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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn9709Paper 4 · MechanicsEnergy, work and powerWork done

Energy, work and power · Subtopic 1 of 4

Work Done by a Force

A force does work when it moves something: force times the distance moved in the direction of the force. Only the component along the motion counts.

Cambridge 9709 Paper 4Edexcel YMA01 WME02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 30 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Calculate the work done by a constant force, $Fd\cos\theta$ .
Recognise when a force does no work.
Find the work done against friction and against gravity.
Use joules (J) and kilojoules (kJ) correctly.

Pause and think

You carry a heavy suitcase along a level corridor at a steady speed, holding it at the same height. In the physics sense, how much work does your upward force on the suitcase do?

Section 2 of 14

Core idea

Start hereHard work that is not work

Hold a heavy bag still for an hour: exhausting, but in physics you've done no work, because nothing moved. Work means a force moving something along its own direction. It's how energy is transferred.

The big idea

Work done $W=Fd\cos\theta$ : force × distance moved in the force's direction (joules). Work against gravity lifting through height $h$ is $mgh$ .

1. Definition

Pulling a sledge with a rope at angle θ

θ (degrees) = 0

Only the part of the pull along the ground moves the sledge forward. At

\theta=90^\circ

none of it does.

W=Fd\cos\theta

$\theta$: angle between force and motion.
$F\cos\theta$: the useful component.

Work is the area under a force–distance graph

A constant

20

N force over

10

W=20\times10=200

J, the rectangle's area.

Worked exampleSledge

20 N at $30^\circ$ drags a sledge 10 m. Work done?

Goal: $W$ .
Strategy: Component along the motion.

$20\cos30^\circ=17.32$
Useful component.
$W=173\text{ J}$
× 10 m.

Answer

173

✓ CheckLess than

200

J (all horizontal) ✓.

Worked exampleAgainst gravity and friction

A 50 kg crate pulled 8 m up a $20^\circ$ slope at constant speed against 100 N friction. Work done by the pull?

Goal: $W$ .
Strategy: Height gained + friction work.

$h=8\sin20^\circ=2.74;\ mgh=1368$
Gravity.
$100\times8=800$
Friction.

Answer

2168

✓ CheckConstant speed: no KE change, so all work goes to PE and heat ✓.

Common misconceptionSlope length vs height Work against gravity uses the vertical height.

Pause and think

Does the normal reaction do work on a sliding box?

Summary

$W=Fd\cos\theta$ .
$mgh$ uses vertical height.

Work: Energy transferred by a force moving its point of application.

HistoryWho named "work"? The French engineer Gaspard-Gustave de Coriolis gave "work" (travail) its precise meaning in 1829, in a book about measuring the useful output of machines. Engineers wanted a single number for "how much a machine has done", and force times distance was it.

Fun factAn apple a metre. Lifting a

100

g apple through

1

m does about

1

J of work against gravity. Climbing a flight of stairs takes a person roughly

2000

J, and a large chocolate bar contains about a million joules, enough in principle for

500

flights.

Section 3 of 14

Key formulae

Work done by a force

W=Fd\cos\theta

Work done against gravity (raising through height h)

W=mgh

Up a slope at α, distance d along it

h=d\sin\alpha,\qquad W=mgd\sin\alpha

Work done against friction

W=Fd=\mu Rd

TrickGravity only cares about height. The work done against gravity depends only on the vertical height gained, not the route. Carrying a box up a ramp or up a ladder to the same height does the same work against gravity:

mgh

. On a slope, convert the distance along it to a height with

h=d\sin\alpha

Section 4 of 14

Worked examples

Example 1 · Force at an angle0 / 2 steps

A sledge is pulled

10

m across horizontal snow by a rope with tension

40

N at

25^\circ

above the horizontal. Find the work done by the tension.

Example 2 · Against gravity on a slope0 / 2 steps

5

kg particle is pulled

8

m up a smooth slope inclined at

30^\circ

. Find the work done against gravity.

Example 3 · Against friction on a slope0 / 2 steps

4

kg block slides

5

m down a rough slope inclined at

20^\circ

, with

\mu=0.25

. Find the work done against friction.

Section 5 of 14

Explore the graph

Work done as the angle changes

Loading the interactive graph…

Try this

The horizontal axis is the angle between the force and the motion (degrees); the curve is the work done, $fd\cos\theta$ .
At what angle is the work zero? Name a force that always acts at that angle.
For angles over $90^\circ$ the work is negative. Which everyday force always acts at $180^\circ$ to the motion?

Work is greatest when the force points along the motion, zero when perpendicular, and negative when it opposes the motion.

