Physics · 1 Motion, forces and energy

1.5.3 Centre of gravity

Before you start

Weight is the gravitational force on an object. It acts vertically downwards and is measured in newtons (1.3).
The moment of a force is force × perpendicular distance from the pivot, in N m, and you must recall this equation (1.5.2).
An object with no resultant moment on it does not start to turn (1.5.2).
A weight hanging freely on a thread hangs vertically, straight down.

Syllabus objectives covered (3)

State what is meant by centre of gravity
Describe an experiment to determine the position of the centre of gravity of an irregularly shaped plane lamina
Describe, qualitatively, the effect of the position of the centre of gravity on the stability of simple objects

Part 1 of 7

One point for the whole weight

Gravity pulls on every part of an object, not on one spot. A ruler has a small downward force on every millimetre of its length.
You cannot work with thousands of tiny forces. You do not have to. For every object there is one point where you can put a single force, equal to the whole weight, and it has the same effect as all the tiny forces together. That includes the same turning effect.
That point has a name. The centre of gravity of an object is the point through which its whole weight can be considered to act.
This sentence is your answer to 'State what is meant by centre of gravity'. The key word is weight. The centre of gravity is defined by where the weight acts, so 'the middle of the object' is not a definition.
In practice this means you draw the weight of an object as one arrow. It starts at the centre of gravity and points vertically down.

Part 2 of 7

Where the centre of gravity is for simple shapes

An object is uniform if it is made of the same material with the same thickness all the way through. For a uniform object with a regular shape, the centre of gravity is at its geometric centre.
So for a uniform rectangular sheet it is where the two diagonals cross. For a uniform circular disc it is at the centre of the circle. For a uniform metre rule it is at the $50$ cm mark.
The centre of gravity does not have to be inside the material. For a uniform ring it is at the centre of the ring, where there is no material at all. The weight still acts as if it were there.
forces diagram — not drawn yet
Three uniform flat shapes side by side: a rectangle with both diagonals drawn as dashed lines and a dot where they cross; a circular disc with a dot at its centre; a ring with a dot at its centre in the empty middle. Each dot is labelled 'centre of gravity' and has one downward arrow from it labelled 'weight'.
For uniform regular shapes the centre of gravity is at the geometric centre. For the ring, that point is empty space.
If an object is not uniform, its centre of gravity is nearer to the heavier part. A hammer's centre of gravity is nearer its heavy metal head than the end of its handle, not halfway along its length.
This is why you can use the centre of gravity in a moments calculation. The whole weight of a uniform beam acts at its midpoint, so the moment of the weight about a pivot is the weight × the distance from the pivot to the midpoint.

Part 3 of 7

Why a hanging object settles with its centre of gravity below the pivot

Hang a piece of card on a pin through a hole, so that it can swing. Its whole weight acts at its centre of gravity.
Suppose the centre of gravity is to one side of the pin. Then the line of action of the weight is a perpendicular distance from the pin, so the weight has a moment about the pin. The card turns.
The card swings and then comes to rest. It can only stay at rest when the weight has no moment about the pin. That happens when the line of action of the weight passes through the pin, which means the centre of gravity is vertically below the pin.
So when a freely hanging object is at rest, its centre of gravity is somewhere on the vertical line through the point it hangs from. This one fact is the whole basis of the experiment in the next section.
The same idea works for balancing. A ruler balances on a knife-edge only when its centre of gravity is directly above the knife-edge, because only then does its weight have no moment about the knife-edge.

