Several parts of a system and its CM

[brotherbobby]

Homework Statement

Prove that the center of mass (CM) of a system composed of several parts can be determined by assuming that all the parts are particles located at their (respective) center of mass.

Relevant Equations

The CM of a system of particles each having masses $m_i$ and position vectors $\mathbf{r_i}$ is given by : $\mathbf{r_{\text{CM}}} = \frac{\Sigma m_i \mathbf{r_i}}{\Sigma m_i}$

I have known and used this theorem for a long time solving problems ("Calculate the CM of the some given shape"). I took the theorem to be "obvious" and didn't know it could be proved (and that indeed it was a theorem at all).

I can make no attempt at the proof. Any help would be welcome.

 

[Delta2]

You can prove it easily by using the additive property of volume integrals. More specifically if we have a volume $V$, that we can break to sub volumes $V_1,V_2,...,V_n$ such that the volumes $V_i$ do not overlap and such that $V=V_1+V_2+...V_n$ then the following holds:
$$\int_V \vec{f}(\vec{r})d^3\vec{r}=\int_{V_1}\vec{f}(\vec{r})d^3\vec{r}+\int_{V_2}\vec{f}(\vec{r})d^3\vec{r}+...+\int_{V_n}\vec{f}(\vec{r})d^3\vec{r}$$
The volume $V$ is the volume of the big system and the volumes $V_i$ are the volumes of the system's parts.
If $V>V_1+V_2+...+V_n$ then we also need that $\vec{f}(\vec{r})=0$ for $\vec{r}\in V-(V_1+V_2+...+V_n)$

 

[DEvens]

Try it explicitly for the simple case of two masses in one group and one mass in the other group. So you have $D_1 = \frac{m_1R_1 + m_2R_2}{m_1+m_2}$ and then you bring in $m_3$ located at $R_3$. Show that it does not matter which order you do it, either all three as one big system, or first the first two then the third. After that it's a question of how you proceed to a general proof.