Physics - Rotation Question with Solution | TestHub

The moment of inertia of a solid sphere about its diameter is $I_d=\frac{2}{5}M{R}^2$.

The radius of gyration about the diameter is $K_d=\sqrt{\frac{2}{5}}R$.

Using the parallel axis theorem, the moment of inertia of the solid sphere about a tangent is $I_t=I_d+M{R}^2=\frac{2}{5}M{R}^2+M{R}^2=\frac{7}{5}M{R}^2$.

The radius of gyration about the tangent is $K_t=\sqrt{\frac{7}{5}}R$.

The ratio of $K_d$ to $K_t$ is $\frac{K_d}{K_t}=\frac{\sqrt{\frac{2}{5}}R}{\sqrt{\frac{7}{5}}R}={\left(\frac{2}{7}\right)}^{1/2}$.

Answer: ${\left(\frac{2}{7}\right)}^{1/2}$