Answer
$$
\tau=1.52 \mathrm{~s}
$$
Work Step by Step
$$
\begin{aligned}
& \bar{y}=\frac{1(8)(2)+2(8)(2)}{8(2)+8(2)}=1.5 \mathrm{ft} \\
& I_O=\frac{1}{32.2}\left[\frac{1}{12}(2)(8)(2)^2+2(8)(1)^2\right] \\
& +\frac{1}{32.2}\left[\frac{1}{12}(2)(8)(2)^2+2(8)(2)^2\right]=2.8157 \text { slug } \cdot \mathrm{ft}^2 \\
& h=\bar{y}(1-\cos \theta) \\
& T+V=\text { const } \\
& T=\frac{1}{2}(2.8157)(\dot{\theta})^2=1.4079 \dot{\theta}^2 \\
& V=8(4)(1.5)(1-\cos \theta)=48(1-\cos \theta) \\
& T+V=1.4079 \dot{\theta}^2+48(1-\cos \theta) \\
& 1.4079(2 \dot{\theta}) \ddot{\theta}+48(\sin \theta) \dot{\theta}=0
\end{aligned}
$$
For small $\theta, \sin \theta=\theta$, then
$$
\begin{aligned}
& \ddot{\theta}+17.047 \theta=0 \\
& \tau=\frac{2 \pi}{\omega_n}=\frac{2 \pi}{\sqrt{17.047}}=1.52 \mathrm{~s}
\end{aligned}
$$
