Q.The density of a non-uniform rod of length 1m is given by $\rho(x) = a(1+bx^2)$ where a and b are constants and $0 \le x \le 1$. The centre of mass of the rod will be at

What is the Center of Mass?

Imagine you pick up a broom by the handle and try to balance it horizontally on one finger. You instinctively slide your finger along the handle until the broom stays level. That point — the one where the broom doesn't tip — is its center of mass.

Now think about throwing a cricket bat. It spins and wobbles in the air, but there is one point on the bat that follows a smooth, parabolic path, as if all the bat's mass were concentrated there. That point is also the center of mass.

The core idea is simple: the center of mass is the average position of all the mass in an object. It's the point where you could imagine the entire mass of the object being concentrated, and the object would behave the same way under the influence of external forces.

Why does this matter?

When you push an object at its center of mass, it moves in a straight line without rotating. Push it anywhere else, and it will both move and spin. This is why:

• A car's stability depends on where its center of mass is (lower = safer).
• A tightrope walker holds a long pole — moving the pole shifts their combined center of mass back over the rope.
• In projectile motion, the center of mass of a system (like an exploding firework) continues along the original parabolic path, even though the fragments scatter.

The precise definition

For a system of particles, the center of mass is the weighted average of their positions, where the weight is the mass of each particle.

Here:

• $\vec{R}_{\text{CM}}$ is the position vector of the center of mass
• $m_i$ is the mass of the $i$-th particle
• $\vec{r}_i$ is the position vector of that particle

For a continuous object (like a rod or a sphere), the sum becomes an integral:

where $M$ is the total mass and $dm$ is an infinitesimal mass element.

Breaking it down with an example

Take two masses on a light rod: $m_1 = 2 \text{ kg}$ at $x = 0$, and $m_2 = 3 \text{ kg}$ at $x = 5 \text{ m}$.

The center of mass is:

So the center of mass is at $x = 3 \text{ m}$, closer to the heavier mass. That makes intuitive sense — the heavier mass "pulls" the average toward itself.

Key properties to remember

1. External forces only — Internal forces (like collisions between parts of the system) do not affect the motion of the center of mass. Only external forces can change its velocity.

2. If no external force acts, the center of mass moves with constant velocity (or stays at rest). This is the law of conservation of momentum applied to the whole system.

3. For symmetric objects with uniform density, the center of mass coincides with the geometric center. For irregular shapes, it shifts toward the region with more mass. …