Answer
$9.8\times10^6rad/s$ or $\approx$ 1.6 million rev/s
Work Step by Step
In the time $\Delta t$ it takes for the light to cross the room and come back, the eight-sided mirror should rotate 1/8 of a revolution. Then, the next mirror in the device is in the correct position to reflect the returning light. Find the angular speed.
Note that the time $\Delta t$ is the roundtrip distance (twice the room length of 12m) divided by the speed of light.
$$\omega=\frac{\Delta \theta}{\Delta t}=\frac{(2 \pi rad)/8}{(2 \Delta x)/c}=\frac{(\pi rad)c}{8 \Delta x}$$
$$\omega=\frac{(\pi rad)(3.00\times10^8m/s)}{8 (12m)}=9.8\times10^6rad/s$$
This is about 1.6 million revolutions per second.
