# [SOLVED]center of mass of a two particle system

#### dwsmith

##### Well-known member
How does one prove the center of mass of a two particle system lies on the line joining them?

Would we do this by contradiction?
Suppose on the contrary that the CM doesn't lie on the line joining the two particles. Where do I go from here though?

#### Ackbach

##### Indicium Physicus
Staff member
Definition of CM of two particles of mass $m_{1}$ and $m_{2}$:
$$\mathbf{r}_{ \text{cm}}= \frac{m_{1} \mathbf{r}_{1}+m_{2} \mathbf{r}_{2}}{m_{1}+m_{2}}.$$
We can view the line segment from $\mathbf{r}_{1}$ to $\mathbf{r}_{2}$ as follows:
$$\{\mathbf{r}| \exists\,t\in[0,1] \; \text{s.t.} \; \mathbf{r}=t \mathbf{r}_{1}+(1-t)\mathbf{r}_{2} \}.$$
You can see that $t=0$ means $\mathbf{r}=\mathbf{r}_{2}$ and $t=1$ corresponds to $\mathbf{r}=\mathbf{r}_{1}$. As $t$ varies in the interval $[0,1]$, the vector $\mathbf{r}$ sweeps out the line segment from $\mathbf{r}_{2}$ to $\mathbf{r}_{1}$. Now compare this expression to the expression for the center of mass.