If a tunnig fork of frequency `(f_(0)) 340 Hz` and tolerance `pm 1%` is used in resonance coulmn method `[v=2f_(0)(l_(2)-l_(1))]`, the first and the second resonance are measured at `l_(1)=24.0 cm` and `l_(2)=74.0 cm`. Find max. permissible error in speed of sound.

Options

A. `1%`
B. `1.2 %`
C. `1.4%`
D. `1.6%`

Explanation

To find the maximum permissible error in the speed of sound using the resonance column method, we can follow these steps:

### Step 1: Write the formula for speed of sound
The speed of sound \( V \) is given by the formula:
\[ V = 2 f_0 (l_2 - l_1) \]
where \( f_0 = 340 \, \text{Hz} \), \( l_1 = 24.0 \, \text{cm} \), and \( l_2 = 74.0 \, \text{cm} \).

### Step 2: Calculate \( l_2 - l_1 \)
Calculate the difference \( l_2 - l_1 \):
\[ l_2 - l_1 = 74.0 \, \text{cm} - 24.0 \, \text{cm} = 50.0 \, \text{cm} \]

### Step 3: Substitute values into the speed of sound formula
Substituting the values into the formula for \( V \):
\[ V = 2 \times 340 \, \text{Hz} \times 50.0 \, \text{cm} \]
Convert \( 50.0 \, \text{cm} \) to meters:
\[ 50.0 \, \text{cm} = 0.50 \, \text{m} \]
Now substituting:
\[ V = 2 \times 340 \times 0.50 = 340 \, \text{m/s} \]

### Step 4: Determine the errors
The frequency \( f_0 \) has a tolerance of \( \pm 1\% \):
\[ \delta f_0 = 0.01 \times 340 = 3.4 \, \text{Hz} \]
For the lengths \( l_1 \) and \( l_2 \), since they are measured to one decimal place, the maximum permissible error in each is:
\[ \delta l_1 = \delta l_2 = 0.1 \, \text{cm} = 0.001 \, \text{m} \]

### Step 5: Calculate the total error in \( l_2 - l_1 \)
The total error in \( l_2 - l_1 \) is:
\[ \delta(l_2 - l_1) = \delta l_1 + \delta l_2 = 0.001 + 0.001 = 0.002 \, \text{m} \]

### Step 6: Use the formula for maximum permissible error in \( V \)
The formula for the maximum permissible error in \( V \) is given by:
\[ \frac{\delta V}{V} = \frac{\delta f_0}{f_0} + \frac{\delta(l_2 - l_1)}{(l_2 - l_1)} \]
Substituting the values:
\[ \frac{\delta V}{340} = \frac{3.4}{340} + \frac{0.002}{0.50} \]

### Step 7: Calculate each term
Calculating the first term:
\[ \frac{3.4}{340} = 0.01 \]
Calculating the second term:
\[ \frac{0.002}{0.50} = 0.004 \]

### Step 8: Add the errors
Now, adding these two contributions:
\[ \frac{\delta V}{340} = 0.01 + 0.004 = 0.014 \]

### Step 9: Calculate the maximum permissible error in percentage
To find the maximum permissible error in percentage:
\[ \delta V = 0.014 \times 340 = 4.76 \, \text{m/s} \]
Expressing this as a percentage:
\[ \text{Percentage error} = 0.014 \times 100 = 1.4\% \]

### Final Answer
The maximum permissible error in the speed of sound is \( 1.4\% \).

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