Center of mass (Kleppner 3.3)

1. Dec 9, 2014

geoffrey159

1. The problem statement, all variables and given/known data
Suppose that a system consists of several bodies, and that the position of the center of mass of each body is known. Prove that the center of mass of the system can be found by treating each body as a particle concentrated at its center of mass.

2. Relevant equations
$\vec R = \frac{1}{M} \int \vec r \ dm$

3. The attempt at a solution
Suppose that there are $n$ bodies of mass ${(M_i)}_{i = 1...n}$ with center of mass ${(\vec R_i)}_{i = 1...n}$ and volume ${(V_i)}_{i = 1...n}$ all disjoint.

By a change of variable : $M_i \vec R_i = \int_{V_i} \vec r \rho \ dV$

The total mass is $M = M_1 + ... + M_n$, and the total center of mass is

$\vec R = \frac{1}{M} \int \vec r \ dm = \frac{1}{M} \int_V \vec r\rho \ dV = \frac{1}{M} \sum_{i=1}^n \int_{V_i} \vec r \rho \ dV = \frac{1}{M} \sum_{i=1}^n M_i \vec R_i$

Which proves that the whole system can be treated as a particle system concentrated on its centers of mass.

Is that correct?

2. Dec 9, 2014

Correct.

3. Dec 9, 2014

geoffrey159

Thanks for looking at it :-)