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

\(\mathrm{I}_1=\mathrm{M}_1 \mathrm{R}_1^2, \mathrm{I}_2=\mathrm{M}_1 \mathrm{R}_2^2\)
If \(\mathrm{m}\) is mas per unit length then
\( \begin{aligned} & \mathrm{M}_1=2 \pi \mathrm{R}_1 \mathrm{~m} \text { and } \mathrm{M}_2=2 \pi \mathrm{R}_2 \mathrm{~m} \\ & \therefore \frac{\mathrm{M}_1}{\mathrm{M}_2}=\frac{\mathrm{R}_1}{\mathrm{R}_2} \\ & \frac{\mathrm{I}_1}{\mathrm{I}_2}=\frac{\mathrm{M}_1}{\mathrm{M}_2}\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^2=\frac{\mathrm{R}_1}{\mathrm{R}_2}\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^2=\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^3=\frac{1}{8} \\ & \therefore \frac{\mathrm{R}_1}{\mathrm{R}_2}=\frac{1}{2} \text { or } \frac{\mathrm{R}_2}{\mathrm{R}_1}=2 \end{aligned} \)