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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:
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Verified Answer
The correct answer is:
$\sqrt{5}: \sqrt{6}$
The radius of gyration of disc about a tangential axis in the plane of disc is $k_1$
$$
\therefore \quad k_1=\frac{\sqrt{5}}{2} R
$$
And radius of gyration of circular ring of same radius about tangential axis is given by:
$$
\begin{aligned}
k_2 & =\frac{\sqrt{3}}{2} \mathrm{R} \\
\therefore \quad \frac{k_1}{k_2} & =\frac{\sqrt{5}}{2} R \times \frac{\sqrt{2}}{\sqrt{3} R} \\
& =\frac{\sqrt{5}}{\sqrt{6}}
\end{aligned}
$$
$$
\therefore \quad k_1=\frac{\sqrt{5}}{2} R
$$
And radius of gyration of circular ring of same radius about tangential axis is given by:
$$
\begin{aligned}
k_2 & =\frac{\sqrt{3}}{2} \mathrm{R} \\
\therefore \quad \frac{k_1}{k_2} & =\frac{\sqrt{5}}{2} R \times \frac{\sqrt{2}}{\sqrt{3} R} \\
& =\frac{\sqrt{5}}{\sqrt{6}}
\end{aligned}
$$
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