning, then a deeper connection — so a wrong answer opens a door, never a dead end. 🟢 received from a board-tagged question bank · seed toward 2,000
Pick your board — the set re-tunes to its wording and emphasis. Competitive draws the JEE / NEET / Olympiad lane.
Four efforts, no numbers yet. Tap them in order, most work first. Then read what W = F·d·cosθ gives — and see what fooled you.
Holding a 30 kg crate steady at chest height for a full ten minutes, arms shaking.
Carrying that same 30 kg crate fifty metres along a flat corridor.
Sliding a 10 kg box four metres across a floor with a steady 25 N push.
Lifting the 30 kg crate from the floor onto a bench one metre up.
Tap the four in the order you believe, most work first.
Effort is what you feel. Work is what arrives. The gap between those two words is most of what this page has been about.