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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

Learn9MA0Paper 3 · Statistics and MechanicsForces and Newton's lawsForces, components and resultants

Forces and Newton's laws · Subtopic 1 of 8

Forces, Components and Resultants

Every force has a size and a direction. Split each force into a horizontal and a vertical part, add the parts, and Pythagoras gives you the overall effect.

Cambridge 9709 Paper 4Edexcel 9MA0 Paper 3AQA 7357 Paper 2≈ 30 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Name and draw the forces on a particle: weight, normal reaction, tension, friction and driving force.
Resolve a force into two perpendicular components.
Find the magnitude and direction of the resultant of several forces.
Use Newton's third law to identify pairs of forces.

Pause and think

You pull a suitcase with a force of $50$ N along a handle tilted at $40^\circ$ to the ground. Is all $50$ N moving the suitcase forwards?

Section 2 of 14

Core idea

Start herePulling a sledge

Pull a sledge with a rope at an angle. Part of your effort drags it forward; part lifts it slightly. How much of each? A force at an angle is equivalent to two perpendicular forces, its components, and splitting forces this way makes every mechanics problem a pair of simple one-direction problems.

The big idea

A force $F$ at angle $\theta$ to the horizontal has components $F\cos\theta$ (horizontal) and $F\sin\theta$ (vertical). To combine several forces, add all horizontal components and all vertical components, then use Pythagoras and $\tan$ for the resultant.

1. Components

A 10 N force and its components as the angle changes

θ (degrees) = 10

Small angle: almost all of the force acts horizontally. Large angle: almost all of it acts vertically. The two components and the force make a right-angled triangle.

Why

F\cos\theta

and

F\sin\theta

$\text{right-angled triangle: hypotenuse }F,\ \text{angle }\theta$
The force and its two components.
$\cos\theta=\frac{F_x}{F}\Rightarrow F_x=F\cos\theta$
Adjacent side.
$\sin\theta=\frac{F_y}{F}\Rightarrow F_y=F\sin\theta$
Opposite side.

"Cos goes with the angle": the component next to the angle uses

\cos

2. Resultants

Adding the two forces head to tail

The resultant is the single force that would have the same effect as both together.

Worked example10 N at $30^\circ$ and 8 N at $120^\circ$

Angles from the positive $x$ -axis. Find the resultant.

Goal: Magnitude and direction.
Strategy: Resolve, add, recombine.

$x:\ 10\cos30^\circ+8\cos120^\circ=8.660-4=4.660$
$\cos120^\circ=-0.5$ .
$y:\ 10\sin30^\circ+8\sin120^\circ=5+6.928=11.928$
Vertical.
$R=\sqrt{4.660^2+11.928^2}=12.8\text{ N}$
Pythagoras.
$\tan^{-1}\tfrac{11.928}{4.660}=68.7^\circ$
Direction.

Answer

12.8

N at

68.7^\circ

to the

x

-axis.

✓ CheckThe forces are

90^\circ

apart:

\sqrt{10^2+8^2}=12.8

✓.

Common misconceptionMixing sin and cos Check which angle you're given: from the horizontal or the vertical. The component adjacent to the angle is

\cos

Pause and think

50 N at $40^\circ$ below the horizontal. Components?

Summary

$F\cos\theta$ , $F\sin\theta$ .
Add components; recombine.

Component: Part of a force in a direction.
Resultant: Single equivalent force.

HistoryNewton's parallelogram. In the Principia (1687), Isaac Newton stated as a corollary to his laws that two forces acting together have the effect of the diagonal of the parallelogram they form. Resolving is the same idea run backwards: one force becomes the two sides of a rectangle.

Fun factHow heavy is a newton? A force of

1

N is roughly the weight of a small apple (about

100

g). It is a fitting unit, given the story that a falling apple set Newton thinking about gravity, a tale he told himself in old age.

