Physics - Rotation Question with Solution | TestHub

Moment of inertia of a thin uniform rod about an axis passing through its centre and perpendicular to its length is $I_1=\frac{M{L}^2}{12}$.

Radius of gyration in the first case is $K_1=\sqrt{\frac{I_1}{M}}=\frac{L}{\sqrt{12}}$.

Moment of inertia of the rod about an axis passing through one end and perpendicular to its length is $I_2=\frac{M{L}^2}{3}$.

Radius of gyration in the second case is $K_2=\sqrt{\frac{I_2}{M}}=\frac{L}{\sqrt{3}}$.

The ratio of radius of gyration in the first case to the second case is $\frac{K_1}{K_2}=\frac{\frac{L}{\sqrt{12}}}{\frac{L}{\sqrt{3}}}=\sqrt{\frac{3}{12}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$.