What is Center of mass; momentum of a system? — Physics

Definition 19.1 University Physics — Year 1 · Chapter 19 — Systems of Points; Introduction to Rigid Bodies

For particles of masses $m_i$ at $M_i$ , total mass $M = \sum m_i$ , the center of mass (barycenter) $G$ is defined by

\sum_i m_i\,\vect{GM_i} = \vect 0, \qquad\text{i.e.}\qquad \vect{OG} = \frac{1}{M}\sum_i m_i\,\vect{OM_i}

for any origin $O$ . The momentum of the system is $\vect P = \sum_i m_i\vect v_i = M\vect v_G$ .

Examples

Example 19.3 (Diver, recoil, collision)

The diver’s $G$ follows a parabola under her weight alone, however she twists — her muscles are internal forces. A rifle of $4.0\,\mathrm{kg}$ firing a $10\,\mathrm{g}$ bullet at $800\,\mathrm{m}/\mathrm{s}$ : $\vect P = \vect 0$ before and after, so the rifle recoils at $2.0\,\mathrm{m}/\mathrm{s}$ . Two cars, $1000\,\mathrm{kg}$ at $20\,\mathrm{m}/\mathrm{s}$ and $1500\,\mathrm{kg}$ at rest, that lock together: $v = 20000/2500 = 8.0\,\mathrm{m}/\mathrm{s}$ — momentum conserved, while $60\%$ of the kinetic energy has gone into crumpled metal and heat. Momentum is conserved in every collision; kinetic energy only in elastic ones (Exercise 19.12).

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