Have you ever tried to balance a broom on your finger? Or wondered why a racing car is so low to the ground? Or why a tightrope walker carries a long pole? Or why a bus full of standing passengers feels like it might tip over on a sharp turn? All of these are stories about equilibrium and the centre of gravity.
Equilibrium is the state of balance — when all the forces and all the turning effects on an object cancel out. The centre of gravity is the point where the entire weight of an object appears to act. Together, these two ideas explain why some objects are stable and others topple at the slightest touch. Understanding them is essential for engineers, architects, athletes, and anyone who has ever tried to stack a tower of blocks. Let us begin.
I once challenged a class to balance a ruler on one finger. Most students tried to balance it at the centre. But one student balanced it by placing her finger at the exact point where it did not tip either way. She had found the centre of gravity by instinct. That is what we are going to learn to do — with understanding, not just instinct.By the end of this chapter, you should be able to:
Equilibrium is a state in which an object experiences no net force and no net moment. In other words, all the forces acting on it cancel out, and all the turning effects cancel out. The object does not accelerate and does not rotate.
There are two conditions for equilibrium:
If both conditions are met, the object is in equilibrium. It stays at rest or moves with constant velocity without rotating.
Think of a book resting on a table. The forces on it (weight downward, normal force upward) balance. The moments also balance. So the book stays still. It is in equilibrium.
Conditions for equilibrium:
1. Σ F = 0 (sum of forces is zero)
2. Σ M = 0 (sum of moments is zero)
Equilibrium can be classified into three types, depending on what happens when an object is slightly displaced:
| Type | Behaviour When Displaced | Example |
|---|---|---|
| Stable equilibrium | Returns to its original position | A ball at the bottom of a bowl |
| Unstable equilibrium | Moves further away from its original position | A ball on top of a hill |
| Neutral equilibrium | Stays in its new position | A ball on a flat table |
Stable equilibrium: When displaced slightly, the object returns to its original position. This happens when the centre of gravity rises and then falls back.
Unstable equilibrium: When displaced slightly, the object moves further away. This happens when the centre of gravity falls and continues to fall.
Neutral equilibrium: When displaced, the object stays in its new position. The centre of gravity does not change height.
Think of a cone. If you balance it on its base, it is stable — it returns to its original position if nudged. If you balance it on its tip, it is unstable — it falls over at the slightest touch. If you lay it on its side, it is in neutral equilibrium — it rolls but does not return or fall further. The same object can exhibit all three types of equilibrium depending on how it is placed.The centre of gravity (CG) of an object is the point where the entire weight of the object appears to act. It is the balance point of the object.
For a uniform object, the centre of gravity is at its geometric centre. For a uniform metre rule, it is at the 50 cm mark. For a uniform sphere, it is at its centre. For a uniform triangle, it is at the centroid.
For irregular objects, the centre of gravity may not be at the geometric centre. It depends on how the mass is distributed.
| Object | Centre of Gravity |
|---|---|
| Uniform metre rule | 50 cm mark |
| Uniform sphere | Geometric centre |
| Uniform cylinder | Midpoint of axis |
| Triangular lamina | Centroid (intersection of medians) |
| Irregular shape | Found by experiment (plumb line method) |
Think of the centre of gravity as the "average position" of all the mass in an object. If you support the object at this point, it will balance perfectly.
The centre of gravity is a powerful idea. It allows us to treat an entire object as if all its weight were concentrated at a single point. This simplifies calculations enormously. Without this concept, analysing the stability of complex objects would be almost impossible.For a regular object: Use symmetry. The centre of gravity lies at the geometric centre.
For an irregular object: Use the plumb line method.
This method works because when an object hangs freely, its centre of gravity is directly below the point of suspension. The plumb line shows the vertical direction, so the line you draw passes through the centre of gravity.
The plumb line method is simple but elegant. It uses gravity itself to find the centre of gravity. Every time you hang the object from a different point, gravity pulls it into a new position, and the plumb line reveals the line along which the centre of gravity lies. Two lines are enough to find the point.The stability of an object depends on two things:
Rule for stability: An object is stable as long as its centre of gravity lies above its base of support. If the centre of gravity moves outside the base, the object topples.
| Factor | Effect on Stability |
|---|---|
| Lower centre of gravity | More stable |
| Higher centre of gravity | Less stable |
| Wider base | More stable |
| Narrower base | Less stable |
Think of a bus. It is wide and low, so its centre of gravity is low and its base is wide. It is stable. A double-decker bus has a higher centre of gravity, so it is less stable and must drive more carefully around corners.
Think of a racing car. It is very low to the ground, so its centre of gravity is very low. This makes it extremely stable, even at high speeds. A tall, narrow vehicle like a van has a higher centre of gravity and is more likely to tip over.
