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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

LearnYMA01WME01 · Mechanics 1Statics of a particleResolving forces and equilibrium

Statics of a particle · Subtopic 1 of 2

Resolving Forces and Equilibrium

Any force can be split into two perpendicular components: $F\cos\theta$ along a direction and $F\sin\theta$ across it. A particle is in equilibrium when the components in every direction add to zero, so you get two equations from two directions.

Edexcel YMA01 WME01≈ 35 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Resolve a force into perpendicular components.
Find a resultant from several forces.
Solve equilibrium problems with strings and smooth slopes.
Use $\mathbf i,\mathbf j$ forces in equilibrium.

Pause and think

A picture hangs from two strings. Why does the tension get larger when the strings are nearly horizontal?

Section 2 of 14

Core idea

Start hereA hanging sign

A shop sign hangs from a string pushed sideways by a strut. Nothing moves, so every push is cancelled by a pull. Resolve horizontally and vertically, set each to zero, and two equations give two unknowns.

The big idea

Equilibrium: resultant force zero. Resolve in two perpendicular directions; each total is zero. Three forces in equilibrium form a closed triangle.

1. String and horizontal force

A 4 kg particle held by a string and a horizontal force

The string makes an angle

\theta

with the vertical, where

\tan\theta=0.5

Worked example4 kg on a string, $\tan\theta=0.5$

String at $\theta$ to the vertical, held by horizontal force $P$ . Find $P$ and $T$ .

Goal: $P$ , $T$ .
Strategy: Resolve, then divide.

$T\cos\theta=39.2$
Vertical.
$T\sin\theta=P$
Horizontal.
$P=39.2\tan\theta=19.6\text{ N}$
Divide: $T$ cancels.
$T=\sqrt{39.2^2+19.6^2}=43.8\text{ N}$
Square and add.

Answer

P=19.6

T=43.8

✓ Check

T

is bigger than either component ✓.

Worked exampleOn a smooth $30^\circ$ plane

5 kg held by a force along the plane. Force and normal reaction?

Goal: $F$ , $R$ .
Strategy: Resolve along and perpendicular to the plane.

$F=49\sin30^\circ=24.5\text{ N}$
Along.
$R=49\cos30^\circ=42.4\text{ N}$
Perpendicular.

Answer

24.5

42.4

✓ Check

\sqrt{24.5^2+42.4^2}=49

: they balance the weight ✓.

TrickResolve perpendicular to an unknown That direction's equation won't contain it.

Pause and think

Three forces in equilibrium drawn head to tail form?

Summary

Resolve two ways; each sum zero.

Equilibrium: Resultant force zero.

HistoryStevin's chain. Simon Stevin, around 1586, imagined a loop of chain draped over a triangular block. Since it can't spin forever by itself, the forces along each slope must balance, which proved how forces resolve on an incline. He was so proud he put the picture on his book's title page.

Fun factTightrope walkers. A tightrope is never perfectly straight. If it were, the tension needed to support even a light walker would be infinite. Rope engineers always leave a slight sag.

Section 3 of 14

Key formulae

Components

F\cos\theta\ \text{(along)},\qquad F\sin\theta\ \text{(perpendicular)}

Equilibrium

\sum F_x=0,\qquad\sum F_y=0

TrickCos for the side next to the angle. The component of a force along a line that touches the angle uses

\cos

; the other uses

\sin

. Mark the angle on the diagram first, then read which component is "next to" it.

Section 4 of 14

Worked examples

Example 1 · Horizontal holding force0 / 2 steps

3

kg particle rests on a smooth

30^\circ

slope, held by a horizontal force

P

. Find

P

Example 2 · Two strings0 / 2 steps

3

kg particle hangs from strings at

30^\circ

and

60^\circ

to the horizontal. Find the tensions.

Section 5 of 14

Explore the graph

Components of a 10 N force

Loading the interactive graph…

Try this

A $10$ N force at $a^\circ$ above the horizontal and its two components. The values are $10\cos a$ and $10\sin a$ .
At what angle are the two components equal?
Check that the components always satisfy $x^2+y^2=100$ .

Horizontal component $F\cos\theta$ , vertical component $F\sin\theta$ .

In the real worldSuspension bridges. Each hanger on a suspension bridge pulls on the main cable. Engineers resolve those forces at every point to find the cable's tension, which can be tens of thousands of tonnes on long bridges.

Section 6 of 14

Common mistakes

IncorrectSwapping

\sin

and

\cos

CorrectCheck against the angle on the diagram

The component next to the angle uses

\cos

IncorrectResolving along the slope with

R

included

Correct

R

is perpendicular to the slope

It has no component along the slope.

