A solid sphere and a solid cylinder of the same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radii of gyration respectively is then the value of is _________ . [2023]

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Solution

Q.
A solid sphere and a solid cylinder of the same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyration respectively $(k_{sph} : k_{cyl})$ is $2 : \sqrt{x},$ then the value of $x$ is _________ . [2023]

Ans.
(5)

$For solid sphere, \frac{2}{5} m R^{2} = m k_{sph}^{2}$

$k_{sph} = \sqrt{\frac{2}{5}} R$

$For solid cylinder, \frac{m R^{2}}{2} = m k_{cyl}^{2}$

$\Rightarrow k_{cyl} = \frac{R}{\sqrt{2}}$

$\frac{k_{sph}}{k_{cyl}} = \frac{\sqrt{\frac{2}{5}}}{\frac{1}{\sqrt{2}}} = \frac{2}{\sqrt{5}} = \frac{2}{\sqrt{x}}$

$∴$ $x = 5$

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