Moment of inertia in case \((ii)\) is \(I_{2}\)

\(I_{1}=2 M R^{2}\)

\(I _{2}=\frac{3}{2} MR ^{2}\)

\(T _{1}=2 \pi \sqrt{\frac{ I _{1}}{ Mgd }} ; T _{2}=2 \pi \sqrt{\frac{ I _{2}}{ Mgd }}\)

\(\frac{ T _{1}}{ T _{2}}=\sqrt{\frac{ I _{1}}{ I _{2}}}=\sqrt{\frac{2 MR ^{2}}{\frac{3}{2} MR ^{2}}}=\frac{2}{\sqrt{3}}\)