A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340Hz ) when the tube length is 25.5cm or 79.3cm. Estimate the speed of sound in air at the temperature of the experiment. The edge effects amy be neglected.

Answer

To estimate the speed of sound in air based on the given information, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Lengths of the Tube**:
- The tube has two lengths where resonance occurs:
- \( L_1 = 25.5 \, \text{cm} = 0.255 \, \text{m} \)
- \( L_2 = 79.3 \, \text{cm} = 0.793 \, \text{m} \)

2. **Use the Resonance Condition for Open Pipes**:
- For a pipe open at one end, the fundamental frequency \( f \) is given by:
\[ f = \frac{V}{4L} \]
- Where \( V \) is the speed of sound and \( L \) is the length of the tube.

3. **Set Up the Equations for Both Lengths**:
- For the first length \( L_1 \):
\[ f_n = \frac{V}{4L_1} \]
- For the second length \( L_2 \):
\[ f_{n+1} = \frac{V}{4L_2} \]

4. **Relate the Frequencies**:
- The frequencies for the nth and (n+1)th modes can be expressed as:
\[ f_n = (2n - 1) \cdot f \]
\[ f_{n+1} = (2n + 1) \cdot f \]
- Given that both frequencies are equal to 340 Hz:
\[ (2n - 1) \cdot 340 = (2n + 1) \cdot 340 \]

5. **Equate the Two Expressions**:
- Since both expressions equal the same frequency:
\[ (2n - 1) \cdot \frac{V}{4L_1} = (2n + 1) \cdot \frac{V}{4L_2} \]
- Cancel \( V \) and \( 4 \):
\[ \frac{2n - 1}{2n + 1} = \frac{L_1}{L_2} \]

6. **Substitute the Lengths**:
- Substitute \( L_1 = 0.255 \, \text{m} \) and \( L_2 = 0.793 \, \text{m} \):
\[ \frac{2n - 1}{2n + 1} = \frac{0.255}{0.793} \]
- Calculate the right side:
\[ \frac{0.255}{0.793} \approx 0.321 \]

7. **Cross-Multiply and Solve for n**:
- Cross-multiplying gives:
\[ (2n - 1) \cdot 793 = (2n + 1) \cdot 255 \]
- Expanding and simplifying:
\[ 1586n - 793 = 510n + 255 \]
\[ 1076n = 1048 \]
\[ n \approx 0.974 \quad \text{(round to 1)} \]

8. **Calculate the Speed of Sound**:
- Substitute \( n = 1 \) back into the equation for \( f_n \):
\[ 340 = \frac{V}{4 \cdot 0.255} \]
- Rearranging gives:
\[ V = 340 \cdot 4 \cdot 0.255 \]
- Calculate \( V \):
\[ V \approx 340 \cdot 1.02 \approx 346.8 \, \text{m/s} \]

### Final Answer
The estimated speed of sound in air at the temperature of the experiment is approximately **346.8 m/s**.