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

Assume a system of three particles having mass $m_1, m_2\text{ and} m_3$located at a distance of $r_1, r_2 \text{and} r_3$from the centre of mass.

If centre of mass of the system is at origin, $\overrightarrow{r}=\frac{m_1\overrightarrow{r_1}+m_2\overrightarrow{r_2}+m_3\overrightarrow{r_3}}{m_1+m_2+m_3}=0$then, $m_1\overrightarrow{r_1}+m_2\overrightarrow{r_2}+m_3\overrightarrow{r_3}=0$

Sum of moments of particles in a system about centre of mass is, $$\begin{gathered} {\tau }_{\mathrm{com}}=m_1{r}_1\left(a\right)+m_2{r}_2\left(a\right)+m_3{r}_3\left(a\right) \\ =a\left(m_1{r}_1+m_2{r}_2+m_3{r}_3\right) \\ =0 \end{gathered}$$

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