Question

All the particles of a body are situated at a distance R from the origin. The distance of the centre of mass of the body from the origin is
(a) = R
(b) ≤ R
(c) > R
(d) ≥ R

Answer 1

(b) ≤ R

Distance of the centre of mass from the origin is given by,
$R\text{'}=\frac{1}{M}{\sum }_{i=1}^n{m}_i{\overrightarrow{r}}_i$

Let half of the particles lie on +Y-axis and the rest of the particles lie on +X-axis.
$$\begin{gathered} Y_{\mathrm{CM}}=\frac{m_{3}R{\displaystyle \overrightarrow{j}}+m_{4}R{\displaystyle \overrightarrow{j}}}{m_1+m_2+m_3+m_4+...} \\ =\frac{R{\displaystyle \overrightarrow{j}}}{2} \end{gathered}$$

Similarly, $X_{\mathrm{CM}}=\frac{R\overrightarrow{i}}{2}$

Therefore, the coordinates of centre of mass are $\left(\frac{R}{2},\frac{R}{2}\right)$.

Distance of the centre of mass from the origin$=\frac{R}{\sqrt{2}}$
For the given situation, $R\text{'}<R$.

In general, R' ≤ R.