Obtain the general expression of centre of mass for distributed $n$ particles of system in three dimension. 

Suppose the coordinates of $n$ particles of masses $m_{1}, m_{2}, \ldots, m_{n}$ are $\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right), \ldots$ $\left(x_{n}, y_{n}, z_{n}\right)$ respectively.

$\therefore$ The position of centre of mass,

$\begin{aligned}(x, y, z) &=\frac{m_{1}\left(x_{1}, y_{1}, z_{1}\right)+m_{2}\left(x_{2}, y_{2}, z_{2}\right)+\ldots+m_{n}\left(x_{n}, y_{n}, z_{n}\right)}{m_{1}+m_{2}+\ldots+m_{n}} \\ &=\frac{\sum m_{i}\left(x_{i}, y_{i}, z_{i}\right)}{\Sigma m_{i}} \text { where } i=1,2,3, \ldots, n \\ \text { OR } \end{aligned}$

X-coordinate, Y-coordinate and Z-coordinate of centre of mass of a system,

$\mathrm{X}=\frac{m_{1} x_{1}+m_{2} x_{2}+\ldots+m_{n} x_{n}}{m_{1}+m_{2}+\ldots+m_{n}}=\frac{\Sigma m_{i} x_{i}}{\mathrm{M}}$

$\mathrm{Y}=\frac{m_{1} y_{1}+m_{2} y_{2}+\ldots+m_{n} y_{n}}{m_{1}+m_{2}+\ldots+m_{n}}=\frac{\Sigma m_{i} y_{i}}{\mathrm{M}}$

$\mathrm{Z}=\frac{m_{1} z_{1}+m_{2} z_{2}+\ldots+m_{n} z_{n}}{m_{1}+m_{2}+\ldots+m_{n}}=\frac{\Sigma m_{i} z_{i}}{\mathrm{M}}$

where $i=1,2,3, \ldots, n$ and

$m_{1}+m_{2}+\ldots+m_{n}=\Sigma m_{i}=\mathrm{M}$ is the total mass of system.

If we want to find the position vector of centre of mass in $i$ then position vector of $\vec{r}=x_{i} \hat{i}+y_{i} \hat{j}+z_{i} \hat{k}$ and the position vector of centre of mass

$ \overrightarrow{\mathrm{R}}=(\mathrm{X} \hat{i}+\mathrm{Y} \hat{j}+\mathrm{Z} \hat{k})$

$\therefore  \overrightarrow{\mathrm{R}}=\frac{\Sigma m_{i} \overrightarrow{r_{i}}}{\mathrm{M}}$

If the origin of frame of reference (coordinate system) taken as centre of mass of a system then for the system of particles $\Sigma m_{i} \vec{r}_{i}=0$