PhysicsICSE Class 10 · Moment of a Force and Centre of Gravity

Centre of Gravity: Definition, Positions and Plumb-line Method (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

The centre of gravity of a body is the point through which its whole weight acts: the point about which the algebraic sum of the moments of the weights of all its particles is zero. It depends on shape and on how mass is distributed, and it may lie outside the material, as in a ring.

Notice the red dots: a ring's C.G. is in its empty centre, and a solid cone's C.G. (h/4) is lower than a hollow cone's (h/3).

Illustration drawn for QuestionBuddy.

Understand Centre of Gravity

Every particle of a body is pulled down by gravity. These many small parallel forces can be replaced by one force, the weight of the body, acting at one point: the centre of gravity (C.G.).

Definition (write all of it). The centre of gravity of a body is the point about which the algebraic sum of the moments of the weights of all its particles is zero. Equivalently, it is the point through which the whole weight of the body acts, whatever the position of the body. A body supported at its C.G. balances, because the weight then has no moment about the support.

What it depends on. The C.G. depends on the shape of the body and on the distribution of its mass. Moving mass to one side moves the C.G. towards that side. It does not depend on where the body is placed, but it may lie outside the material: the C.G. of a ring is at its centre, where there is no material.

Positions for regular, uniform bodies (the C.G. lies at the geometric centre):

Uniform rod: its mid-point.
Circular ring or disc: its centre.
Hollow or solid sphere: its centre.
Rectangular or square lamina, parallelogram: the point where the diagonals meet.
Triangular lamina: the centroid, where the medians meet.
Cylinder: the mid-point of its axis.
Hollow cone of height $h$ : on the axis at $\frac{h}{3}$ from the base. Solid cone: on the axis at $\frac{h}{4}$ from the base. A solid cone has more of its mass near the base, so its C.G. is lower.

Non-uniform bodies. If the mass is not evenly spread, the C.G. is not at the geometric centre: it shifts towards the heavier part. A cricket bat's C.G. is nearer the blade than the handle; a hammer's is near the head.

Freely suspended body. A body hanging freely from a point comes to rest with its C.G. vertically below the point of suspension. In any other position the weight has a moment about the support and turns the body until that moment is zero.

Finding the C.G. of an irregular lamina (plumb-line method).
1. Make two or three small holes near the edge of the lamina (for example, a piece of cardboard).
2. Hang it freely from a pin through one hole, with a plumb line (a thread with a small bob) from the same pin.
3. When both are at rest, mark the vertical line of the thread on the lamina.
4. Repeat from a second hole. The C.G. is where the two lines cross.
5. A third line from a third hole passes through the same point: this is the check.

The syllabus treats the C.G. qualitatively: you need definitions, positions for regular bodies, reasoning about mass distribution, and the plumb-line method, not calculations of the C.G. of combined bodies.

Centre of Gravity: formulas and key facts

C.G. of a hollow cone: $\frac{h}{3} from the base = \frac{2 h}{3} from the vertex$; On the axis.
C.G. of a solid cone: $\frac{h}{4} from the base = \frac{3 h}{4} from the vertex$; On the axis.
Condition at the C.G.: $Σ (w_{i} \times d_{i}) = 0 about the C.G.$; Sum of moments of the particles' weights is zero.
Freely suspended body: $C.G. lies vertically below the point of suspension$; The weight then has zero moment about the support.

How to solve Centre of Gravity questions

C.G. questions test definitions, positions and reasons; answer precisely, with values when dimensions are given.

1For a definition, include 'point', 'moments of the weights of all particles' and 'sum is zero' (or 'whole weight acts').
2For a regular uniform body, name the geometric point (mid-point, centre, diagonals, centroid).
3For a cone, decide hollow ( $\frac{h}{3}$ ) or solid ( $\frac{h}{4}$ ), substitute $h$ , and say 'from the base' (or convert to 'from the vertex').
4For a non-uniform body, say the C.G. shifts towards the heavier part.
5For a freely suspended body, the C.G. is vertically below the point of suspension; give the moment reason.
6For an irregular lamina, describe the plumb-line method with a diagram and the third-hole check.

Centre of Gravity in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Keeping the C.G. over your feet

Suppose you carry a heavy bag in your right hand on the way to school.

