In an experiment to measure the speed of light by Fizeau's apparatus, following data are used :
Distance between the mirrors = 12.0 km,
Number of teeth in the wheel = 180.
Find the minimum angular speed of the wheel for which the image is not seen.

Explanation

To solve the problem of finding the minimum angular speed of the wheel in Fizeau's apparatus, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Data:**
- Distance between the mirrors, \( D = 12 \, \text{km} = 12 \times 10^3 \, \text{m} \)
- Number of teeth in the wheel, \( N = 180 \)
- Speed of light, \( c = 3 \times 10^8 \, \text{m/s} \)

2. **Use the Formula for Speed of Light in Fizeau's Experiment:**
The formula relating the speed of light \( c \), the distance \( D \), the number of teeth \( N \), and the angular speed \( \omega \) is given by:
\[ c = \frac{2 D N \omega}{\pi} \]

3. **Rearranging the Formula to Solve for Angular Speed \( \omega \):**
We can rearrange the formula to find \( \omega \):
\[ \omega = \frac{\pi c}{2 D N} \]

4. **Substituting the Values:**
Now, substitute the values into the equation:
\[ \omega = \frac{\pi \times (3 \times 10^8)}{2 \times (12 \times 10^3) \times 180} \]

5. **Calculating the Denominator:**
Calculate \( 2 \times 12 \times 10^3 \times 180 \):
\[ 2 \times 12 = 24 \]
\[ 24 \times 180 = 4320 \]
\[ 4320 \times 10^3 = 4.32 \times 10^6 \]

6. **Calculating \( \omega \):**
Now, calculate \( \omega \):
\[ \omega = \frac{\pi \times 3 \times 10^8}{4.32 \times 10^6} \]
Using \( \pi \approx 3.14 \):
\[ \omega = \frac{3.14 \times 3 \times 10^8}{4.32 \times 10^6} \approx \frac{9.42 \times 10^8}{4.32 \times 10^6} \]
\[ \omega \approx 21875 \, \text{rad/s} \]

7. **Convert to Degrees per Second:**
To convert from radians per second to degrees per second, use the conversion factor \( \frac{180}{\pi} \):
\[ \omega \approx 21875 \times \frac{180}{3.14} \approx 1250000 \, \text{degrees/s} \]

8. **Final Result:**
The minimum angular speed of the wheel for which the image is not seen is approximately:
\[ \omega \approx 1.25 \times 10^4 \, \text{degrees/s} \]