This is why you should never overload the roof of a car. Adding weight to the roof raises the centre of gravity, making the car less stable. It is also why passengers on a bus are asked to move to the lower deck on a ferry — a high centre of gravity can be dangerous in rough seas. Physics is not just theory — it is safety.| Application | How It Works |
|---|---|
| Balancing a broom on a finger | The finger must be placed directly under the centre of gravity |
| Loading a cargo ship | Heavy cargo is placed low to keep the centre of gravity low |
| Designing racing cars | Low centre of gravity improves stability at high speeds |
| Stacking boxes | Stack heavier boxes at the bottom to keep the centre of gravity low |
| Tightrope walking | A long pole lowers the centre of gravity and helps balance |
| High jump (Fosbury flop) | Athletes arch their back to keep their centre of gravity below the bar |
Equilibrium is not just a physics concept — it is a way of understanding the world. Here are some examples:
Understanding equilibrium helps engineers design safe structures, vehicles, and machines. It is one of the most practical concepts in physics.
Every time you stand up, you are unconsciously adjusting your centre of gravity to stay balanced. Your body is a master of equilibrium. It is a beautiful thing to realise that the same principles that govern bridges and cranes also govern your own body.These two terms are often used interchangeably, but they are slightly different:
In a uniform gravitational field (like near the Earth's surface), the centre of mass and the centre of gravity are at the same point. For most everyday purposes, we can treat them as the same thing.
The distinction between centre of mass and centre of gravity becomes important in very large objects or in non-uniform gravitational fields. For example, the centre of gravity of a very tall building is slightly lower than its centre of mass because gravity is slightly stronger at the bottom. But for everyday objects, they are the same.Conditions for equilibrium:
1. Σ F = 0 (sum of forces is zero)
2. Σ M = 0 (sum of moments is zero)
Stability: An object is stable if its centre of gravity lies above its base of support.
Centre of gravity of a uniform object: at its geometric centre
Centre of gravity of an irregular object: found by the plumb line method
Question: A uniform metre rule balances at the 50 cm mark. Where is its centre of gravity?
Answer: The centre of gravity is at the 50 cm mark — the balance point. For a uniform rule, this is also its geometric centre.
Units check: the answer is in cm. Notice that the balance point is the centre of gravity. This is a simple but important observation. If you support an object at its centre of gravity, it balances.Question: A box has a base of 0.5 m × 0.5 m and a height of 1.0 m. Its centre of gravity is at its geometric centre, 0.5 m above the base. Is it stable?
Answer: The centre of gravity is 0.5 m above the base, which is within the base (the base extends 0.25 m on each side from the centre). So the object is stable. If tilted, it will return to its original position as long as the centre of gravity remains above the base.
Units check: the answer is in metres. Notice that the stability depends on the height of the centre of gravity relative to the base. A lower centre of gravity or a wider base increases stability.Question: A uniform rectangular block is 0.2 m wide and 0.6 m tall. At what angle of tilt will it topple?
Answer: The block will topple when its centre of gravity passes directly above the edge of its base. The critical angle θ is given by tan θ = (width/2) / (height/2) = 0.1 / 0.3 = 0.333. So θ = tan⁻¹(0.333) = 18.4°.
Units check: the answer is in degrees. Notice that a taller, narrower object has a smaller critical angle, so it topples more easily. This is why tall, narrow objects are less stable.Question: A uniform rod of length 1.0 m has a 2 kg mass attached at one end and a 3 kg mass at the other. Where is the centre of gravity?
Answer: Taking moments about the 2 kg end: (2 kg × 0) + (3 kg × 1.0 m) = (2 + 3) kg × d. So 3 = 5d, d = 0.6 m. The centre of gravity is 0.6 m from the 2 kg end (or 0.4 m from the 3 kg end).
Units check: the answer is in metres. Notice that the centre of gravity is closer to the heavier mass. This makes sense — the balance point is always closer to the heavier side.Question: A see-saw is balanced. A 50 kg person sits 2 m from the pivot. Where must a 40 kg person sit to balance the see-saw?
Answer: Weight of 50 kg = 500 N. Weight of 40 kg = 400 N. Clockwise moment = Anticlockwise moment. 500 N × 2 m = 400 N × d. 1000 = 400d. d = 2.5 m. The 40 kg person must sit 2.5 m from the pivot.
Units check: the answer is in metres. Notice that the lighter person sits farther from the pivot. This is the same principle we studied in moments. The see-saw is a beautiful example of equilibrium in action.Graphs can help us understand stability and the centre of gravity.
Height of centre of gravity vs stability: If you plot the height of the centre of gravity on the vertical axis and the width of the base on the horizontal axis, you can see how stability changes. A lower centre of gravity and a wider base give greater stability.
Angle of tilt vs toppling: If you plot the angle of tilt on the horizontal axis and the restoring moment on the vertical axis, you get a curve that decreases as the angle increases. When the restoring moment reaches zero, the object topples.
Graphs help us visualise the conditions for stability. They turn abstract concepts into pictures.
Think of a graph as a map of a relationship. The shape tells you how one quantity changes with another.
