Answer
$R=\sqrt{\frac{2 \kappa}{k}}$
Work Step by Step
We know that
Angular frequency for vertical oscillation is $\omega_v=\sqrt{\frac{k}{m}}$
Angular frequency for torsional oscillation is $\omega_t=\sqrt{\frac{K}{I}}$
As we are given that $T_v=T_t$
so, $\omega_v=\omega_t$
$\implies\sqrt{\frac{k}{m}}=\sqrt{\frac{K}{I}} $
Squaring both sides, we obtain:
$\frac{k}{m}=\frac{K}{I}$
As $I=\frac{mR^2}{2}$
$\implies \frac{k}{m}=\frac{K}{\frac{mR^2}{2}}$
This simplifies to:
$R=\sqrt{\frac{2K}{k}}$