Why do you lean to the left, and why is it easier to carry two half-loads, one in each hand?

Try it first, then show the solutionSolution

The bag shifts the C.G. of 'you plus bag' to the right, so its weight acts to the right of your feet and tends to topple you.
Leaning left moves your body's mass left until the combined C.G. is back above your feet.
With equal loads in both hands, the C.G. stays in the middle, so no lean is needed.

Answer: Leaning brings the combined C.G. back over your feet; equal loads keep it there without leaning.

Memory hook: Your feet chase your centre of gravity.

C.G. below the support

Imagine a toy bird that balances on the tip of its beak on your finger, because small heavy weights are hidden in its wing tips, which curve down below the beak.

Where is its C.G., and why does it swing back when you tilt it?

Try it first, then show the solutionSolution

The heavy wing tips pull the C.G. down to a point below the beak, outside the material.
When tilted, the weight acting at the C.G. has a moment about the beak that turns the bird back until the C.G. is again vertically below the support.

Answer: Below the beak; the weight's moment restores it.

Memory hook: Hang the weight low and the toy comes home.

Tricks to remember Centre of Gravity

Short memory aids. Say each one aloud once, then use it on a question the same day.

Hollow three, solid four
Hollow cone: h/3 from the base. Solid cone: h/4 from the base. Solid has more 'floor', so it sits lower.
Use it when: Cone questions like 2024 Q6(i).
PMZ definition
Point, Moments of weights of all particles, Zero sum. Miss one and lose the mark.
Use it when: Defining C.G.
Plumb points down to G
A freely hanging body always settles with G straight below the pin.
Use it when: Suspended-lamina and kite questions.
Mass pulls G
Wherever extra mass goes, the C.G. follows.
Use it when: Non-uniform bodies, filled containers.

Centre of Gravity: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: C.G. of a triangular lamina · 1 mark · MCQBoard level

The centre of gravity of a uniform triangular lamina lies at the point where:
(a) the angle bisectors meet
(b) the medians meet
(c) the perpendicular bisectors of the sides meet
(d) one side meets the median from the opposite vertex

Try it first, then show the solutionSolution

Each median divides a uniform triangle into two parts that balance about it, so the C.G. lies on every median.
It is therefore at the centroid, where the medians meet.

Answer: (b)

Example 2: Definition and three regular bodies · 3 marks · Short answerBoard level

(a) Define centre of gravity.
(b) State the position of the C.G. of (i) a uniform rod 80 cm long, (ii) a solid uniform cylinder of height 10 cm, (iii) a hollow sphere.

Try it first, then show the solutionSolution

(a) The point about which the algebraic sum of the moments of the weights of all the particles of the body is zero; the whole weight of the body acts through it. (1)
(b)(i) At its mid-point, 40 cm from either end. (½) (ii) On its axis, 5 cm from either flat face. (½)
(iii) At its centre, where there is no material. (1)

Answer: (a) As above. (b) (i) Mid-point, 40 cm from each end (ii) Mid-point of the axis, 5 cm from each face (iii) Its centre.

Example 3: Plumb-line method · 3 marks · Long answerHarder board level

Describe, with the help of a labelled diagram, how you would find the centre of gravity of an irregular cardboard lamina. Why must the C.G. lie on the plumb line?

Try it first, then show the solutionSolution

Diagram: lamina hanging from a pin through hole A, plumb line from the same pin, vertical line drawn; second position from hole B; lines cross at G. (1)
Method: suspend from A, let lamina and plumb line come to rest, mark the line; repeat from B; the crossing point is the C.G.; a line from a third hole C passes through the same point as a check. (1)
Reason: at rest, the weight acting at G has zero moment about the pin, so its line of action, the vertical through G, passes through the pin: G lies on the plumb line. (1)

Answer: G is where the vertical lines from two suspension points cross; it lies on each plumb line because the weight's moment about the pin must be zero.

Example 4: Hollow cone and solid cone · 3 marks · Short answerHarder board level

A hollow cone and a solid cone, both uniform, each have height 9 cm.
(a) Find the position of the C.G. of each from its base.
(b) How far is the C.G. of the hollow cone from its vertex?
(c) Which has the lower C.G., and why?

