Question

The ratio of radius of gyration of a solid sphere and hollow sphere of the same mass and radius about a tangential axis is

• A. $\sqrt{7}:\sqrt{3}$
• B. $5:\sqrt{21}$
• C. $\sqrt{5}:\sqrt{6}$
• D. $\sqrt{21}:5$

Answer 1

The correct option is D $\sqrt{21}:5$

Let the mass and radius of both sphere are $M$ and $R$ respectively.
Moment of inertia of solid sphere about its COM, $I_{com}=\frac{2}{5}M{R}^2$
Moment of inertia of solid sphere about tangential axis, applying parallel axis theorem,
$I_s=I_{com}+M{R}^2=\frac{2}{5}M{R}^2+M{R}^2{I}_s=\frac{7}{5}M{R}^2$
Radius of gyration of solid sphere about its tangential axis is given by $K_s$
$K_s=\sqrt{\frac{I_s}{M}}=\sqrt{\frac{7M{R}^2}{5M}}=\sqrt{\frac{7}{5}}R{K}_s=\sqrt{\frac{7}{5}}R$

Moment of inertia of hollow sphere about its COM, $I_{com}=\frac{2}{3}M{R}^2$
Moment of inertra of hollow sphere about its tangential axis, applying parallel axis theorem,
$I_h=I_{com}+M{R}^2=\frac{2}{3}M{R}^2+M{R}^2{I}_h=\frac{5}{3}M{R}^2$
Now, radius of gyration of hollow sphere about its tangential axis is given by $K_h$
$K_h=\sqrt{\frac{I_h}{M}}=\sqrt{\frac{5M{R}^2}{3M}}=\sqrt{\frac{5}{3}}R{K}_h=\sqrt{\frac{5}{3}}R$
Dividing $K_s$ by $K_h$,
$\frac{K_s}{K_h}=\frac{\sqrt{\frac{7}{5}}R}{\sqrt{\frac{5}{3}}R}=\frac{\sqrt{21}}{5}$