Part 4 of 7

The plumb-line experiment

A plane lamina is a thin, flat sheet, such as a piece of card. 'Irregularly shaped' means it has no symmetry, so you cannot find its centre of gravity by drawing diagonals. You have to find it by experiment.
Apparatus: the card, a clamp stand holding a pin (or a thin nail) horizontally, a plumb line (a thread with a small weight on the end, which hangs vertically), a pencil and a ruler.
Step 1. Make a small hole near the edge of the card. Hang the card on the pin through this hole, so that it can swing freely.
Step 2. Hang the plumb line from the same pin, in front of the card. Wait until the card and the plumb line are both at rest.
Step 3. Mark two crosses on the card along the plumb line, one near the pin and one near the bottom edge. Take the card down and join the crosses with a ruler and pencil. The centre of gravity is somewhere on this line, because it was vertically below the pin.
Step 4. Make a second hole near a different part of the edge, well away from the first. Repeat steps 1 to 3 to draw a second line.
Step 5. The centre of gravity is the point where the two lines cross. One line only tells you the line the centre of gravity is on. The second line fixes the point.
apparatus diagram — not drawn yet
A clamp stand with a pin held horizontally in the clamp; an irregular flat card hangs from the pin through a small hole near its edge; a plumb line (thread with a small weight) hangs from the same pin in front of the card; two pencil crosses on the card along the thread. An inset shows the same card taken down, with two straight lines drawn from two different holes crossing at a point labelled 'centre of gravity'.
Each hanging position gives one line through the centre of gravity. Where two lines cross is the centre of gravity.

Part 5 of 7

Making the result trustworthy

The card must swing freely on the pin. If the pin grips the card, or the card rubs on the stand, friction can stop it before the centre of gravity is directly below the pin. Then your line misses the centre of gravity.
Wait until the card and the plumb line are completely still before you mark the line. A moving plumb line is not vertical.
Put the holes near the edge and well apart. Then the two lines cross at a large angle and the crossing point is sharp. Two lines that are nearly parallel cross at a point that is hard to locate.
Check with a third hole. The third line should pass through the same point. If the three lines make a small triangle instead, take the middle of the triangle as the centre of gravity. A smaller triangle means more careful work.
A second check: hold the card flat and put the tip of a pencil under the point you marked. The card should balance there, because its centre of gravity is directly above the support.

Part 6 of 7

Falling back or toppling over

An object is stable if, when you tilt it slightly and let go, it returns to its original position.
When you tilt an object, it turns about one edge of its base. That edge is the pivot. The weight still acts vertically down from the centre of gravity.
Small tilt: the vertical line through the centre of gravity still passes through the base. The weight has a moment about the edge that turns the object back. It falls back to where it started.
Large tilt: the vertical line through the centre of gravity passes outside the base, beyond the edge. Now the moment of the weight about the edge turns the object further over. It topples.
forces diagram — not drawn yet
Two drawings of the same tall block on a horizontal floor, each tilted about its bottom right edge. Left: a small tilt, a dot at the centre of gravity, a vertical weight arrow from the dot meeting the floor inside the base, and a curved arrow showing the block turning back. Right: a larger tilt, the vertical weight arrow from the centre of gravity meeting the floor outside the base beyond the pivot edge, and a curved arrow showing the block toppling over.
What decides it is where the weight's line of action meets the floor. Inside the base, the object falls back. Outside the base, it topples.
The changeover is the tilt at which the centre of gravity is directly above the edge. At that moment the weight has no moment about the edge. Tilt it any further and it topples.
The trap here is the wording. It is not the centre of gravity that goes outside the base: the centre of gravity is still up inside the object. It is the vertical line through the centre of gravity — the line of action of the weight — that passes outside the base.

Part 7 of 7

What makes an object more stable

Two things decide how far you can tilt an object before it topples: how high its centre of gravity is, and how wide its base is.
A lower centre of gravity means you must tilt the object through a larger angle before the vertical line through its centre of gravity passes outside the base. So it is more stable.
A wider base puts the edge further from the centre of gravity. Again you must tilt it through a larger angle before it topples. So it is more stable.
forces diagram — not drawn yet
Two objects of the same height on a horizontal floor, both tilted by the same angle about their bottom right edge. Left: a narrow object with a high centre of gravity, its vertical weight line meeting the floor outside its base. Right: a wide object with a low centre of gravity, its vertical weight line still meeting the floor inside its base.
Same tilt, different result. The lower, wider object can be tilted further before it topples.
This is why a racing car is built low and wide, why a Bunsen burner has a heavy, wide base, and why you stand with your feet apart on a moving bus.
The reverse is also true. Heavy things put high up raise the centre of gravity and make an object less stable: heavy luggage on a car roof, or heavy books on the top shelf of a tall bookcase.
Adding mass helps only if it lowers the centre of gravity. A heavy base helps because it brings the centre of gravity down, not simply because the object is heavier.
A pencil balanced on its point is the extreme case. Its base is so small that the smallest tilt puts the line of action of its weight outside the base.