In the real worldRamps and wheelchairs. A wheelchair ramp does not reduce the work needed to reach a doorway; the height is the same. What it does is spread that work over a longer distance, so the force needed is much smaller. Building regulations limit ramps to gentle slopes (often about

1

12

) for exactly this reason.

Section 6 of 14

Common mistakes

Incorrect

W=Fd

for a force at an angle

Correct

W=Fd\cos\theta

Only the component along the motion does work.

IncorrectWork against gravity

=mgd

for

d

along a slope

Correct

mgd\sin\alpha

Gravity only cares about the vertical height.

IncorrectThe normal reaction does work

CorrectIt does none on a fixed surface

It is perpendicular to the motion.

IncorrectAnswers in kJ without saying so

CorrectState the units

1

=1000

carrying this heavy box across the room is so much work

Physics

force is vertical, motion is horizontal

Physics

work done $=0$ J. Your back disagrees.

Section 7 of 14

Exam tip

Exam tipName the force. Say which force's work you are finding: "work done by the tension", "work done against friction". Energy questions often need several of these, and labelling each line prevents double counting or leaving one out.

Section 8 of 14

Guided practice

SA-9709-P4-EW-G012 marksL1 FoundationCalculatorWork

A crate is pulled

5

m along a horizontal floor by a force of

20

N at

30^\circ

above the horizontal. Find the work done by the force, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-9709-P4-EW1-0011 markL1 FoundationCalculatorWork

A horizontal force of

25

N pushes a box

6

m across a floor. Find the work done by the force.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-EW1-0022 marksL1 FoundationCalculatorWork

A sledge is pulled

10

m across horizontal snow by a rope with tension

40

N inclined at

25^\circ

above the horizontal. Find the work done by the tension, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-EW1-0031 markL1 FoundationCalculatorWork against gravity

A box of mass

30

kg is lifted vertically through

4

m. Find the work done against gravity.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-EW-0092 marksL1 FoundationCalculatorWork

A box of mass

10

kg is pulled

8

m across a rough horizontal floor (

\mu=0.3

). Find the work done against friction.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-EW1-0042 marksL2 StandardCalculatorWork against gravity

A particle of mass

5

kg is pulled

8

m up a smooth slope inclined at

30^\circ

to the horizontal. Find the work done against gravity.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-EW1-0062 marksL1 FoundationCalculatorWork against gravity

A particle of mass

2

kg slides

6

m down a slope inclined at

30^\circ

. Find the work done by its weight.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-EW1-0053 marksL2 StandardCalculatorWork

A block of mass

4

kg slides

5

m down a rough slope inclined at

20^\circ

, with coefficient of friction

0.25

. Find the work done against friction, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-EW1-0092 marksL2 StandardCalculatorWork against gravity

A crane does

9000

J of work lifting a

150

kg load vertically at constant speed. Find the height through which the load is lifted.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-9709-P4-EW1-0074 marksL3 ExamCalculatorWork

A crate is pulled

12

m along a rough horizontal floor by a rope inclined at

30^\circ

above the horizontal, with tension

50

N. The friction force is

30

N. Find the work done by the tension (3 s.f.) and the work done against friction.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$W=Fd\cos\theta$ , in joules.
Perpendicular forces do no work.
Against gravity: $mgh$ , with $h$ the vertical height.
Against friction: friction force $\times$ distance.

Section 12 of 14

Quick quiz

Q1 · SA-9709-P4-EW-Q031 markL1 FoundationCalculatorWork

The work done by the normal reaction on a block sliding along a horizontal floor is

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-9709-P4-EW1-0081 markL1 FoundationCalculatorWork

A force does no work on a moving object when it acts

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 9709 AS & A Level Mathematics
Where: Paper 4, Mechanics · 4.5 Energy, work and power
Expected: Understand the concept of the work done by a force, and calculate the work done by a constant force when its point of application undergoes a displacement not necessarily parallel to the force.
Calculator: Permitted. Use g = 10 m s⁻² unless told otherwise.
Typical style: Usually the first part of a longer energy question: find the work done by a pulling force or against friction.
Open in: 9709 · Paper 4

Section 14 of 14

Past paper questions

4 questions (18 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

9709/42 · May/June 2025Question 14 marks
Constant speed from distance and time, then work done by an angled force and power.
Question paper, page 3 Mark scheme
9709/42 · Feb/March 2025Question 36 marks
Power and resistance for an aeroplane at constant speed, and average power.
Question paper, page 5 Mark scheme
9709/42 · May/June 2024Question 13 marks
Work done by a cyclist against resistance while speeding up.
Question paper, page 2 Mark scheme
9709/42 · Oct/Nov 2023Question 35 marks
Speed of a block pulled up a rough slope at an angle, by energy.
Question paper, page 4 Mark scheme

All 9709 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 9709, 2026–2027 (official specification).