Work, Energy and Power: Grade 12 Physical Sciences
Hold a heavy box still for ten minutes and you will be exhausted, and you will have done no work at all. In physics, work needs a force and a displacement in the direction of that force. No movement, no work.
This page covers the definition of work, positive and negative work, the work-energy theorem, conservative and non-conservative forces, conservation of mechanical energy, and power. Plus the sign error that quietly wrecks more answers in this chapter than anything else.
What work actually means
Work is done on an object when a force acting on it displaces it in the direction of that force.
W = FΔx cos θ
| Symbol | Means | Unit |
|---|---|---|
| W | Work done | J (joule) |
| F | Magnitude of the force | N |
| Δx | Magnitude of the displacement | m |
| θ | Angle between the force and the displacement | degrees |
Work is a scalar. It has a sign but not a direction, and the sign is not a direction either. One joule is the work done when 1 N moves something 1 m.
Three cases where no work is done, and they all surprise people
| Situation | Why W = 0 |
|---|---|
| Holding a heavy box still | Δx = 0. Your muscles are working. Physics is not |
| Carrying a box horizontally across a room | You lift upwards, you move sideways. θ = 90° and cos 90° = 0 |
| The normal force on a sliding block | Perpendicular to the motion. θ = 90° again |
The second one is the one worth arguing about with a class. Carrying a suitcase across an airport does no work on the suitcase, and everybody's instinct says otherwise. The instinct is measuring tiredness, not physics.
Positive, negative and zero work
The angle decides the sign, and the sign tells you whether energy went in or came out.
| θ | cos θ | Work is | Energy of the object | Example |
|---|---|---|---|---|
| 0° | +1 | Positive, maximum | Increases | Pushing a trolley forwards |
| Between 0° and 90° | Between 0 and 1 | Positive, reduced | Increases | Pulling a case by a handle at an angle |
| 90° | 0 | Zero | Unchanged | The normal force on a sliding block |
| 180° | −1 | Negative, maximum | Decreases | Friction, always |
Friction always does negative work. It always opposes the motion, so θ is always 180°, so cos θ is always −1. If you have written a positive number for the work done by friction, you have made a mistake, every time, without exception.
Negative work does not mean "no work" or "backwards work". It means energy was taken out of the object.
Worked example: force at an angle
A suitcase is pulled by its handle with a force of 60 N at 35° to the horizontal, over 20 m of floor. Find the work done by that force.
- W = FΔx cos θ = (60)(20)cos 35°
- W = 1 200 × 0,819
- W = 983 J
Only the horizontal component of the pull does any work. The vertical component is trying to lift the case off the floor and, since the case does not move upwards, it does nothing.
Worked example: friction on the same suitcase
A frictional force of 18 N acts over the same 20 m.
- W = FΔx cos 180° = (18)(20)(−1)
- W = −360 J
The net work is 983 − 360 = 623 J, and that is the number that tells you what happened to the case's kinetic energy.
The work-energy theorem
Wnet = ΔEk = Ekf − Eki
In words: the net work done on an object equals the change in its kinetic energy.
Since Ek = ½mv², the full form is:
Wnet = ½mvf² − ½mvi²
This is the most useful equation in the chapter, because it connects forces to speeds without going anywhere near time or acceleration.
| If Wnet is | Then |
|---|---|
| Positive | The object speeds up |
| Negative | The object slows down |
| Zero | The speed does not change, though the direction might |
Two ways to find Wnet, and both are marked
- Add up the work done by every force separately, including the negative ones
- Find the resultant force first, then multiply by the displacement
They give the same answer. The first is safer when several forces act at different angles. The second is quicker when everything is along one line.
Worked example: a car braking
A 1 200 kg car slows from 20 m·s-1 to 8 m·s-1. Find the net work done on it.
- Wnet = ½mvf² − ½mvi²
- Wnet = ½(1 200)(8)² − ½(1 200)(20)²
- Wnet = 38 400 − 240 000
- Wnet = −2,02 × 105 J
Negative, because the car slowed down. If your answer came out positive here, you subtracted the wrong way round: it is final minus initial, always.
Conservative and non-conservative forces
A conservative force does work that does not depend on the path taken, only on the start and end points.
| Conservative | Non-conservative (dissipative) | |
|---|---|---|
| Work depends on | Start and end points only | The path taken |
| Examples | Gravity, spring force, electrostatic force | Friction, air resistance, applied forces, tension |
| Associated with | A potential energy | No potential energy |
| Round trip back to the start | Total work is zero | Total work is not zero |
The round-trip test is the quickest way to tell them apart. Carry a box up a hill and back down: gravity has done zero net work, because the box is where it started. Drag it up and back: friction has taken energy out of you both ways, and none of it comes back.
Which route you take up a mountain does not change the work gravity does on you. It very much changes the work friction does.
Conservation of mechanical energy
Mechanical energy is the sum of kinetic and gravitational potential energy.
Emech = Ek + Ep = ½mv² + mgh
When only conservative forces act, mechanical energy is conserved. Energy moves between kinetic and potential and the total never changes.
Ek(i) + Ep(i) = Ek(f) + Ep(f)
A ball thrown up trades kinetic for potential on the way and gets all of it back on the way down. Ignore air resistance and it returns to your hand at exactly the speed it left.
