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

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answer is 5.

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Detailed Solution

Moment of inertia for ring $I = m R_{1}^{2} = m R_{1}^{2}$
Radius of gyration $K_{1} = R_{1}$
Moment of inertia for solid sphere
$I' = \frac{2}{5} m' R_{2}^{2}$
Radius of gyration $K_{2} = \sqrt{\frac{I'}{m'}} = \sqrt{\frac{2}{5}} R_{2}$
$∴ K_{1} = K_{2} (G i v e n) R_{1} = \sqrt{\frac{2}{5}} R_{2} ∴ \frac{R_{1}}{R_{2}} = \sqrt{\frac{2}{5}} ∴ x = 5$

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