Suppose, there are two particles in a system. Their masses are m1 and m2 respectively and their position vectors are \(\vec{r}_{1}\) and \(\vec{r}_{2}\) respectively. If the position vector of the center of mass of this system be \(\vec{r}_{\mathrm{cm}}\), then.
Proof: Suppose, external forces acting on these particle are \(\vec{F}_{1 \mathrm{ext}}\) and \(\vec{F}_{2\mathrm{ext}}\) respectively and internal forces are \(\vec{F}_{12}\) and \(\vec{F}_{21}\) respectively. Therefore, net force acting on particle no.1
Suppose, at any moment the position vector, velocity vector and acceleration of center of mass are, \(\vec{r}_{\mathrm{cm}}, \vec{v}_{\mathrm{cm}} \text { and } \vec{a}_{\mathrm{cm}}\) respectively, then
Total mass of the system M = (m1 + m2). Since, according to concept of center of mass, all external forces act at the center of mass. Therefore, according to Newton’s second law of motion,
or \(\overrightarrow{r_{\mathrm{cm}}}=\frac{m_{1} \overrightarrow{r_{1}}+m_{2} \overrightarrow{r_{2}}}{m_{1}+m_{2}}\)