Try it first, then show the solutionSolution

(a) Hollow: $\frac{h}{3} = \frac{9}{3} = 3$ cm from the base. Solid: $\frac{h}{4} = \frac{9}{4} = 2.25$ cm from the base. (1)
(b) $9 - 3 = 6$ cm from the vertex. (1)
(c) The solid cone: more of its mass lies near the broad base, so the C.G. is nearer the base. (1)

Answer: (a) Hollow 3 cm, solid 2.25 cm from the base (b) 6 cm (c) The solid cone.

Example 5: A freely hanging cricket bat · 3 marks · Short answerHOTS / competency

A cricket bat is hung freely from a nail through a hole at the top of its handle.
(a) Where does its C.G. lie when it is at rest?
(b) Is the C.G. at the middle of its length? Explain.
(c) The bat is now hung from a hole at the bottom of the blade instead. Does the position of its C.G. in the bat change?

Try it first, then show the solutionSolution

(a) On the vertical line through the nail, directly below it. (1)
(b) No. The blade is much heavier than the thin handle, so the C.G. is nearer the blade, below the middle of the length as it hangs. (1)
(c) No. The C.G. is a fixed point in the bat; it depends on shape and mass distribution, not on how the bat is hung. It now lies vertically below the new hole. (1)

Answer: (a) Vertically below the nail (b) No, it is nearer the heavier blade (c) No, it is fixed in the bat.

Example 6: Case study: filling a water bottle · 4 marks · Case-basedHOTS / competency

A tall cylindrical plastic bottle stands upright on a table. The bottle alone is uniform, and its C.G. is at the middle of its height. Water is poured in slowly until the bottle is full.
(a) Where is the C.G. of the empty bottle?
(b) What happens to the C.G. of 'bottle + water' when a little water is poured in? Why?
(c) Where is the C.G. when the bottle is completely full?
(d) So how does the height of the C.G. change from empty to full?

Try it first, then show the solutionSolution

(a) At the middle of its height, on its axis. (1)
(b) It moves down: the added mass is all near the bottom, so the C.G. shifts towards it. (1)
(c) A full bottle is a uniform cylinder of water inside a uniform shell, so the C.G. is back at the middle of the height. (1)
(d) It first falls, then rises again to the middle: it is lowest at some part-filled level. (1)

Answer: (a) Middle of the height (b) It moves down (c) Middle again (d) It falls, then rises back to the middle.

Where students lose marks in Centre of Gravity

The slips examiners see most often, and what to write instead.

Mistake: An incomplete definition
The definition needs 'point', 'moments of the weights of the particles' and 'algebraic sum is zero' (or 'through which the whole weight acts').
Wrong working
C.G. is where the sum of the weights is zero.
Right working
C.G. is the point about which the algebraic sum of the moments of the weights of all particles is zero.
Do this: Learn the definition word for word with its three key terms.
Usually costs: Reported for 2024 Q6(i)(a).
Mistake: Writing h/3 without a value or a reference point
When a height is given, the examiner wants the number and where it is measured from.
Wrong working
'At $\frac{h}{3}$ ' or 'at the centre'.
Right working
For a 6 cm hollow cone: 2 cm above the broad base (or 4 cm from the vertex).
Do this: Substitute $h$ and add 'from the base' or 'from the vertex'.
Usually costs: Reported for 2024 Q6(i)(b), where answers such as 1.5 cm, 3 cm or '4 cm' without 'from the top' lost the mark.
Mistake: Taking the geometric centre for a non-uniform body
Non-uniform mass shifts the C.G. towards the heavier part.
Wrong working
A non-uniform kite: C.G. at its geometric middle.
Right working
C.G. at the point that lies vertically below the point of suspension.
Do this: Ask 'is the mass evenly spread?' before naming a geometric point.
Usually costs: Reported for 2025 Q2(ii).
Mistake: Giving no reason for the C.G. lying below the support
The reason is about moments: only then does the weight have zero moment about the point of suspension.
Wrong working
'Because it hangs straight.'
Right working
The weight acts through the C.G.; at rest its moment about the support is zero, so the C.G. is vertically below it.
Do this: Always bring in 'moment about the point of suspension'.
Usually costs: Reported for 2025 Q2(ii)(b).