Section 3 of 14

Key formulae

Components of F at angle θ to the horizontal

F_x=F\cos\theta,\qquad F_y=F\sin\theta

Resultant

|R|=\sqrt{X^2+Y^2},\qquad \tan\phi=\frac{Y}{X}

Weight

W=mg,\qquad g=10\ \text{m s}^{-2}

Weight $mg$ acts vertically downwards.
Normal reaction $R$ acts at right angles to the surface, pushing away from it.
Tension acts along a string, pulling away from the particle. A light string has the same tension throughout.
Newton's third law: if $A$ pushes or pulls $B$ , then $B$ pushes or pulls $A$ with an equal and opposite force, of the same type.

TrickCos goes next to the angle. The component adjacent to the marked angle uses

\cos

; the other one uses

\sin

. If the angle is given to the vertical, the vertical component is

F\cos\theta

. Check with an extreme: at

\theta=0

the whole force is along the adjacent direction, and

\cos0=1

Section 4 of 14

Worked examples

Example 1 · Resolving0 / 2 steps

Find the horizontal and vertical components of a force of

50

N acting at

40^\circ

above the horizontal.

Example 2 · Resultant of three forces0 / 4 steps

Forces of

10

N at

0^\circ

8

N at

90^\circ

and

6

N at

225^\circ

(angles measured anticlockwise from the positive

x

-axis) act on a particle. Find the magnitude and direction of the resultant.

Example 3 · Normal reaction0 / 2 steps

6

kg block rests on a horizontal floor. It is pulled by a force of

20

N at

30^\circ

above the horizontal but does not move. Find the normal reaction.

Section 5 of 14

Explore the graph

A force and its components

Loading the interactive graph…

Try this

The slanted line is a force of $f$ newtons at $a^\circ$ . The horizontal and vertical lines are its components.
Set $a=90$ . What happens to the horizontal component? Why?
Move $a$ into the second quadrant ( $90<a<180$ ). Which component becomes negative?

The force is the hypotenuse; its components are the other two sides: $f\cos a$ and $f\sin a$ .

In the real worldWhy tow ropes are kept low. When a car is towed, only the horizontal component of the rope's tension pulls it forward. A rope tied high at a steep angle wastes tension lifting the car's front, so recovery trucks keep the tow line as close to horizontal as possible.

Section 6 of 14

Common mistakes

IncorrectVertical component

=F\cos\theta

when

\theta

is to the horizontal

Correct

F\sin\theta

Cos goes with the side next to the angle.

IncorrectAdding magnitudes:

10\text{ N}+6\text{ N}=16

CorrectResolve, then use Pythagoras

Forces only add like numbers if they act in the same direction.

IncorrectNormal reaction

=mg

always

CorrectResolve perpendicular to the surface

Other forces with a vertical component change

R

IncorrectWeight and normal reaction are a third-law pair

CorrectThey are not

A third-law pair acts on two different bodies; both of these act on the same body.

Add the forces as numbers
Draw them and guess
Resolve every force into $\rightarrow$ and $\uparrow$ components, then Pythagoras

Section 7 of 14

Exam tip

Exam tipDraw and label every force. A clear force diagram, with each force drawn as an arrow from the particle and the angles marked, is often worth a mark on its own and makes the resolving almost automatic. Give angles to

1

decimal place and forces to

3

significant figures.

Section 8 of 14

Guided practice

SA-9709-P4-FE-G012 marksL1 FoundationCalculatorResolving

A force of

20

N acts at

30^\circ

above the horizontal. Find its horizontal and vertical components, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-9709-P4-FE1-0012 marksL1 FoundationCalculatorResolving

A force of

50

N acts at

40^\circ

above the horizontal. Find its horizontal and vertical components, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-FE-0013 marksL1 FoundationCalculatorResolving

Forces of

3

N due east and

4

N due north act on a particle. Find the magnitude of the resultant and the angle it makes with east, in degrees to 1 decimal place.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-FE1-0063 marksL2 StandardCalculatorResolving

