Centre of mass

The centre of mass is the single point that represents the average position of a system's mass, each part weighted by how much mass it contributes: x_cm = (m₁x₁ + m₂x₂ + …)/(m₁ + m₂ + …). For two equal masses it sits exactly halfway between them, but change the masses and it shifts toward the heavier one — a small moon and a large planet have their mutual centre of mass much closer to the planet's centre than to the midpoint between them.

Nothing requires the centre of mass to sit on any actual material. A ring or a hollow sphere has uniform mass only around its rim or shell, and by symmetry its centre of mass sits at the geometric centre — empty space, with no mass there at all. The definition is purely about the weighted average of position, not about where matter happens to be.

The reason the centre of mass earns a name of its own, rather than being a mere bookkeeping convenience, is how it moves: exactly as if the system's entire mass were a single point particle, subject only to the external forces acting on the system as a whole. Internal forces — a spring pushing two blocks apart, a collision between billiard balls, an exploding firework — always come in equal and opposite pairs by Newton's third law, so their effects on the centre of mass cancel exactly, whatever chaos unfolds between the individual parts. A firework that explodes in mid-air scatters fragments in every direction, yet their combined centre of mass keeps falling along the same smooth parabola the unexploded shell would have followed, right up until a fragment hits the ground and an genuinely external force finally acts.