Center of mass

The center of mass isolates translation

A many-body system can move and deform internally, but its center of mass responds only to the net external force. Internal pushes rearrange matter without accelerating the system’s center of mass.

Weighted position, then differentiate

For particles m_i at positions r_i, define R_cm = (Σm_i r_i)/M with M = Σm_i. Differentiating twice for constant masses gives M a_cm = Σm_i a_i = ΣF_ext because internal forces cancel in pairs. The center-of-mass equation is Newton’s law for the entire system.

Two masses 2 kg at x = 0 and 6 kg at x = 3 m have x_cm = (2·0 + 6·3)/8 = 2.25 m. If an external 16 N force acts right, a_cm = 16/8 = 2 m/s², regardless of internal tension.

Internal energy can still grow

A firework’s center of mass follows the same external-force path while fragments fly apart. Center-of-mass motion does not describe the internal kinetic energy or the shape of the system.

Work it: Three masses 1, 2, and 3 kg sit at x = 0, 1, and 4 m. Find x_cm and its acceleration under a 12 N external force; compare with the acceleration of the 3 kg mass alone.