Two loops ' \(A\) ' and ' \(B\) ' of radii ' \(R_1\) ' and ' \(R_2\) ' are made from uniform wire. If moment of inertia of ' \(A\) ' is ' \(I_A\) ' and that ' \(B\) ' is ' \(\mathrm{B}_{\mathrm{B}}\) ', then \(R_2 / R_1\) is \(\left[\frac{I_A}{I_B}=27\right]\)

\(\mathrm{I}_{\mathrm{A}}=\mathrm{M}_1 \mathrm{R}_1^2, \mathrm{I}_{\mathrm{B}}=\mathrm{M}_2 \mathrm{R}_2^2...(i)\)
If \(m\) is mass per unit length then \(\mathrm{M}_1=2 \pi \mathrm{R}_1 \mathrm{~m}\) and \(\mathrm{M}_2=2 \pi \mathrm{R}_2 \mathrm{~m}\)
\(\therefore \frac{\mathrm{M}_1}{\mathrm{M}_2}=\frac{\mathrm{R}_1}{\mathrm{R}_2}...(ii)\)
\(\therefore \frac{\mathrm{I}_{\mathrm{A}}}{\mathrm{I}_{\mathrm{B}}}=\frac{\mathrm{M}_1}{\mathrm{M}_2}\left(\frac{\mathrm{R}_1}{\mathrm{R}_2}\right)^2 \ldots\) [From(i)]
\(\frac{I_A}{I_B}==\frac{R_1}{R_2}\left(\frac{R_1}{R_2}\right)^2 \ldots\) [From(ii)]
\(\frac{I_A}{I_B}=\left(\frac{R_1}{R_2}\right)^3\)
\(\begin{array}{ll}\therefore \quad 27=\left(\frac{R_1}{R_2}\right)^3 \quad \ldots\left(\text { given, }, \frac{\mathrm{I}_A}{\mathrm{I}_{\mathrm{B}}}=27\right) \\ \therefore\frac{\mathrm{R}_2}{\mathrm{R}_1}=\frac{1}{3}\end{array}\)