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- Thread starter dwsmith
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- Jan 26, 2012

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$$ \mathbf{r}_{ \text{cm}}= \frac{m_{1} \mathbf{r}_{1}+m_{2} \mathbf{r}_{2}}{m_{1}+m_{2}}.$$

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.