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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

Learn7367Paper 3 · Mechanics optionCentres of mass and momentsCentres of mass of laminas

Centres of mass and moments · Subtopic 1 of 1

Centres of Mass of Laminas

The centre of mass is the point where the whole weight acts. For particles $\bar x=\frac{\sum m_ix_i}{\sum m_i}$ . Split a composite lamina into simple shapes, treat each as a particle at its own centre, and subtract for holes. A suspended body hangs with its centre of mass vertically below the pivot.

Edexcel YMA01 WME02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 50 minRevision 1 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find the centre of mass of particles in a line and in a plane.
Use standard results and composite or cut-out laminas.
Solve problems on suspended laminas and toppling.

Pause and think

Where would you hold a hammer so it balances on one finger?

Section 2 of 14

Core idea

Start hereBalancing on one finger

You can balance a ruler on one finger at its middle, but a hammer only near its head. Every object has one point where its whole weight seems to act. Find it, and you can predict how it hangs, balances and topples.

The big idea

The centre of mass is the mass-weighted average position: $\bar x=\frac{\sum m_ix_i}{\sum m_i}$ . Composite shapes are split into simple parts placed at their own centres; holes count as negative mass.

1. Particles and moments

Why the weighted average

$\sum m_igx_i$
Total moment of the weights about the $y$ -axis.
$=\left(\sum m_i\right)g\,\bar x$
A single total weight at $\bar x$ must have the same moment.
$\bar x=\frac{\sum m_ix_i}{\sum m_i}$
Cancel $g$ and solve.

Worked exampleThree particles

$2$ , $3$ , $5$ kg at $(0,0)$ , $(4,0)$ , $(2,6)$ . Find $(\bar x,\bar y)$ .

Goal: Coordinates.
Strategy: Weighted averages.

$\bar x=\frac{0+12+10}{10}=2.2$
$x$ moments.
$\bar y=\frac{0+0+30}{10}=3$
$y$ moments.

Answer

(2.2,3)

✓ CheckInside the triangle of particles ✓.

2. Composite laminas and suspension

\bar x=\frac{\sum A_ix_i}{\sum A_i}\quad(\text{uniform: mass}\propto\text{area})

$A_i$: area of each part; negative for a hole.

Worked exampleRectangle with a corner removed

A $6\times4$ rectangle has a $2\times2$ square removed from its bottom-right corner ( $4\le x\le6$ , $0\le y\le2$ ). Find $\bar x$ .

Goal: $\bar x$ .
Strategy: Whole minus hole.

$\text{whole: }A=24,\ x=3$
Centre of the rectangle.
$\text{hole: }A=-4,\ x=5$
Negative area.
$\bar x=\frac{72-20}{20}=2.6$
Weighted average.

Answer

\bar x=2.6

✓ CheckMass removed on the right moves

\bar x

left of

3

✓.

Worked exampleHanging from a corner

The lamina with $G=(2.2,3)$ (origin at corner $O$ ) hangs from $O$ . Find the angle between the $y$ -axis edge and the vertical.

Goal: Angle.
Strategy: $OG$ is vertical.

$\tan\theta=\frac{\bar x}{\bar y}=\frac{2.2}3$
Angle between $OG$ and the $y$ -axis.
$\theta=36.3^\circ$
Inverse tan.

Answer

36.3^\circ

✓ Check

\bar x<\bar y

so less than

45^\circ

✓.

Composite method

Drawing diagram…

A table keeps composite problems organised.

Key takeawayToppling A body on a plane topples when the vertical through

G

passes outside its base.

Common misconceptionTriangle centre A triangle's centre is

\frac13

of the height above the base, not half.

Pause and think

A uniform wire bent into an L has arms $4$ and $2$ . Is its centre of mass on the wire?

Summary

$\bar x=\frac{\sum mx}{\sum m}$ .
Composite: table, holes negative.
Suspended: $G$ directly below the pivot.

Centre of mass: Point where the total weight effectively acts.
Lamina: Flat object with negligible thickness.
Uniform: Mass spread evenly.

HistoryArchimedes. Archimedes found the centres of mass of triangles and parabolic segments around 250 BC in "On the Equilibrium of Planes".

Fun factHigh jump. In the Fosbury flop the jumper's body curves over the bar while their centre of mass passes under it.

