Answer
$$ \omega=5.92 \mathrm{rad} / \mathrm{s}
$$
Work Step by Step
$$
\begin{aligned}
& T_1+\Sigma U_{1-2}=T_2 \\
& 2\left[\frac{1}{2}\left\{\frac{1}{3}\left(\frac{8}{32.2}\right)(2)^2\right\}(2)^2\right]+\frac{1}{2}\left(\frac{10}{32.2}\right)(4)^2 \\
& +\left[20\left(2 \sin 45^{\circ}\right)+15\left(\frac{\pi}{4}\right)-2(8)\left(1-\cos 45^{\circ}\right)-10\left(2-2 \cos 45^{\circ}\right)\right] \\\\
& =2\left[\frac{1}{2}\left\{\frac{1}{3}\left(\frac{8}{32.2}\right)(2)^2\right\} \omega^2\right]+\frac{1}{2}\left(\frac{10}{32.2}\right)(2 \omega)^2 \\
& \omega=5.92 \mathrm{rad} / \mathrm{s}
\end{aligned}
$$
