Answer
$$
\omega=\sqrt{\omega_0^2+\frac{g}{r^2} s \sin \theta}
$$
Work Step by Step
$$
\begin{aligned}
& T_1+\Sigma U_{1-2}=T_2 \\
& \frac{1}{2}\left(m r^2+m r^2\right) \omega_0^2+m g(s \sin \theta)=\frac{1}{2}\left(m r^2+m r^2\right) \omega U^2 \\
& \omega=\sqrt{\omega_0^2+\frac{g}{r^2} s \sin \theta}
\end{aligned}
$$
