If a tuning fork of frequency (`f_(0)`) 340 Hz and tolerance `pm1%` is used in the resonance column method for determining the speed of sound. If the first and the second resonance are measured at `l_(1) = 24.0 cm and l_(2) = 74.70 cm`, then the permissible error in speed of sound is

Options

A. `1.2%`
B. `1.8%`
C. `1%`
D. `0.8%`

Answer

To determine the permissible error in the speed of sound using the resonance column method, we will follow these steps:

### Step 1: Calculate the difference in lengths
We need to find the difference between the second resonance length \( l_2 \) and the first resonance length \( l_1 \).

\[ l_2 - l_1 = 74.70 \, \text{cm} - 24.0 \, \text{cm} = 50.70 \, \text{cm} \]

### Step 2: Determine the errors in measurements
Next, we need to identify the errors in the measurements of \( l_1 \) and \( l_2 \).

- For \( l_1 = 24.0 \, \text{cm} \), the least count is \( 0.1 \, \text{cm} \), so the error \( \delta l_1 = \pm 0.1 \, \text{cm} \).
- For \( l_2 = 74.70 \, \text{cm} \), since it has two decimal places, the least count is \( 0.01 \, \text{cm} \), so the error \( \delta l_2 = \pm 0.01 \, \text{cm} \).

### Step 3: Calculate the total error in \( l_2 - l_1 \)
The total error in \( l_2 - l_1 \) is the sum of the individual errors:

\[ \delta (l_2 - l_1) = \delta l_1 + \delta l_2 = 0.1 \, \text{cm} + 0.01 \, \text{cm} = 0.11 \, \text{cm} \]

### Step 4: Calculate the speed of sound
The formula for the speed of sound \( v \) is given by:

\[ v = 2 f_0 (l_2 - l_1) \]

Substituting the values:

\[ v = 2 \times 340 \, \text{Hz} \times 50.70 \, \text{cm} = 2 \times 340 \times 0.5070 \, \text{m} = 344.58 \, \text{m/s} \]

### Step 5: Determine the percentage error in speed of sound
The percentage error in speed \( v \) can be calculated using the formula:

\[ \frac{\delta v}{v} = \frac{\delta f}{f} + \frac{\delta (l_2 - l_1)}{(l_2 - l_1)} \]

Where:
- \( \delta f = 1\% \) of \( f_0 \)
- \( \delta (l_2 - l_1) = 0.11 \, \text{cm} \)
- \( l_2 - l_1 = 50.70 \, \text{cm} \)

Calculating the percentage error for \( l_2 - l_1 \):

\[ \frac{\delta (l_2 - l_1)}{(l_2 - l_1)} = \frac{0.11 \, \text{cm}}{50.70 \, \text{cm}} \approx 0.00217 \text{ or } 0.217\% \]

### Step 6: Combine the percentage errors
Now we can find the total percentage error in the speed of sound:

\[ \delta v \% = 1\% + 0.217\% \approx 1.217\% \]

### Step 7: Final permissible error in speed of sound
Rounding this to two decimal places, we find:

\[ \delta v \% \approx 1.22\% \]

### Conclusion
The permissible error in the speed of sound is approximately **1.22%**.

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