Question

Ratio of radii of gyration of a hollow & solid sphere of the same radii about the axis which is tangent to the sphere is

• A. $\sqrt{\frac{7}{3}}$
• B. $\frac{5}{\sqrt{21}}$
• C. $\sqrt{\frac{4}{5}}$
• D. $\frac{25}{9}$

Answer 1

Answer: (B)

$\frac{5}{\sqrt{21}}$

$\begin{array}{c}\text{We know that moment of Inertia of a Solid}\\ \text{sphere about its axis is -}\\ I_{\text{solid sphere}}=\frac{2}{5}{MR}^2\\ \text{Theretore moment of Inertia about an axis}\\ \text{Tangent to the sphere-}\\ I^{\prime }=\frac{2}{5}M{R}^2+M{R}^2=\frac{7}{5}M{R}^2\\ {\text{I}}_{\text{Hollow sphere}}=\frac{2}{3}M{R}^2\end{array}$

$\begin{array}{c}\text{Therefore Moment of Inertia about an axis tangent}\\ \text{to the sphere -}\\ I^{\prime }=\frac{2}{3}{MR}^2+{MR}^2=\frac{5}{3}{MR}^2\\ \text{Radius of gyration of solid sphere be}{K}_5\\ \Rightarrow M\cdot K_3^2=\frac{7}{5}M{R}^2\\ \Rightarrow k_5=\sqrt{\frac{7}{5}}R\end{array}$$\begin{array}{c}\text{Radius of gyration of Hollow sphere be}{k}_H\\ \Rightarrow M{K}_H^2=\frac{5}{3}M{R}^2\\ \Rightarrow K_H=\sqrt{\frac{5}{3}}R\\ \Rightarrow \frac{k_H}{K_S}=\frac{5}{\sqrt{21}}\\ \Rightarrow \text{Hence, option (B) is correct.}\end{array}$