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(B) Let the positions of the particles be represented by vectors $\vec{r}_i$ such that $|\vec{r}_i| = R$ for all $i = 1, 2, ..., n$.The position of th

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less than or equal to $R$

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(B) Let the positions of the particles be represented by vectors $\vec{r}_i$ such that $|\vec{r}_i| = R$ for all $i = 1, 2, ..., n$.The position of the center of mass $\vec{R}_{cm}$ is given by $\vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$.Using the triangle inequality for vectors, the magnitude of the center of mass is $|\vec{R}_{cm}| = \frac{|\sum m_i \vec{r}_i|}{\sum m_i} \le \frac{\sum m_i |\vec{r}_i|}{\sum m_i}$.Since $|\vec{r}_i| = R$ for all particles, we have $|\vec{R}_{cm}| \le \frac{\sum m_i R}{\sum m_i} = R$.Thus, the distance of the center of mass from the origin is always less than or equal to $R$.

$A$ large number of particles are placed around the origin, each at a distance $R$ from the origin. The distance of the center of mass of the system from the origin is

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