Question

The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is

• A. $1:\sqrt{2}$
• B. $1:3$
• C. $2:1$
• D. $\sqrt{5}:\sqrt{6}$

Answer 1

The correct option is D $\sqrt{5}:\sqrt{6}$


Moment of inertia of a circular disc about the line $y_1$

$I_{y_1}=\frac{M{R}^2}{4}$
$\therefore I_{y_1}^{\prime }=\frac{M{R}^2}{4}+M{R}^2=\frac{5}{4}M{R}^2$
$($By parallel axis theorem$)$

$K_1=\sqrt{\frac{5}{4}}R$ (As $I=M{K}^2$)

Moment of inertia of a circular ring about the line $y_2$

$I_{y_2}=\frac{M{R}^2}{2}$

$\therefore I_{y_2}^{\prime }=\frac{M{R}^2}{2}+M{R}^2=\frac{3}{2}M{R}^2$
$K_2=\sqrt{\frac{3}{2}}R$

$I_{y_1}^{\prime }=M{K}_1^2,I_{y_2}^{\prime }=M{K}_2^2$

$\therefore \frac{K_1^2}{K_2^2}=\frac{I_{y_1}^{\prime }}{I_{y_2}^{\prime }}\Rightarrow K_1:K_2=\sqrt{5}:\sqrt{6}$

Hence option D is the correct answer