Answer
$34.19\;rad/s$
Work Step by Step
The angular displacement of the wheel can be written as
$\theta(t)=\theta_0\sin(\frac{2\pi t}{T})$
Therefore, the angular velocity of the wheel is
$\Omega=\frac{d\theta(t)}{dt}$
or, $\Omega=\theta_0\frac{2\pi}{T}\cos(\frac{2\pi t}{T})$
Now, $\theta_0=\pi$ and $\theta=\frac{\pi}{2}$
Therefore, $\sin(\frac{2\pi t}{T})=\frac{\theta}{\theta_0}=\frac{\pi}{2\pi}=\frac{1}{2}$
$\therefore\; \cos(\frac{2\pi t}{T})=\sqrt {1- \sin^2(\frac{2\pi t}{T})}=\sqrt {1-\frac{1}{4}}=\frac{\sqrt 3}{2}$
Therefore,
$\Omega=\theta_0\frac{2\pi}{T}\cos(\frac{2\pi t}{T})$
or, $\Omega=\pi\times\frac{2\pi}{0.500}\times\frac{\sqrt 3}{2}\;rad/s$
or, $\Omega=34.19\;rad/s$
Therefore, the angular speed at displacement $\frac{\pi}{2}\;rad$ is $34.19\;rad/s$
