At first glance, replacing an extended body — a rugby ball in flight, a high jumper, a galaxy — with a single mathematical point seems a brutal simplification. The surprise is that, as far as the motion of the centre of gravity is concerned, this simplification is exact: holds rigorously, regardless of what the internal forces do, of deformation, of the body’s own rotation, of the vibrations of its various parts. It is a theorem, not an approximation.
The physical meaning is profound: internal forces, by the third law, always occur in equal and opposite pairs and therefore cancel in the vector sum; only external forces contribute to the motion of the centre of gravity. This “democracy of internal forces” is what allows a gymnast to twist in mid-air while still falling with their centre of gravity on a perfect parabola, and what allows us to treat the solar system as a collective gravitational pendulum around its centre of mass (which falls inside the Sun, but not exactly at the Sun’s centre).
At a more abstract level, this is the first manifestation of a principle that runs throughout the whole of physics: separating the collective degrees of freedom (the motion of the centre of gravity, usually easy) from the internal degrees of freedom (rotations, oscillations, deformations, often complicated). The same idea, applied to a crystal, will give sound waves (normal modes of the lattice) and the phonons of statistical mechanics. Applied to a cloud of galaxies, it will separate the Hubble flow from the peculiar motion of individual galaxies. Learning to recognise and isolate the motion of the centre of gravity is therefore a mental exercise that pays off well beyond the single problem of collisions or explosions.
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Topics: Centre of mass Concepts: Centre of mass · Newton’s third law
Related exercises: Worked exercise — The cat on the cart · Rocket that explodes mid-trajectory · Disc with a tangent hole