## Essential University Physics: Volume 1 (3rd Edition)

$R=\sqrt{\frac{2 \kappa}{k}}$
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: $R=\sqrt{\frac{2K}{k}}$