Worked examples

Example 1. A metre rule is not uniform. It balances on a knife-edge at the $46.0$ cm mark. (a) Deduce the position of its centre of gravity. (b) The rule has a weight of $1.5$ N. It is now pivoted at the $20.0$ cm mark. Calculate the moment of its weight about this pivot.

Show the working

(a) The rule balances, so its weight has no moment about the knife-edge. That means the line of action of the weight passes through the knife-edge, so the centre of gravity is directly above it: at the $46.0$ cm mark.
Notice it is not at the $50$ cm mark. The rule is not uniform, so its centre of gravity is nearer its heavier end.
(b) The whole weight acts at the centre of gravity. So the distance you need is from the pivot to the centre of gravity.
$d = 46.0 - 20.0 = 26.0 cm = 0.260 m$
Use moment = force × perpendicular distance from the pivot. This is not given on Paper 4; you recall it from 1.5.2.
$moment = 1.5 \times 0.260 = 0.39 N m$
Give the unit N m. You converted the distance to metres first for exactly this reason: with the distance in cm, the same working gives $39$ N cm, which is a different unit.

Answer: (a) At the $46.0$ cm mark. (b) $0.39$ N m.

Example 2. You are given a flat piece of card with an irregular shape, a clamp stand with a pin, a plumb line, a pencil and a ruler. Describe how you find the position of the centre of gravity of the card.

Show the working

Start with the fact the method depends on: a freely hanging object at rest has its centre of gravity vertically below the point it hangs from. Every step below follows from it.
Make a hole near the edge of the card. Hang the card on the pin through the hole so it swings freely. 'Freely' matters: friction could stop it in the wrong place.
Hang the plumb line from the same pin. The plumb line shows you the vertical line through the pin, and the centre of gravity is on that line.
When the card and plumb line are at rest, mark two points along the plumb line on the card. Take the card down and join the points with a ruler.
Repeat from a second hole, well away from the first, and draw a second line. One line gives a line, not a point.
The centre of gravity is where the lines cross. Add a check if the question gives space: a third hole whose line passes through the same point.

Answer: Hang the card from a hole near its edge on the pin so it swings freely; hang the plumb line from the same pin; when both are at rest, mark the plumb line's position on the card and draw the line; repeat from a second hole; the centre of gravity is where the two lines cross (check with a third hole).

Example 3. A tall cupboard stands on a level floor. Its base is $0.60$ m deep from front to back. When upright, its centre of gravity is $0.90$ m above the floor and $0.30$ m behind the front edge. Someone tilts the cupboard forwards about its front bottom edge. (a) It is released when its centre of gravity is $0.10$ m (horizontally) behind the front edge. Deduce what happens. (b) It is released when its centre of gravity is $0.05$ m in front of the front edge. Deduce what happens. (c) Heavy boxes are put on the top shelf. Explain the effect on the stability of the cupboard.

Show the working

The pivot is the front bottom edge. Look at where the vertical line through the centre of gravity meets the floor, relative to that edge.
(a) The centre of gravity is $0.10$ m behind the front edge, so the vertical line through it meets the floor inside the base. The weight has a moment about the front edge that turns the cupboard backwards. It falls back to its upright position.
(b) The centre of gravity is $0.05$ m in front of the edge, so the line of action of the weight is outside the base. The moment of the weight about the edge now turns the cupboard further forwards. It topples.
(c) Heavy boxes high up raise the centre of gravity. A higher centre of gravity reaches the position directly above the front edge after a smaller tilt.
So the cupboard topples at a smaller angle of tilt: it is less stable. Putting the heavy boxes on the bottom shelf would have the opposite effect.

Answer: (a) It falls back upright, because the weight's line of action is inside the base, so its moment turns the cupboard back. (b) It topples, because the weight's line of action is outside the base. (c) The centre of gravity is higher, so the cupboard topples after a smaller tilt: it is less stable.

Where people go wrong

Mistake: Defining the centre of gravity as 'the middle of the object' or 'where the object is heaviest'.

Instead: It is the point through which the whole weight of the object can be considered to act.

