A ring and a solid sphere rotating about an axis passing through their centers have the same radius of gyration. The axis of rotation is perpendicular to the plane of the ring. The ratio of radius of the ring to that of the sphere is The value of is ___

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Solution

Q.
A ring and a solid sphere rotating about an axis passing through their centers have the same radii of gyration. The axis of rotation is perpendicular to plane of the ring. The ratio of radius of the ring to that of the sphere is $\sqrt{\frac{2}{x}} .$ The value of $x$ is ________ . [2023]

Ans.
(5)

$For ring, I = m R_{1}^{2} = m K_{1}^{2}$

$∴$ $Radius of gyration K_{1} = R_{1}$

$For solid sphere, I' = \frac{2}{5} m' R_{2}^{2} = m' K_{2}^{2}$

$∴$ $Its radius of gyration K_{2} = \sqrt{\frac{2}{5}} R_{2}$

$∴$ $K_{1} = K_{2}$

$∴$ $R_{1} = \sqrt{\frac{2}{5}} R_{2}$

$∴$ $\frac{R_{1}}{R_{2}} = \sqrt{\frac{2}{5}}$

$∴$ $x = 5$

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