Finding the Center of Mass of a System Using Particle Concentration

[geoffrey159]

Homework Statement

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.

Homework Equations

$\vec R = \frac{1}{M} \int \vec r \ dm$

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?

 

[mfb]

Correct.

 

[geoffrey159]

Thanks for looking at it :-)