A force has horizontal component

12

N and vertical component

5

N. Find its magnitude and the angle it makes with the horizontal, in degrees to 1 decimal place.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-FE1-0042 marksL1 FoundationCalculatorTypes of force

A box of mass

4

kg rests on a horizontal table. A string attached to the top of the box pulls vertically upwards with tension

15

N, but the box stays on the table. Find the normal reaction.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-FE1-0024 marksL2 StandardCalculatorResolving

Forces

(5\mathbf{i}+2\mathbf{j})

(3\mathbf{i}-8\mathbf{j})

N and

(-2\mathbf{i}+3\mathbf{j})

N act on a particle. Find the magnitude of the resultant, and the angle it makes with

\mathbf{i}

(in degrees, to 1 decimal place, below the

\mathbf{i}

direction).

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-FE1-0082 marksL2 StandardCalculatorTypes of force

A block of mass

6

kg rests on a horizontal floor. It is pulled by a force of

20

N at

30^\circ

above the horizontal and does not move. Find the normal reaction.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-FE-0113 marksL2 StandardCalculatorResolving

Two forces of

10

N and

6

N act at a point, with an angle of

60^\circ

between their directions. Find the magnitude of their resultant.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P4-FE1-0094 marksL3 ExamCalculatorResolving

Two forces, each of magnitude

10

N, act on a particle. One acts due east and the other at

60^\circ

north of east. Find the magnitude of the resultant (3 s.f.) and its direction, as an angle north of east.

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-FE1-0036 marksL4 ChallengingCalculatorResolving

Forces of

10

8

N and

6

N act on a particle in directions

0^\circ

90^\circ

and

225^\circ

, measured anticlockwise from the positive

x

-axis. Find the magnitude of the resultant (3 s.f.) and the angle it makes with the positive

x

-axis (1 d.p.).

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Components: $F\cos\theta$ next to the angle, $F\sin\theta$ opposite it.
Resultant: add all the components, then $\sqrt{X^2+Y^2}$ and $\tan^{-1}\dfrac YX$ .
Weight $mg$ down; normal reaction perpendicular to the surface; tension along the string.
Third-law pairs act on different bodies.

Section 12 of 14

Quick quiz

Q1 · SA-9709-P4-FE-Q011 markL1 FoundationCalculatorTypes of force

Taking

g=10\ \text{m s}^{-2}

, the weight of a

2

kg mass is

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-9709-P4-FE1-0051 markL1 FoundationCalculatorTypes of force

A book rests on a table. By Newton's third law, the force paired with the weight of the book is

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-9709-P4-FE1-0072 marksL1 FoundationCalculatorResolving

Two forces of

8

N and

P

N act at right angles. Their resultant has magnitude

17

N. Find

P

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics
Where: WME01 (Mechanics 1) · Vectors and statics of a particle
Expected: Resolve forces, find resultants (often with i, j vectors), and model with particles, light strings and smooth surfaces.
Calculator: Permitted. Edexcel uses g = 9.8 m s⁻².
Typical style: Forces often given as i, j vectors; directions sometimes asked as a bearing.
Open in: 9MA0 · Paper 3

Section 14 of 14

Past paper questions

4 questions (27 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

9709/42 · Oct/Nov 2025Question 711 marks
Resultant of four coplanar forces, then a coefficient of friction for a ring on a wire.
Question paper, page 10 Mark scheme
9709/42 · Feb/March 2025Question 14 marks
Three coplanar forces in equilibrium: an unknown angle and force.
Question paper, page 3 Mark scheme
9709/42 · May/June 2024Question 36 marks
Four coplanar forces in equilibrium: find an angle and a force.
Question paper, page 4 Mark scheme
9709/42 · May/June 2022Question 26 marks
Magnitude and direction of the resultant of coplanar forces.
Question paper, page 3 Mark scheme

All 9709 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel 9MA0, First teaching 2017 (official specification).