Exam tips

When a dimension is given, give the C.G. as a number with 'from the base' or 'from the vertex'; 2024 Q6(i) expected '2 cm above the broad base' for a 6 cm hollow cone.
Uniform solids such as a sphere or cylinder keep the same C.G. whether hollow or filled, but a cone does not: filling a hollow cone moves its C.G. towards the base. The 2024 paper tested this with honey in a cone.
For any freely suspended body (a lamina or a kite, as in 2025 Q2(ii)), the C.G. is on the vertical line through the point of suspension.
Draw the plumb-line diagram with labels: point of suspension, plumb line, vertical lines, G.

Practice questions on Centre of Gravity

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The C.G. of a uniform hollow cone of height 15 cm lies on its axis at a distance from the vertex of:
(a) 5 cm
(b) 10 cm
(c) 3.75 cm
(d) 11.25 cm
Hint
$\frac{h}{3}$ is from the base.
Answer
(b) 10 cm ( $\frac{15}{3} = 5$ cm from the base).
Q2 · 2 marks · Very short answerBoard level
Fill in the blanks: (a) The C.G. of a uniform rectangular lamina lies at the point where its __________ meet. (b) The C.G. of a uniform circular disc lies at its __________.
Hint
Think of symmetry.
Answer
(a) diagonals (b) centre
Q3 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The C.G. of a hollow sphere lies at its centre.
Reason (R): The C.G. of a body must always lie within its material.
(a) Both true and R explains A (b) Both true but R does not explain A (c) A true, R false (d) A false, R true
Hint
Is there any material at the centre of a hollow sphere?
Answer
(c) A is true; R is false, since the C.G. can lie where there is no material.
Q4 · 2 marks · Short answerHarder board level
A uniform wire is bent at its mid-point into a V shape. Does its C.G. lie on the wire? Explain where it lies.
Hint
Each half has its C.G. at its own mid-point.
Answer
No. Each half's weight acts at its own mid-point; the C.G. of the V lies midway between these two points, on the line of symmetry inside the V, where there is no wire.
Q5 · 2 marks · Short answerHarder board level
In the plumb-line method for an irregular lamina, why are at least two suspension points needed, and why is a third used?
Hint
One line gives a line, not a point.
Answer
One suspension only shows that the C.G. lies somewhere on one vertical line; a second line crosses it at the C.G. The third line should pass through the same point, which checks the result.
Q6 · 2 marks · Short answerHarder board level
A picture frame hangs from a single nail by a string fixed at one top corner. In what position does it come to rest? Explain using moments.
Hint
Where must the weight act for zero moment about the nail?
Answer
It hangs tilted, with its C.G. (the centre of the frame) vertically below the nail. Only then does the weight's line of action pass through the nail, so its moment about the nail is zero.
Q7 · 3 marks · Short answerHOTS / competency
A heavy metal clip is fixed to the 0 cm end of a uniform metre rule. A student says the C.G. is still at the 50 cm mark because the rule is uniform. Is he right? Where does the C.G. move, and how could you locate it with a knife edge?
Hint
The C.G. belongs to the whole body: rule plus clip.
Answer
No. The body is now rule plus clip, which is non-uniform; the C.G. moves from 50 cm towards the 0 cm end. Move a knife edge under the rule until it balances horizontally: the C.G. lies directly above the knife edge.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Where is the C.G. of a uniform ring?Show answer
At its centre, outside the material.
2.C.G. of a solid cone of height 8 cm, from its base?Show answer
2 cm.
3.C.G. of a hollow cone of height 18 cm, from its base?Show answer
6 cm.
4.Where does a freely suspended body's C.G. settle?Show answer
Vertically below the point of suspension.
5.Does the C.G. depend on how the body is placed?Show answer
No. It depends on shape and mass distribution only.

Centre of Gravity: frequently asked questions

Can the centre of gravity lie outside a body?

Yes. For a ring, a hollow sphere, a V-shaped wire or a boomerang, the C.G. lies at a point where there is no material.

Why is the C.G. of a solid cone lower than that of a hollow cone?

A solid cone has much more of its mass near the broad base, so its C.G. is at h/4 from the base, below the hollow cone's h/3.

Do I need to calculate the C.G. of combined bodies for ICSE Class 10?

No. The 2027 syllabus treats the centre of gravity qualitatively: definitions, positions for regular bodies, reasoning, and the plumb-line method.