IncorrectAdding magnitudes of perpendicular forces

CorrectPythagoras

5

N and

12

N at right angles give

13

IncorrectOne equation for two unknowns

CorrectResolve in two directions

Two unknowns need two equations.

Guess the forces balance
Resolve in two directions
Lami's theorem or a triangle of forces when there are exactly three

Section 7 of 14

Exam tip

Exam tipState the directions. Write "Resolving vertically:" and "Resolving parallel to the plane:" before each equation. Examiners award method marks per direction, and labelled equations make them easy to give.

Section 8 of 14

Guided practice

SA-IAL-M1-ST-G011 markL1 FoundationCalculatorEquilibrium

A particle of mass

2

kg hangs at rest on a vertical string. Find the tension.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-IAL-M1-ST-0012 marksL1 FoundationCalculatorResolving

Forces of

5

N due east and

12

N due north act on a particle. Find the magnitude of the resultant.

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-M1-ST1-0021 markL1 FoundationCalculatorResolving

A force of

10

N acts at

30^\circ

above the horizontal. Find its horizontal component, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-M1-ST-0022 marksL1 FoundationCalculatorEquilibrium

A particle of mass

3

kg is held at rest on a smooth plane inclined at

30^\circ

by a force

P

acting up the line of greatest slope. Find

P

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-M1-ST1-0032 marksL1 FoundationCalculatorEquilibrium

4

kg particle is held at rest on a smooth plane inclined at

25^\circ

by a force acting up the line of greatest slope. Find the force, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-M1-ST-0033 marksL2 StandardCalculatorEquilibrium

A particle of mass

3

kg is held at rest on a smooth plane inclined at

30^\circ

by a horizontal force

P

. Find

P

, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-M1-ST1-0012 marksL2 StandardNon-calculatorEquilibrium

Forces

(p\mathbf i+2\mathbf j)

(-3\mathbf i+q\mathbf j)

N and

(\mathbf i-5\mathbf j)

N act on a particle in equilibrium. Find

p

and

q

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-M1-ST-0083 marksL2 StandardCalculatorEquilibrium

Three forces

\mathbf F_1=(3\mathbf i+4\mathbf j)

\mathbf F_2=(-5\mathbf i+\mathbf j)

N and

\mathbf F_3

act on a particle in equilibrium. Find the magnitude of

\mathbf F_3

, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-M1-ST1-0042 marksL2 StandardCalculatorResolving

A particle of weight

10

N hangs from two strings at right angles to each other. One string makes

30^\circ

with the vertical. Find the tension in that string, to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-IAL-M1-ST-0096 marksL3 ExamCalculatorEquilibrium

A particle of mass

3

kg hangs in equilibrium from two light strings. One string makes an angle of

30^\circ

with the horizontal and the other makes an angle of

60^\circ

with the horizontal, on the other side. Find the tension in each string. (

g=9.8

)

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-IAL-M1-ST-0066 marksL3 ExamCalculatorEquilibrium

A particle of mass

5

kg hangs in equilibrium from two light strings inclined at

30^\circ

and

60^\circ

to the horizontal. Find the tensions

T_1

(the

30^\circ

string) and

T_2

(the

60^\circ

string), to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Components: $F\cos\theta$ along, $F\sin\theta$ across.
Equilibrium: components sum to zero in every direction.
Resolve in two directions for two unknowns.
Choose directions to avoid unknowns.

Section 12 of 14

Quick quiz

Q1 · SA-IAL-M1-ST-Q021 markL1 FoundationCalculatorEquilibrium

A particle is in equilibrium. Which must be true?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: IAL Mathematics, unit WME01 (Mechanics 1)
Where: M1 · 5 Statics of a particle
Expected: Resolve forces in two perpendicular directions; a particle is in equilibrium when the resultant is zero; problems with strings, smooth slopes and forces given in i, j form.
Calculator: Calculators are permitted, but exact answers (surds, fractions, π) are expected where the question asks.
Typical style: "A particle hangs in equilibrium from two strings… find the tension in each string."
Open in: YMA01 · WME01

Section 14 of 14

Past paper questions

1 question (12 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

WME01/01 · January 2025Question 612 marks
Particle on a rough slope in limiting equilibrium, and whether it stays at rest when the string breaks.
Question paper, page 18 Mark scheme

All YMA01 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Pearson Edexcel YMA01, 2018 specification (official specification).