Question

Centre of mass of n-particles system

Answer 1

For a system of n-particles of masses m,m,m,....,m having the position vectors \(\vec r_1\), \(\vec r_2\),....,\(\vec r_n\) respectively with the respect to co-ordinate system, The position of the centre of mass is given by position vector

\(​​\vec{R}_{\text {com }}=\frac{m_{1} \vec{r}_{1}+m_{2} \vec{r}_{2}+\ldots m_{n} \vec{r}_{n}}{m_{1}+m_{2}+\ldots .+m_{n}}=\frac{\sum_{i=1}^{n} m_{i} \vec{r}_{i}}{M}\)

​​​​Where, M is total mass of the system.

The coordinates of the centre of mass is given by

\(x_{\text {com }}=\frac{\sum_{i=1}^{n} m_{i} x_{i}}{M}\)

\(y_{\text {com }}=\frac{\sum_{i=1}^{n} m_{i} y_{i}}{M}\) and

\(z_{\text {com }}=\frac{\sum_{i=1}^{n} m_{i} z_{i}}{M}\)