Answer
$\omega = 3800~rad/s$
Work Step by Step
We can find the time it takes the beam of light to travel a distance of $2L$:
$t = \frac{2L}{c}$
$t = \frac{(2)(500~m)}{3.0\times 10^8~m/s}$
$t = 3.33\times 10^{-6}~s$
In this time, the wheel would complete $\frac{1}{500}$ of a rotation. Thus, the wheel would make one complete rotation in a time of $500~t$
We can find the angular speed:
$\omega = \frac{2\pi~rad}{(500)(3.33\times 10^{-6}~s)}$
$\omega = 3800~rad/s$
