Newton's second law for the CM

The centre of mass is not just a convenient midpoint: it obeys a surprisingly simple law of motion, Newton’s second law for the CM.

Law — Second law for the centre of mass

$\ev{\sum \vec{F}_{\text{esterne}} = M_{\text{tot}}\,\vec{a}_{\text{CM}}}$ Internal forces do not affect the motion of the CM.

This result is remarkable: the centre of mass of a system moves as if the whole mass were concentrated there and all external forces were applied at that point. Internal forces — collisions between parts, explosions, tensions, deformations — change nothing about the CM’s motion. The deep reason is Newton’s third law: internal forces always come in equal and opposite pairs, so they cancel in the vector sum. Only external forces “survive”.

Example — Two balls falling and colliding

Two balls ($3\;\text{kg}$ and $9\;\text{kg}$) fall from a height of $100\;\text{m}$ with opposite horizontal velocities. During the fall they collide.

The system’s CM has $\vec{a}_{\text{CM}} = (0,\; -9{,}81)$, because only weight is an external force. The collision is an internal force: the CM continues its parabolic motion undisturbed, as if the two balls had never touched. Where the CM lands can be found using nothing but the 2D kinematics equations, completely ignoring the collision.

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Topics: Centre of mass Concepts: Centre of mass · Newton’s second law · Newton’s third law

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