The centre of mass of a continuous body is the single point where we can imagine all the mass of an object is concentrated.

What is a Continuous Body?

In physics, a continuous body is an object where mass is spread out smoothly. Unlike a group of separate particles, you cannot point to one single dot of mass. Examples include a metal rod, a solid sphere, or a wooden cube. Because the material is everywhere inside the shape, we use special math called calculus to find the balance point.

The Mathematical Approach

To find the centre of mass, we divide the object into tiny, microscopic pieces. We call these tiny pieces dm (differential mass). We then find the position of every tiny piece. The formula for the centre of mass (X-coordinate) is:

X = (1/M) ∫ x dm

• M is the total mass of the object.
• dm is a tiny slice of the mass.
• x is the distance of that slice from a starting point.
• The symbol ∫ is an integral, which means we add up all the tiny slices to see the total effect.

Real-World Examples

Imagine a uniform wooden stick. Because the material is the same everywhere, the centre of mass is exactly in the middle. If you put your finger right under that spot, the stick will balance perfectly. If you have a triangular piece of metal, the centre of mass shifts toward the wider base. For a perfectly solid, uniform shape, the centre of mass is often at its geometric centre.

Why Does It Matter?

The centre of mass helps us predict how objects move. If you throw a spinning wrench into the air, the wrench flips and turns in a complicated way. However, the centre of mass of that wrench will follow a simple, smooth path called a parabola. By focusing on this single point, we make hard physics problems much easier to solve.