Section 3 of 14

Key formulae

Particles

\bar x=\frac{\sum m_ix_i}{\sum m_i},\quad\bar y=\frac{\sum m_iy_i}{\sum m_i}

Triangle

\tfrac23\ \text{of the way along each median from the vertex}

Sector (angle 2α)

\frac{2r\sin\alpha}{3\alpha}\ \text{from the centre}

TrickTable method. Make a table: shape, mass (or area),

x

y

. For a hole use a negative area. Then

\bar x=\frac{\sum Ax}{\sum A}

Section 4 of 14

Worked examples

Example 1 · Particles0 / 2 steps

Masses

2

3

5

kg at

(0,0)

(4,0)

(2,6)

. Find the centre of mass.

Example 2 · Suspension0 / 2 steps

A lamina has centre of mass

(2.2,3)

relative to corner

O

. It hangs from

O

. Find the angle between the

y

-axis edge and the vertical.

Section 5 of 14

Explore the graph

Moving one mass moves the centre

Loading the interactive graph…

Try this

Masses $2$ at $(0,0)$ , $3$ at $(4,0)$ , $5$ at $(p,q)$ . The fourth point is the centre of mass.
It moves half as far as the $5$ kg mass: $5$ out of $10$ kg.

Weighted average of positions.

In the real worldBuses on a tilt test. Double-decker buses must not topple when tilted to

28^\circ

: engineers keep the centre of mass low with heavy parts under the floor.

Section 6 of 14

Common mistakes

IncorrectAdding the hole's area

CorrectSubtract it

A hole is negative mass.

IncorrectTriangle centre at half height

CorrectOne third of the height from the base

Medians meet

\frac23

from each vertex.

IncorrectSector angle

\alpha

as the full angle

Correct

\alpha

is half the angle

Read the formula carefully.

Cut a hole in the lamina

Centre of mass: moving away from the hole.

Section 7 of 14

Exam tip

Exam tipChoose axes on edges. Put the origin at a corner and axes along edges so all coordinates are positive.

Section 8 of 14

Guided practice

SA-FM-CL-G012 marksL1 FoundationNon-calculatorParticles

Masses of

2

kg and

6

kg are at

x=0

and

x=4

. Find

\bar x

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-FM-CL-0012 marksL2 StandardNon-calculatorParticles

Masses

2

3

5

kg at

(0,0)

(4,0)

(2,6)

. Find

\bar x

Salam Academy Original Practice · Proposed mark scheme

SA-FM-CL-0023 marksL2 StandardNon-calculatorComposite laminas

An L-shape is made from a

4\times2

rectangle occupying

0\le x\le4

0\le y\le2

and a

2\times2

square occupying

0\le x\le2

2\le y\le4

. Find

\bar x

Salam Academy Original Practice · Proposed mark scheme

SA-FM-CL-0033 marksL3 ExamNon-calculatorComposite laminas

A uniform

6\times4

rectangle (

0\le x\le6

0\le y\le4

) has a

2\times2

square removed from

4\le x\le6

0\le y\le2

. Find

\bar x

Salam Academy Original Practice · Proposed mark scheme

SA-FM-CL-0041 markL2 StandardNon-calculatorComposite laminas

A uniform triangular lamina has height

9

cm above its base. How far above the base is its centre of mass?

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-FM-CL-0052 marksL3 ExamCalculatorSuspension

A lamina with centre of mass at

(2.2,3)

relative to corner

O

hangs freely from

O

. Find, in degrees, the angle between the edge along the

y

-axis and the vertical.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Weighted mean of positions.
Composite: table; holes negative.
Suspended: $G$ below the pivot.
Topples when $G$ passes outside the base.

Section 12 of 14

Quick quiz

Q1 · SA-FM-CL-Q011 markL1 FoundationNon-calculatorSuspension

A lamina hangs freely from a point

A

. In equilibrium,

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 7367 A-level Further Mathematics
Where: Mechanics · Centres of mass and moments
Expected: Centres of mass of systems of particles, uniform laminas and composite bodies; suspension; moments.
Calculator: Permitted on all papers.
Typical style: AQA mixes short multiple-choice questions with longer structured ones.
Open in: 7367 · Paper 3

Section 14 of 14

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Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7367, First teaching 2017 (official specification).