In a Michelson experiment intended for measuring the speed of...

The time taken by light to travel the given distance $D$ is

$t = \dfrac{D}{c} = \dfrac{4.8 \times 10^3}{3 \times 10^8} = 1.6 \times 10^{-5} \text{ s}$

For the image to be formed at the exact position as a stationary mirror, the regular octagonal mirror must rotate by an angle equal to the angle between its adjacent faces in time $t$.

The angle between adjacent faces of a regular octagon is $\dfrac{1}{8}$ of a revolution.

The minimum non-zero angular speed $f$ in revolutions per second is given by

$f = \dfrac{1/8}{t}$

$f = \dfrac{1}{8 \times 1.6 \times 10^{-5}}$

$f = \dfrac{10^5}{12.8} = 7812.5 \text{ rev/s}$