The radius of gyration of a solid sphere of mass 5 kg about XY is 5 m as shown in figure. The radius of the sphere is , then the value of is [2024]

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Solution

Q.
The radius of gyration of a solid sphere of mass 5 kg about XY is 5 m as shown in figure. The radius of the sphere is $\frac{5 x}{\sqrt{7}} m$ , then the value of $x$ is [2024]

1 5

2 $\sqrt{2}$

3 $\sqrt{3}$

4 $\sqrt{5}$

Ans.
(4)

As we know that the moment of inertia of a solid sphere about XY is,

$I_{X Y} = \frac{7}{5} M R^{2}$ $Also,$ $I_{X Y} = M K^{2}$

$So, M K^{2} = \frac{7}{5} M R^{2}$

$\Rightarrow R^{2} = \frac{5}{7} K^{2}$

$or R = K \sqrt{\frac{5}{7}} or R = \frac{5 \sqrt{5}}{\sqrt{7}} (∵ K = 5 m)$

$On comparing with R = \frac{5 x}{\sqrt{7}}, the value of x is \sqrt{5} .$

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