center of mass

The center of mass of a system of particles is the mass-weighted average of their positions: R_CM = (Σ m_i r_i) / (Σ m_i). For a rigid body it is the single special point whose motion summarises the motion of the whole: the total momentum of the body equals M·V_CM, where M is the total mass, and the net external force equals M·A_CM. In other words, the center of mass moves as if it were a point particle carrying all the mass, subject to all the external forces.

This has a remarkable consequence: in the absence of external forces, the center of mass moves at constant velocity no matter how complicated the internal motion is. A thrown grenade that explodes mid-flight into a hundred fragments has a center of mass that continues along the original parabola as if the explosion never happened — the internal forces cancel in pairs. A tuck-somersaulting diver's CM follows a perfect parabola from the springboard to the water, even though every limb traces a complicated curve. Every trampoline routine, every trapeze act, every cat landing on its feet is choreography around the fact that the CM obeys its own private Newton's laws.

For continuous bodies the sum becomes an integral: R_CM = ∫r dm / M. For simple uniform shapes the CM lies at the geometric center (the centroid); for composite bodies, like a hammer, it shifts toward the heavier part. It is why a hammer thrown end-over-end appears to rotate about a specific point on its handle rather than its geometric center — the rotation axis passes through the CM, and that is nearer the head.