The sum of moments of all the particles in a system about its center of mass is always______

Question

Aicharya · Accepted Answer

Assume a system of three particles having mass $m_{1}, m_{2} and m_{3}$ located at a distance of $r_{1}, r_{2} and r_{3}$ from the centre of mass.

If centre of mass of the system is at origin, $\vec{r} = \frac{m_{1} \vec{r_{1}} + m_{2} \vec{r_{2}} + m_{3} \vec{r_{3}}}{m_{1} + m_{2} + m_{3}} = 0$ then, $m_{1} \vec{r_{1}} + m_{2} \vec{r_{2}} + m_{3} \vec{r_{3}} = 0$

Sum of moments of particles in a system about centre of mass is, $τ_{com} = m_{1} r_{1} (a) + m_{2} r_{2} (a) + m_{3} r_{3} (a) = a (m_{1} r_{1} + m_{2} r_{2} + m_{3} r_{3}) = 0$

Hence, sum of moments of particles of a system is always zero about centre of mass is zero.

The sum of moments of all the particles in a system about its center of mass is always______

Solution