A tuning fork of frequency 480 Hz is used in an experiment of measuring speed of sound (v) in air by resonance tube method. Resonance is observed to occur at two successive lengths of the air column, `l_(1)=30cm` and `l_(2)=70cm`. Then, v is equal to:

Options

A. `379ms^(-1)`
B. `332 ms^(-1)`
C. `338 ms^(-1)`
D. `384 ms^(-1)`

Explanation

To find the speed of sound in air using the resonance tube method, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the lengths of the air column**:
We have two lengths of the air column where resonance occurs:
- \( l_1 = 30 \, \text{cm} \)
- \( l_2 = 70 \, \text{cm} \)

2. **Understand the resonance condition**:
In a resonance tube, the first resonance occurs at a length \( l_1 \) corresponding to \( \frac{\lambda}{4} \) (where \( \lambda \) is the wavelength) plus an end correction \( e \). The second resonance occurs at length \( l_2 \) corresponding to \( \frac{3\lambda}{4} \) plus the same end correction \( e \).

Thus, we can write:
\[ l_1 + e = \frac{\lambda}{4} \quad \text{(1)} \]
\[ l_2 + e = \frac{3\lambda}{4} \quad \text{(2)} \]

3. **Subtract the two equations**:
By subtracting equation (1) from equation (2), we eliminate \( e \):
\[ (l_2 + e) - (l_1 + e) = \frac{3\lambda}{4} - \frac{\lambda}{4} \]
This simplifies to:
\[ l_2 - l_1 = \frac{2\lambda}{4} = \frac{\lambda}{2} \]

4. **Calculate the wavelength \( \lambda \)**:
Now, substituting the values of \( l_1 \) and \( l_2 \):
\[ l_2 - l_1 = 70 \, \text{cm} - 30 \, \text{cm} = 40 \, \text{cm} \]
Therefore:
\[ \frac{\lambda}{2} = 40 \, \text{cm} \implies \lambda = 80 \, \text{cm} \]

5. **Use the frequency to find the speed of sound**:
The frequency \( f \) of the tuning fork is given as \( 480 \, \text{Hz} \). The speed of sound \( v \) is given by the formula:
\[ v = f \cdot \lambda \]
First, convert \( \lambda \) to meters:
\[ \lambda = 80 \, \text{cm} = 0.8 \, \text{m} \]
Now, substituting the values:
\[ v = 480 \, \text{Hz} \cdot 0.8 \, \text{m} = 384 \, \text{m/s} \]

### Final Answer:
The speed of sound \( v \) in air is \( 384 \, \text{m/s} \).

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