The centre of gravity is at the middle only for a uniform, regular object, and it can even be where there is no material, as for a ring. The definition is about where the weight acts.

Mistake: Thinking the centre of gravity must be somewhere inside the material of the object.

Instead: It can be in empty space. A uniform ring has its centre of gravity at the centre of the ring.

The centre of gravity is a point where the weight can be considered to act, not a piece of the object.

Mistake: Hanging the card from only one hole and marking the centre of gravity somewhere on that line.

Instead: Hang it from at least two holes, well apart. The centre of gravity is where the lines cross.

One hanging position tells you only that the centre of gravity is on the vertical line through the pin. A second line is needed to fix the point.

Mistake: Saying an object topples when 'the centre of gravity goes outside the base'.

Instead: Say it topples when the vertical line through the centre of gravity (the line of action of the weight) passes outside the base.

The centre of gravity is always up in the object, never in the base. What matters is where its line of action meets the ground.

Mistake: Thinking a heavier object is always more stable.

Instead: Mass helps only if it lowers the centre of gravity. Mass added high up makes an object less stable.

The tilt at which an object topples depends on where its centre of gravity is compared with the edge of its base, not on how heavy it is.

Check yourself

State what is meant by the centre of gravity of an object.
Answer
The point through which the whole weight of the object can be considered to act.
Where is the centre of gravity of a uniform ring?
Answer
At the centre of the ring, in the empty space where there is no material.
A uniform beam is $2.4$ m long and has a weight of $300$ N. It is pivoted $0.40$ m from one end. Calculate the moment of its weight about the pivot.
Answer
The weight acts at the midpoint, $1.2$ m from the end, so $0.80$ m from the pivot. Moment $= 300 \times 0.80 = 240$ N m.
In the plumb-line experiment, why must the card be able to swing freely on the pin?
Answer
So that it comes to rest with its centre of gravity vertically below the pin. Friction could stop it in a position where the centre of gravity is not below the pin.
Why do you hang the card from a second hole?
Answer
One line only shows that the centre of gravity is somewhere on that line. The second line crosses it at the centre of gravity.
Give one way to check the position of the centre of gravity you found.
Answer
Hang the card from a third hole: the third line should pass through the same point. Or balance the card flat on a pencil tip at the marked point.
A box is tilted slightly and released, and it falls back. Explain why.
Answer
The vertical line through its centre of gravity still passes through its base, so the weight has a moment about the edge that turns the box back.
State two features that make an object more stable.
Answer
A low centre of gravity and a wide base.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. A student has a thin piece of card with an irregular shape. She also has a clamp stand holding a pin, a plumb line, a pencil and a ruler.

(a) State what is meant by the centre of gravity of the card. 1 mark No calculator

Mark scheme

B1 the point through which the (whole) weight of the card (can be considered to) act(s); insufficient: 'the middle of the card'

(b) Describe how the student finds the position of the centre of gravity of the card. 4 marks No calculator

Mark scheme

B1 makes a hole near the edge of the card and hangs the card on the pin so that it can swing freely
B1 hangs the plumb line from the same pin and, when the card and plumb line are at rest, marks the line of the plumb line on the card
B1 repeats with the card hung from a second hole (in a different position)
B1 the centre of gravity is where the (two) lines cross

(c) Explain why the centre of gravity of the card lies on the line of the plumb line when the card hangs at rest. 2 marks No calculator

Mark scheme

B1 if the centre of gravity is not below the pin, the weight has a moment about the pin, so the card turns, owtte
B1 at rest the weight has no moment about the pin, so its line of action passes through the pin / the centre of gravity is vertically below the pin

(d) Suggest one way the student can check her result. 1 mark No calculator

Mark scheme

B1 hang the card from a third hole and check that the third line passes through the same point / balance the card flat on a pointed support at the marked point

Question 2. A tall, narrow box stands upright on a horizontal floor. Its centre of gravity is at its centre. The box is tilted about one of its bottom edges and then released.

