Physics (9203) | Centre of mass

Welcome to the Chapter: The Centre of Mass!

Hello Physics students! This chapter, Centre of Mass (CoM), is super important because it helps us understand how objects balance, move, and—crucially—why they sometimes fall over! Since we are studying Forces and their effects, understanding CoM allows us to determine exactly where the force of gravity (weight) effectively acts on an object.
Don't worry if this sounds complicated; we will break down the concepts into easy, bite-sized pieces. Let's get started!

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1. Defining the Centre of Mass (CoM)

What is the Centre of Mass?

The Centre of Mass (CoM) is a special point in an object where the entire mass of the object appears to be concentrated.
You can think of it as the object’s 'average' position of mass.

When we talk about CoM in relation to forces, we often use another term: Centre of Gravity (CG).

CoM vs. CG: Are they the same?

In the context of the International GCSE, CoM and CG are usually used interchangeably.

• The Centre of Mass (CoM) relates to the distribution of mass.
• The Centre of Gravity (CG) relates to the force of gravity (weight) acting on the object.

Since the force of gravity is practically uniform across all parts of an object (unless the object is huge!), the CoM and CG are located at the exact same point.

Crucial Point: When calculating the effect of weight, we treat the entire weight (W) of the object as acting downwards through the Centre of Gravity.
\( \text{Weight } (W) = \text{Mass } (m) \times \text{Gravitational Field Strength } (g) \)

Real-World Analogy: Balancing a Ruler

Try balancing a normal wooden ruler on the tip of your finger. You must place your finger exactly in the middle (at the 50 cm mark, if it’s a 100 cm ruler). This balancing point is the Centre of Mass. If you push the ruler, it acts as if all its mass were concentrated right at that spot!

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2. Locating the Centre of Mass

Where the CoM is located depends entirely on the shape and the distribution of mass within the object.

A. Symmetrical Objects (The Easy Way)

For objects with a uniform density and a regular, symmetrical shape, the CoM is located exactly at the geometric centre of the object.

• A square or rectangle: The CoM is where the diagonals cross.
• A uniform rod or ruler: The CoM is halfway along its length.
• A sphere (like a football): The CoM is at the very centre of the sphere.

Did You Know? The Centre of Mass doesn't always have to be inside the object! For a doughnut or a horseshoe shape, the CoM is located in the empty space!

B. Irregular Objects (The Plumb Line Method)

How do we find the CoM of something irregularly shaped, like a piece of cardboard cut into a strange shape (a lamina)? We use an experimental technique involving a plumb line.

Step-by-Step: Finding the CoM of an Irregular Lamina

1. Preparation: Take the irregular piece of cardboard (the lamina) and mark at least three small holes (A, B, C) around its edge.
2. Hang It Up: Hang the lamina freely from one of the holes (e.g., Hole A) using a pin or clamp.
3. Use the Plumb Line: A plumb line is simply a mass (a bob) tied to a string. When hung, the string always hangs perfectly vertical (due to gravity). Hang the plumb line from the same pin used to hold the lamina.
4. Draw the Line: Once the lamina and the plumb line have stopped swinging, use a pencil to draw a line on the lamina exactly along the path of the plumb line string.
5. Repeat: Unhook the lamina and repeat steps 2-4, hanging the lamina from a different hole (e.g., Hole B). Draw the second line.
6. Find the Intersection: The point where the two drawn lines cross is the Centre of Mass (CoM) of the lamina.

Why does this work? When an object is hanging freely and is stable, its Centre of Mass must be directly below the point of suspension. By hanging it from two different points, we ensure that the CoM lies on both straight lines. Therefore, the only point common to both lines is the CoM itself!

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3. Centre of Mass and Stability

One of the most important applications of the Centre of Mass is understanding stability—how resistant an object is to being toppled over.

The Two Rules for High Stability

An object is more stable (harder to knock over) if it meets two criteria:

1. A Wide Base Area

The base area is the surface area that an object rests on. A wider base provides more support and makes the object more stable.
Example: Think about standing normally versus standing with your feet wide apart. Which position is harder for someone to push you over from? Standing with a wider base is much more stable!

2. A Low Centre of Mass (CoM)

If the CoM is close to the ground, the object is more stable.

• Example: Racing cars (like Formula 1 cars) are designed to be extremely flat and low to the ground. This gives them a very low CoM, allowing them to take corners at high speed without rolling over.
• Example: A pyramid is incredibly stable because its CoM is very low compared to its massive base.

Common Mistake Alert! Stability is NOT about how heavy something is. It’s about how the weight is distributed. A heavy, tall object can be less stable than a lighter, squat object.

The Condition for Toppling

An object will only topple (fall over) if the line of action of its weight falls outside its base area.

1. The line of action of weight is the imaginary vertical line that drops straight down from the Centre of Mass.

2. As you tilt an object, the position of the CoM stays fixed relative to the object itself.

3. As the object tilts further, the vertical line from the CoM moves closer and closer to the edge of the base.

4. Toppling Occurs: The moment this vertical line falls outside the base, the object loses its turning force (moment) to return to its original position, and instead gains a turning force that makes it fall over.

Step-by-Step Toppling Process

1. Start with a stable object: The line of action of weight falls well within the base.
2. Tilt slightly: The CoM is still above the base. The weight creates a moment that pulls the object back upright (restoring moment).
3. Reach the critical angle: The CoM is exactly above the edge of the base. The object is balanced precariously.
4. Tilt further: The CoM moves outside the base. The weight now creates a moment that rotates the object and makes it fall over (toppling moment).

Therefore, having a low CoM means you have to tilt the object much further (a greater angle) before the line of action of weight falls outside the base, significantly increasing its stability.