Centre of Mass
Throw a spanner spinning across a room and it tumbles in a complicated way. Yet hidden inside that chaos, one special point traces a smooth, simple arc — a perfect parabola. That point is the centre of mass, and it is one of the most useful ideas in all of mechanics.
The single point that represents the whole
The centre of mass is the average position of all the mass in an object or system, weighted by how much mass sits at each location. For motion under external forces, the entire object behaves exactly as if all its mass were concentrated at this one point and all external forces acted there.
For two masses m₁ and m₂ at positions x₁ and x₂ along a line, the centre of mass sits at:
It always lies closer to the heavier mass — which is exactly what “weighted average” means. A heavy adult and a light child balance a see-saw only when the pivot sits much nearer the adult.
Finding it for a system of particles
For many particles, the same idea extends naturally. Each coordinate of the centre of mass is the mass-weighted average of that coordinate across all the particles:
You compute the y and z coordinates the same way. For a solid object with continuous mass, the sums become integrals over the body, but the principle is identical: add up position weighted by mass, then divide by total mass.
External forces move the centre of mass; internal forces never can. A diver can twist and somersault wildly, but their centre of mass still falls along a single parabola, because their own muscles are internal and cannot shift it.
Centre of mass versus centre of gravity
People often use these interchangeably, and in everyday situations they coincide. The centre of gravity is the point where the total weight effectively acts. They differ only when gravity varies across the object — for a tall mountain or a satellite-sized structure, the lower parts feel slightly stronger gravity. For anything human-scale in a uniform gravitational field, centre of mass and centre of gravity are the same point.
Why it makes physics simpler
The centre of mass lets us replace a complicated, extended object with a single point particle. When you study Newton’s laws applied to a thrown brick, you do not track every atom — you track the centre of mass, and Newton’s second law applies to it directly:
Here M is the total mass and a_cm is the acceleration of the centre of mass. This is why projectile problems treat a ball as a dot: only the centre of mass matters for the trajectory.
Balance, stability and toppling
Whether an object stays upright depends on its centre of mass. An object is stable as long as a vertical line dropped from its centre of mass falls within its base of support. Tip it far enough that the line passes outside the base, and gravity rotates it over.
- Racing cars sit low to keep the centre of mass near the ground, resisting roll.
- Tightrope walkers carry long poles to lower and widen their effective centre of mass control.
- A high jumper using the Fosbury flop can arch so their centre of mass passes below the bar even as their body clears it.
Conservation in collisions and explosions
Because internal forces cannot move the centre of mass, its velocity is conserved whenever no external force acts. When a firework explodes in mid-air, the fragments scatter, but the centre of mass of all the pieces keeps following the original parabolic path until they land. This connects directly to momentum and impulse: the total momentum equals the total mass times the velocity of the centre of mass, and that total momentum is conserved in any isolated system. You can explore related ideas with our centre of mass calculator.
Frequently asked questions
Can the centre of mass lie outside the object?
Yes. A doughnut’s centre of mass is in the hole, where there is no material at all. A boomerang’s and a high-jumper’s arched body are other examples — the centre of mass is a geometric average, not necessarily a physical location.
Does the centre of mass depend on orientation?
No. It is fixed within the object regardless of how the object is turned. Rotating a hammer does not move the centre of mass relative to the hammer itself, only relative to the room.
How is it different from the centroid?
The centroid is the purely geometric centre, ignoring mass. The centre of mass equals the centroid only when the object has uniform density. For objects with varying density, the centre of mass shifts toward the denser regions.
