To determine the minimum angular speed at which Michelson's eight-sided mirror would have to rotate to reflect light into an observer's eye by succeeding mirror faces, we need to understand the underlying physics principles. This involves the speed of light, the distance light travels, and the rotation of the mirror.

Here's a step-by-step explanation:

1. Understanding the Problem:

   • An eight-sided mirror (octagonal mirror) is used to reflect light.
   • The mirror needs to rotate such that the light reflects off one face and then the next, which means the mirror completes $\frac{1}{8}$ of a revolution during the time light travels to a distant object and back.

2. Relevant Information:

   • The speed of light $c$ is $3.00 \times 10^8$ m/s.
   • Let $d$ be the distance to the reflecting surface and back.

3. Calculation of Time:

   • The time $t$ for the light to travel to the distant object and back is: $t = \frac{2d}{c}$

4. Angular Speed:

   • The mirror needs to complete $\frac{1}{8}$ of a revolution in this time.
   • One full revolution is $2\pi$ radians.
   • $\frac{1}{8}$ of a revolution is $\frac{2\pi}{8} = \frac{\pi}{4}$ radians.
   • Let $\omega$ be the angular speed in radians per second. Then: $\omega t = \frac{\pi}{4}$
   • Substituting the expression for $t$: $\omega \left(\frac{2d}{c}\right) = \frac{\pi}{4}$
   • Solving for $\omega$: $\omega = \frac{\pi c}{8d}$

5. Units and Significant Figures:

   • The units for angular speed $\omega$ are radians per second.
   • To express the answer to two significant figures, we can simplify the formula based on the required parameters.

Therefore, the minimum angular speed is:

$\omega = \frac{\pi c}{8d}$

Given no specific distance $d$ is provided, the expression itself is the answer. However, assuming $d$ is a known quantity, you can plug in the values to get a numeric answer. For the general form:

$\omega = \frac{\pi \cdot (3.00 \times 10^8 \text{ m/s})}{8d}$

If $d$ is provided, you can substitute it into this formula to find the specific angular speed.

Final Answer:

Expressing this in general form, the minimum angular speed $\omega$ is:

$\boxed{\frac{\pi c}{8d}}$

where $c$ is the speed of light ($3.00 \times 10^8$ m/s) and $d$ is the distance to the reflecting surface and back.

Next Steps and Questions:

1. What specific distance $d$ should we consider for practical applications?
2. How does the number of mirror sides affect the minimum angular speed?
3. Can this principle be applied to mirrors with different shapes?
4. What are the typical distances used in Michelson's experiments?
5. How do changes in the speed of light medium affect this calculation?
6. What precision equipment is necessary to measure such high angular speeds?
7. How is the angular speed measured in experimental setups?
8. What other experiments rely on similar principles of mirror rotation?

Tip: Always ensure units are consistent when applying physical formulas to avoid calculation errors.