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 solve the problem, we need to determine the permissible error in the speed of sound using the resonance column method. Let's break down the solution step by step.

### Step 1: Understanding the Formula
The speed of sound (v) can be calculated using the formula:
\[ v = 2 f_0 (L_2 - L_1) \]
where:
- \( f_0 = 340 \, \text{Hz} \) (frequency of the tuning fork)
- \( L_1 = 24.0 \, \text{cm} = 0.24 \, \text{m} \)
- \( L_2 = 74.70 \, \text{cm} = 0.747 \, \text{m} \)

### Step 2: Calculate \( L_2 - L_1 \)
First, we calculate the difference \( L_2 - L_1 \):
\[ L_2 - L_1 = 0.747 \, \text{m} - 0.24 \, \text{m} = 0.507 \, \text{m} \]

### Step 3: Calculate the Speed of Sound
Now we can calculate the speed of sound:
\[ v = 2 \times 340 \, \text{Hz} \times 0.507 \, \text{m} = 344.76 \, \text{m/s} \]

### Step 4: Determine the Errors
We need to consider the errors in frequency and length measurements:
- The tolerance in frequency \( \Delta f = 1\% \) of \( f_0 \):
\[ \Delta f = 0.01 \times 340 = 3.4 \, \text{Hz} \]

- The permissible error in lengths \( L_1 \) and \( L_2 \) is assumed to be \( \Delta L = 0.1 \, \text{cm} = 0.001 \, \text{m} \).

### Step 5: Calculate the Maximum Possible Error in Speed of Sound
The formula for the maximum percentage error in speed of sound is given by:
\[ \frac{\Delta v}{v} \times 100 = \frac{\Delta f}{f_0} \times 100 + \frac{\Delta L_1 + \Delta L_2}{L_2 - L_1} \times 100 \]

Substituting the values:
\[ \frac{\Delta v}{v} \times 100 = \frac{3.4}{340} \times 100 + \frac{0.001 + 0.001}{0.507} \times 100 \]

Calculating each term:
1. For frequency:
\[ \frac{3.4}{340} \times 100 = 1\% \]

2. For lengths:
\[ \frac{0.001 + 0.001}{0.507} \times 100 = \frac{0.002}{0.507} \times 100 \approx 0.394\% \]

### Step 6: Combine the Errors
Now, we combine the errors:
\[ \frac{\Delta v}{v} \times 100 = 1\% + 0.394\% \approx 1.394\% \]

### Step 7: Round the Result
Rounding this to one decimal place gives us approximately:
\[ \text{Permissible error in speed of sound} \approx 1.4\% \]

### Final Result
Thus, the permissible error in the speed of sound is approximately **1.4%**.

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