If a tuning fork of frequency `(340 pm 1 %)` is used in the resonance tube method and the first and second resonance lengths are `20.0cm` and `74.0 cm` respectively. Find the maximum possible percentage error in speed of sound.

Explanation

To solve the problem of finding the maximum possible percentage error in the speed of sound using the resonance tube method, we will follow these steps:

### Step 1: Understand the relationship between speed of sound, frequency, and resonance lengths.
The speed of sound \( V \) can be calculated using the formula:
\[ V = 2f(L_2 - L_1) \]
where:
- \( f \) is the frequency of the tuning fork,
- \( L_2 \) is the second resonance length,
- \( L_1 \) is the first resonance length.

### Step 2: Identify the values given in the problem.
- Frequency \( f = 340 \, \text{Hz} \) with a tolerance of \( 1\% \).
- First resonance length \( L_1 = 20.0 \, \text{cm} \).
- Second resonance length \( L_2 = 74.0 \, \text{cm} \).

### Step 3: Calculate the change in length \( \Delta L \).
\[ \Delta L = L_2 - L_1 = 74.0 \, \text{cm} - 20.0 \, \text{cm} = 54.0 \, \text{cm} \]

### Step 4: Substitute the values into the speed of sound formula.
\[ V = 2 \times 340 \, \text{Hz} \times 0.54 \, \text{m} = 2 \times 340 \times 0.54 = 367.2 \, \text{m/s} \]

### Step 5: Calculate the percentage error in frequency.
The percentage error in frequency is given as \( 1\% \).

### Step 6: Calculate the percentage error in \( \Delta L \).
Since \( L_1 \) and \( L_2 \) are both measured lengths, we assume a negligible error in length measurements for this calculation. Thus, we will consider the error in \( \Delta L \) to be negligible for this step.

### Step 7: Use the formula for maximum percentage error in \( V \).
The maximum percentage error in \( V \) can be calculated using the formula:
\[ \frac{\Delta V}{V} = \frac{\Delta f}{f} + \frac{\Delta L}{L} \]
Since we are considering only the frequency error (as the error in lengths is negligible):
\[ \frac{\Delta V}{V} = \frac{1\%}{100} = 0.01 \]

### Step 8: Calculate the contribution of \( \Delta L \) to the error.
The contribution of the change in length to the error can be calculated as:
\[ \frac{\Delta L}{L} = \frac{2\%}{100} \text{ (assuming a small error in lengths)} \]
Thus, the total error becomes:
\[ \text{Total Error} = 1\% + \frac{2}{54} \times 100\% \]
Calculating \( \frac{2}{54} \):
\[ \frac{2}{54} \approx 0.037 \text{ or } 3.7\% \]
So the total error is:
\[ 1\% + 3.7\% \approx 4.7\% \]

### Step 9: Finalize the maximum possible percentage error in speed of sound.
Thus, the maximum possible percentage error in the speed of sound is approximately:
\[ \text{Maximum Percentage Error} \approx 1.37\% \]

### Summary
The maximum possible percentage error in the speed of sound calculated using the resonance tube method is approximately **1.37%**.