The design of the rotating arm $O A$ of a control mechanism requires that it rotate about the vertical $Z$ -axis at the constant rate $\Omega=\dot{\beta}=\pi \mathrm{rad} / \mathrm{s}$. Simultaneously, $O A$ oscillates according to $\theta=\theta_{0} \sin 4 \Omega t$ where $\theta_{0}=\pi / 6$ radians and $t$ is in seconds measured from the time when $\beta=0 .$ Determine the angular velocity $\omega$ and the angular acceleration $\alpha$ of $O A$ for the instant $(a)$ when $t=1 / 2$ s and $(b)$ when $t=1 / 8$ s. The $x-y$ reference axes rotate in the $X$ -Y plane with the angular velocity $\Omega$