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The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is -
A
$\sqrt{2} : 1$
B
$\sqrt{2} : \sqrt{3}$
C
$\sqrt{3} : \sqrt{2}$
D
$1 : \sqrt{2}$
Detailed Solution
Radius of gyration $K$ is defined by $I = MK^2$.
For the circular disc about its axis: $MK_1^2 = \dfrac{MR^2}{2}$
$K_1 = \dfrac{R}{\sqrt{2}}$
For the circular ring about its axis: $MK_2^2 = MR^2$
$K_2 = R$
$\dfrac{K_1}{K_2} = \dfrac{R/\sqrt{2}}{R} = \dfrac{1}{\sqrt{2}}$
So $K_1 : K_2 = 1 : \sqrt{2}$.
For the circular disc about its axis: $MK_1^2 = \dfrac{MR^2}{2}$
$K_1 = \dfrac{R}{\sqrt{2}}$
For the circular ring about its axis: $MK_2^2 = MR^2$
$K_2 = R$
$\dfrac{K_1}{K_2} = \dfrac{R/\sqrt{2}}{R} = \dfrac{1}{\sqrt{2}}$
So $K_1 : K_2 = 1 : \sqrt{2}$.