The ratio of the radius of gyration of a holl | vedclass

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(C) For a hollow sphere of mass $M$ and radius $R$,the moment of inertia about its diameter axis $AB$ is $I_{	ext{sphere}} = \frac{2}{3} MR^2$.Since

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$67$

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(C) For a hollow sphere of mass $M$ and radius $R$,the moment of inertia about its diameter axis $AB$ is $I_{	ext{sphere}} = \frac{2}{3} MR^2$.Since $I = Mk^2$,the radius of gyration $k_1$ is given by $k_1^2 = \frac{2}{3} R^2$.For a solid cylinder of mass $M$,radius $R_{cyl} = R$,and length $L = 4R$,the moment of inertia about the axis $AB$ passing through the edge is calculated using the parallel axis theorem.The moment of inertia about the central longitudinal axis is $I_{cm} = \frac{1}{2} MR^2$.The distance from the central axis to the edge axis $AB$ is $d = R$.Using the parallel axis theorem,$I_{	ext{cylinder}} = I_{cm} + Md^2 = \frac{1}{2} MR^2 + MR^2 = \frac{3}{2} MR^2$.Wait,re-evaluating the cylinder: The axis $AB$ is at the edge of the circular face. The moment of inertia of a solid cylinder about its central longitudinal axis is $I_{cm} = \frac{1}{2} MR^2$. The distance from the center of mass to the axis $AB$ is $R$. Thus,$I_{AB} = \frac{1}{2} MR^2 + MR^2 = \frac{3}{2} MR^2$.However,based on the provided solution logic: $I_{	ext{cylinder}} = \frac{1}{12} M(4R)^2 + \frac{1}{4} MR^2 + M(2R)^2 = \frac{16}{12} MR^2 + \frac{1}{4} MR^2 + 4MR^2 = (\frac{4}{3} + \frac{1}{4} + 4) MR^2 = (\frac{16+3+48}{12}) MR^2 = \frac{67}{12} MR^2$.This corresponds to a cylinder rotating about an axis perpendicular to its length at the end. $I = \frac{1}{12}ML^2 + \frac{1}{4}MR^2 + M(L/2)^2 = \frac{1}{12}M(4R)^2 + \frac{1}{4}MR^2 + M(2R)^2 = \frac{67}{12}MR^2$.Thus,$k_2^2 = \frac{67}{12} R^2$.The ratio $\frac{k_1}{k_2} = \sqrt{\frac{2/3}{67/12}} = \sqrt{\frac{2}{3} \cdot \frac{12}{67}} = \sqrt{\frac{8}{67}}$.Therefore,$x = 67$.

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 $\sqrt{\frac{8}{x}}$. The value of $x$ is:

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