Apparatus: An irregular cardboard lamina, a pin, a plumb line (weight on a string), a pencil, a ruler, and a retort stand.
Method:
Observation: The three lines intersect at a single point — the centre of gravity.
Explanation: When the lamina hangs freely, its centre of gravity is directly below the point of suspension. The plumb line shows the vertical direction, so each line passes through the centre of gravity. The intersection of two lines is enough to find it; the third line is a check.
Precautions:
Likely source of error: If the holes are too large, the lamina may not hang freely, and the lines may not intersect at a single point. Use small holes for best results.
This experiment is a classic. It shows how to find the centre of gravity of any irregular shape using only gravity and a plumb line. It is simple, elegant, and deeply satisfying. Once you have done it, you understand the concept in a way no textbook can give you.Equilibrium is a state in which an object experiences no net force and no net moment. The object does not accelerate and does not rotate.
1. The sum of all forces acting on the object is zero. 2. The sum of all moments acting on the object is zero.
The centre of gravity is the point where the entire weight of an object appears to act. It is the balance point of the object.
Stable: ball at the bottom of a bowl. Unstable: ball on top of a hill. Neutral: ball on a flat table.
The position of the centre of gravity and the size of the base of support.
Use the plumb line method: hang the object freely from different points and draw lines along the plumb line. The intersection of the lines is the centre of gravity.
A racing car has a lower centre of gravity and a wider base, making it more stable. A van has a higher centre of gravity and a narrower base, making it less stable.
The pole lowers the centre of gravity and allows the walker to make small adjustments to keep their centre of gravity above the rope, improving stability.
(a) Define equilibrium and state the two conditions for equilibrium. [4]
(b) Explain the difference between stable, unstable, and neutral equilibrium. [6]
(c) Give one example of each type of equilibrium. [3]
(a) Equilibrium is a state in which an object experiences no net force and no net moment. (2 marks) Conditions: sum of forces = 0; sum of moments = 0. (2 marks)
(b) Stable: object returns to its original position when displaced. Unstable: object moves further away when displaced. Neutral: object stays in its new position when displaced. (6 marks.)
(c) Stable: ball in a bowl. Unstable: ball on a hill. Neutral: ball on a flat table. (3 marks.)
(a) What is the centre of gravity? [2]
(b) Describe how to find the centre of gravity of an irregular lamina. [5]
(c) Explain why the centre of gravity is important for stability. [3]
(a) The centre of gravity is the point where the entire weight of an object appears to act. (2 marks.)
(b) Make three holes near the edges. Hang the lamina freely from each hole in turn, using a plumb line. Draw a line along the plumb line each time. The intersection of the lines is the centre of gravity. (5 marks.)
(c) An object is stable if its centre of gravity lies above its base of support. If the centre of gravity moves outside the base, the object topples. (3 marks.)
(a) A uniform metre rule balances at the 50 cm mark. A 4 N weight is placed at the 20 cm mark. Where must a 2 N weight be placed to balance the rule? [4]
(b) Explain why a lower centre of gravity increases stability. [3]
(c) Why is it dangerous to overload the roof of a car? [3]
(a) Clockwise moment = Anticlockwise moment. 4 N × 30 cm = 2 N × d. 120 = 2d. d = 60 cm. The 2 N weight must be placed 60 cm from the pivot (at the 110 cm mark, but since the rule is only 100 cm, it would need to be placed beyond the end — or the rule would not balance). (4 marks.)
(b) A lower centre of gravity means the object must be tilted more before its centre of gravity moves outside its base. This makes it more stable. (3 marks.)
(c) Overloading the roof raises the centre of gravity, making the car less stable. It is more likely to tip over, especially on corners. (3 marks.)
A uniform metre rule of weight 2 N is balanced at the 40 cm mark. A 5 N weight is placed at the 10 cm mark. Where must a 3 N weight be placed to balance the rule?
The rule's weight acts at its centre (50 cm mark), 10 cm from the pivot. Its moment is 2 N × 10 cm = 20 N cm anticlockwise. The 5 N weight at 10 cm is 30 cm from the pivot, giving 5 N × 30 cm = 150 N cm anticlockwise. Total anticlockwise moment = 170 N cm. The 3 N weight must produce a clockwise moment of 170 N cm, so its distance from the pivot = 170 / 3 = 56.7 cm. It must be placed at 40 + 56.7 = 96.7 cm.
Explain why a tightrope walker carries a long pole.
The pole increases the walker's moment of inertia, making them more stable. It also allows them to make small adjustments to their centre of gravity by moving the pole, helping to keep their centre of gravity above the rope. This reduces the risk of tipping over.
A crane is lifting a heavy load. Explain why it needs a counterweight on the opposite side.
The load creates a moment about the crane's base. Without a counterweight, this moment could tip the crane over. The counterweight creates an opposite moment that balances the load's moment, keeping the crane stable. This is a direct application of the principle of moments.
Equilibrium is the quiet balance of forces and moments — and the centre of gravity is the point where all that balance is decided.