(a) The box is released while the vertical line through its centre of gravity still passes through its base. Predict what happens to the box, and explain your answer. 2 marks No calculator

Mark scheme

B1 it falls back to its original (upright) position
B1 the weight has a moment about the edge that turns the box back, owtte

(b) State what happens if the box is released when the vertical line through its centre of gravity passes outside its base. 1 mark No calculator

Mark scheme

B1 it topples / falls over (onto its side); insufficient: 'it falls' without saying which way

(c) Heavy objects are now packed into the bottom of the box. Explain why the box is now more stable. 2 marks No calculator

Mark scheme

B1 the centre of gravity is lower
B1 it must be tilted through a larger angle before the vertical line through the centre of gravity passes outside the base, owtte

(d) Suggest one change to the shape of the box that would also make it more stable. 1 mark No calculator

Mark scheme

B1 a wider base (accept: make the box shorter); ignore 'make it heavier' on its own

Question 3. A uniform metre rule has a weight of $1.2$ N. It lies on a horizontal bench, with part of its length pushed out beyond the edge of the bench.

(a) State the position of the centre of gravity of the rule. 1 mark No calculator

Mark scheme

B1 at the 50 cm mark / at its midpoint

(b) Nothing is attached to the rule. Deduce the greatest length of the rule that can project beyond the edge of the bench without the rule tipping. Give a reason. 2 marks No calculator

Mark scheme

B1 50 cm (0.50 m)
B1 the centre of gravity is then directly above the edge; any further and the line of action of the weight is beyond the edge, so the weight has a moment that tips the rule, owtte

(c) A load of weight $0.30$ N is hung from the end of the rule that projects beyond the bench. Calculate the greatest length that can now project without the rule tipping. 3 marks

Mark scheme

C1 moments taken about the edge of the bench, with the rule's weight acting at the 50 cm mark
C1 0.30 × x = 1.2 × (0.50 − x), with x the length projecting; accept any equivalent correct moments equation (e.g. in terms of the distance of the centre of gravity from the edge, or in cm)
A1 0.40 m (accept 40 cm); unit required

How this is examined

Describing the plumb-line method step by step: a three- to five-mark Describe about a flat, irregular shape such as a piece of card, with the apparatus named in the stem or left for you to choose Hole near the edge; hang the card from a pin so it swings freely; plumb line hung from the same pin; mark the line on the card when everything is at rest; repeat from a second hole; the centre of gravity is where the lines cross. Write each as its own sentence so each one can be credited.
Why a low centre of gravity and a wide base make an object hard to topple: a two- or three-mark Explain about an everyday object, such as a vehicle, a lamp or a piece of laboratory apparatus Name the feature: 'lower centre of gravity' or 'wider base'. Then: 'it must be tilted through a larger angle before the line of action of its weight passes outside its base'. For a further mark: 'beyond that, the weight has a moment about the edge that turns it over'.
Deciding whether a tilted object falls back or topples: a one- or two-mark Predict or Deduce about a tilted object with its centre of gravity marked Where the vertical line from the centre of gravity meets the ground. Inside the base: 'it falls back, because the weight's moment about the edge turns it back'. Outside the base: 'it topples'. Say the line of action of the weight passes outside the base, not the centre of gravity.
Why the centre of gravity must lie on the plumb line: a two-mark Explain attached to the experiment 'When the card is at rest, its weight has no moment about the pin' and 'so the line of action of the weight passes through the pin: the centre of gravity is vertically below the pin'.
Putting the weight of a uniform object at its midpoint in a moments calculation: one step inside a moments calculation about a uniform rod, beam or ruler The weight placed at the midpoint; the distance measured from the pivot to the midpoint; moment = force × perpendicular distance, with the unit N m.
Saying what the centre of gravity is: a one-mark State 'The point through which the whole weight of the object can be considered to act.' The word weight must be there. 'The middle of the object' does not earn the mark.

Words to know

Centre of gravity: The point through which the whole weight of an object can be considered to act.
Uniform: Made of the same material with the same thickness all through. A uniform regular shape has its centre of gravity at its geometric centre.
Plane lamina: A thin, flat sheet of material, such as a piece of card.
Plumb line: A thread with a small weight on the end. It hangs vertically, so it shows the vertical line through the point it hangs from.
Line of action: The line along which a force acts. For the weight, it is the vertical line through the centre of gravity.
Stable: Describes an object that returns to its original position when it is tilted slightly and released.
Topple: To fall over. A tilted object topples when the line of action of its weight passes outside its base.

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