Center of Mass

Center of mass (COM) is the mass-weighted average position of a system: R_COM = Σm_i·r_i / Σm_i.

For continuous bodies: R_COM = (1/M)·∫r·dm.

For symmetric uniform bodies: COM lies at the geometric center (rod, sphere, cube, ring).

COM does NOT have to lie INSIDE the body — e.g., COM of a ring or a doughnut is at the geometric center (in the hole).

Behaves as if all external force F_ext acts on a particle of total mass M at the COM: F_ext = M·a_COM.

Internal forces (between parts of the system) DO NOT affect the COM motion. Action-reaction pairs cancel.

If no external force: COM moves with constant velocity (or stays at rest).

Useful for analyzing collisions, explosions, rocket motion — all internal dynamics, no change in COM trajectory.