The ratio of the radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for the moment of inertia about their diameter axis AB as shown in the figure, is . The value of is

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Solution

Q.
Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for the moment of inertia about their diameter axis AB as shown in the figure, is $\sqrt{8 / x}$ . The value of $x$ is [2024].

1 17

2 34

3 67

4 51

Ans.
(3)

Hollow Sphere, $I_{sphere} = \frac{2}{3} M R^{2} = M k_{1}^{2}$

Solid Cylinder, $I_{cylinder} = \frac{1}{12} M (4 R^{2}) + \frac{1}{4} M R^{2} + M {(2 R)}^{2}$

$I_{cylinder} = \frac{67}{12} M R^{2} = M k_{2}^{2}$

$\frac{k_{1}}{k_{2}} = \sqrt{\frac{2}{3} \cdot \frac{12}{67}} = \sqrt{\frac{8}{67}} \Rightarrow x = 67$

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