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(A) The distance of each particle of mass $m$ from the center of the square is $r = \frac{	ext{diagonal}}{2} = \frac{\ell\sqrt{2}}{2} = \frac{\ell}{\

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$\frac{\ell}{\sqrt{2}}$

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(A) The distance of each particle of mass $m$ from the center of the square is $r = \frac{	ext{diagonal}}{2} = \frac{\ell\sqrt{2}}{2} = \frac{\ell}{\sqrt{2}}$.The moment of inertia $I$ of the system about an axis passing through the center and perpendicular to the plane is given by $I = \sum mr^2 = 4 	imes m 	imes \left(\frac{\ell}{\sqrt{2}}ight)^2$.$I = 4 	imes m 	imes \frac{\ell^2}{2} = 2m\ell^2$.The radius of gyration $K$ is defined by $I = MK^2$, where $M$ is the total mass of the system $(M = 4m)$.$4mK^2 = 2m\ell^2$.$K^2 = \frac{2m\ell^2}{4m} = \frac{\ell^2}{2}$.$K = \frac{\ell}{\sqrt{2}}$.

Four particles of mass $m$ each are placed at the four corners of a square of side length $\ell$. The radius of gyration of the system about an axis passing through its center and perpendicular to the plane of the square is .........

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