When friction is present
Mechanical energy is not conserved, but total energy still is. The missing mechanical energy has become heat and sound. It has not vanished.
The equation that handles it:
Wnc = ΔEk + ΔEp
where Wnc is the work done by all the non-conservative forces.
Say "mechanical energy is not conserved" and never "energy is lost". Examiners mark the distinction, and it is a real one: energy is never lost, it just stops being useful.
Which method should you use?
This is the decision the chapter is really testing.
| If the question | Use |
|---|---|
| Mentions friction, or a rough surface | The work-energy theorem, or Wnc = ΔEk + ΔEp |
| Says "ignore friction" or "smooth" | Conservation of mechanical energy. Far quicker |
| Asks for a force or a distance | Work-energy theorem |
| Asks only for a speed or a height | Conservation of mechanical energy, if it is frictionless |
Read the question for the word "friction" before you choose. Half the wasted time in this chapter is learners doing the hard method on an easy question.
Power
Power is the rate at which work is done, or the rate at which energy is transferred.
P = W ÷ Δt and equivalently P = E ÷ Δt
The unit is the watt (W), which is one joule per second. One watt is a very small amount of power, which is why kilowatts turn up everywhere.
Power says nothing about how much work was done. It says how fast. Two learners carrying identical boxes up identical stairs do identical work. The one who runs develops more power.
The other power equation
For an object moving at constant velocity against a constant resistive force:
Pav = Fvav
It comes straight out of the first one: P = W/t = FΔx/t, and Δx/t is velocity.
Worked example: a pump
A pump lifts 250 kg of water through 15 m in 40 s. Find its minimum power output.
- W = mgh = (250)(9,8)(15) = 36 750 J
- P = W ÷ Δt = 36 750 ÷ 40
- P = 919 W
Minimum, because a real pump also fights friction in the pipe and has to give the water some kinetic energy at the outlet. Real pumps are rated well above this.
Worked example: a car at constant speed
A car travels at a constant 25 m·s-1 against a total resistive force of 900 N. Find the power developed by the engine.
- Constant speed means the engine force equals the resistance, so F = 900 N
- P = Fv = (900)(25)
- P = 22 500 W = 22,5 kW
At constant velocity the net work is zero and the car's kinetic energy never changes, and the engine is still working hard. All of its output is going into overcoming resistance. That is a genuinely useful thing to understand about cars.
Where this fits in the curriculum
| Subject | Physical Sciences |
|---|---|
| Grade | 12 |
| Term | 2 |
| Topic | Mechanics |
| Status | Examinable theory. No formal assessment attached |
| Paper | Physics, Paper 1 |
It is the whole of Topic 4 on its own, and it reliably carries a long question in Paper 1, usually a block on a slope with friction and four or five parts.
The mistakes that cost the marks
| The mistake | What to do instead |
|---|---|
| Positive work for friction | Friction opposes motion, so θ = 180° and the work is always negative |
| ΔEk calculated as initial minus final | Final minus initial. Always. A car slowing down must give a negative answer |
| Forgetting cos θ when the force is at an angle | Only the component along the motion does work |
| Saying work is done while holding something still | No displacement, no work, however tired you are |
| Using conservation of mechanical energy when there is friction | Read for the word friction first. If it is there, use the work-energy theorem |
| Saying "energy is lost" | Say "mechanical energy is not conserved". The energy became heat and sound |
| Confusing work and power | Work is how much. Power is how fast. Two different units |
| Treating the normal force as doing work | It is perpendicular to the motion, so it never does work on a horizontal or sloping surface |
Seeing it without apparatus
Get a learner to hold a full schoolbag at arm's length for a minute. Ask how hard they are working, then tell them physics says zero. The argument that follows is the lesson.
Then race two learners of similar size up the same flight of stairs, one walking and one running, and time them. Same mass, same height, same work. Different times, so different power. P = mgh ÷ t with real numbers off a phone stopwatch, and the class has measured their own power output in watts.
Most learners come out somewhere between 200 and 600 W, which is a satisfying number because it is comparable to a household appliance.
How the 50 marks are made up
| Section | Marks |
|---|---|
| Multiple choice | 10 |
| Definitions and terminology | 8 |
| Work calculations, including angles and friction | 12 |
| The work-energy theorem | 10 |
| Conservation of mechanical energy | 6 |
| Power | 4 |
No mark allocation is prescribed for this chapter. The split above is ours, weighted towards the work calculations and the theorem because that is where Paper 1 puts its marks.
Free worksheet and marking memo
Both free, no sign up, straight to the PDF.
- Learner worksheet, 50 marks, with multiple choice, work at an angle, negative work by friction, the work-energy theorem, conservation of mechanical energy and two power calculations
- Marking memorandum, with full working, the mark breakdown line by line and a note on the eight places learners drop marks
Related pages
- Vertical projectile motion, Grade 12. The same ball, solved with equations of motion instead of energy
- Conservation of energy with a pendulum. Mechanical energy conservation you can watch
- Newton's Second Law, Grade 11. Where the forces in these questions come from
Apparatus
This chapter needs a stopwatch and a staircase. That is genuinely the whole list for the demonstration that matters.
A stopwatch and a bathroom or platform scale between them let a class measure their own power output, which is the most memorable thing you can do with this material.
If you are teaching the whole mechanics topic, a ticker tape timer is the piece of equipment that earns its place across the most chapters.