To calculate the speed of light using the data provided from Foucault's apparatus, we can follow these steps:
### Step-by-Step Solution:
1. **Identify the Given Values:**
- Distance between the rotating mirror and the fixed mirror, \( r = 16 \, \text{m} \)
- Distance between the lens and the rotating mirror, \( b = 6 \, \text{m} \)
- Distance between the source and the lens, \( a = 2 \, \text{m} \)
- Speed of the rotating mirror, \( \omega = 356 \, \text{revolutions per second} \)
- Image shift, \( s = 0.7 \, \text{mm} = 0.7 \times 10^{-3} \, \text{m} = 7 \times 10^{-4} \, \text{m} \)
2. **Convert the Angular Speed to Radians:**
\[
\omega = 356 \, \text{rev/s} \times 2\pi \, \text{rad/rev = 356 \times 2\pi \, \text{rad/s}}
\]
\[
\omega \approx 2235.5 \, \text{rad/s}
\]
3. **Use the Formula for Speed of Light:**
The formula derived from Foucault's experiment is:
\[
c = \frac{4 r^2 \omega a}{s (r + b)}
\]
4. **Substitute the Values into the Formula:**
- \( r = 16 \, \text{m} \)
- \( \omega \approx 2235.5 \, \text{rad/s} \)
- \( a = 2 \, \text{m} \)
- \( s = 7 \times 10^{-4} \, \text{m} \)
- \( r + b = 16 + 6 = 22 \, \text{m} \)
Now substituting these values:
\[
c = \frac{4 \times (16)^2 \times 2235.5 \times 2}{7 \times 10^{-4} \times 22}
\]
5. **Calculate Each Component:**
- Calculate \( 4 \times (16)^2 = 4 \times 256 = 1024 \)
- Calculate \( 1024 \times 2235.5 \times 2 = 1024 \times 4471 = 4,577,344 \)
- Calculate \( 7 \times 10^{-4} \times 22 = 0.0007 \times 22 = 0.0154 \)
6. **Final Calculation:**
\[
c = \frac{4,577,344}{0.0154} \approx 297,000,000 \, \text{m/s}
\]
7. **Conclusion:**
The calculated speed of light is approximately:
\[
c \approx 2.97 \times 10^8 \, \text{m/s}
\]
