Two circular loops $\mathrm{P}$ and $\mathrm{Q}$ are made from a uniform wire. The radii of $P$ and $Q$ are…
Two circular loops $\mathrm{P}$ and $\mathrm{Q}$ are made from a uniform wire. The radii of $P$ and $Q$ are $R_1$ and $R_2$ respectively. The moments of inertia about their own axis are $\mathrm{I}_{\mathrm{P}}$ and $\mathrm{I}_{\mathrm{Q}}$ respectively. If $\frac{\mathrm{I}_{\mathrm{P}}}{\mathrm{I}_{\mathrm{Q}}}=\frac{1}{8}$ then $\frac{\mathrm{R}_2}{\mathrm{R}_1}$ is
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Solution
$\mathrm{I}_1=\mathrm{M}_1 \mathrm{R}_1^2, \mathrm{I}_2=\mathrm{M}_1 \mathrm{R}_2^2$
If $\mathrm{m}$ is mas per unit length then
$\begin{aligned}
& \mathrm{M}_1=2 \pi \mathrm{R}_1 \mathrm{~m} \text { and } \mathrm{M}_2=2 \pi \mathrm{R}_2 \mathrm{~m} \\
& \therefore \frac{\mathrm{M}_1}{\mathrm{M}_2}=\frac{\mathrm{R}_1}{\mathrm{R}_2} \\
& \frac{\mathrm{I}_1}{\mathrm{I}_2}=\frac{\mathrm{M}_1}{\mathrm{M}_2}\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^2=\frac{\mathrm{R}_1}{\mathrm{R}_2}\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^2=\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^3=\frac{1}{8} \\
& \therefore \frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{1}{2} \text { or } \frac{\mathrm{R}_2}{\mathrm{R}_1}=2
\end{aligned}$