A tuning fork of frequency `480 Hz` is used in an experiment for 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) = 30 cm` and `l_(2) = 70 cm`. Then, `v` is equal to :

Options

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

Answer

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

### Step 1: Understand the resonance condition
In a closed organ pipe (like the resonance tube), the resonance occurs at odd multiples of \(\frac{\lambda}{4}\). The lengths of the air column at which resonance occurs can be expressed as:
\[ L_n = \frac{(2n - 1) \lambda}{4} \]
where \(L_n\) is the length of the air column for the \(n^{th}\) resonance.

### Step 2: Set up the equations for the two lengths
Given the two successive lengths of the air column where resonance occurs:
- \(L_1 = 30 \, \text{cm}\)
- \(L_2 = 70 \, \text{cm}\)

We can write the equations for these lengths:
1. For \(L_1\) (when \(n = n\)):
\[ L_1 = \frac{(2n - 1) \lambda}{4} = 30 \, \text{cm} \]
2. For \(L_2\) (when \(n = n + 1\)):
\[ L_2 = \frac{(2(n + 1) - 1) \lambda}{4} = 70 \, \text{cm} \]

### Step 3: Simplify the equations
From the first equation:
\[ (2n - 1) \lambda = 120 \, \text{cm} \quad \text{(1)} \]
From the second equation:
\[ (2n + 1) \lambda = 280 \, \text{cm} \quad \text{(2)} \]

### Step 4: Subtract the two equations
Now, we subtract equation (1) from equation (2):
\[ (2n + 1) \lambda - (2n - 1) \lambda = 280 \, \text{cm} - 120 \, \text{cm} \]
This simplifies to:
\[ 2\lambda = 160 \, \text{cm} \]
Thus, we find:
\[ \lambda = 80 \, \text{cm} \]

### Step 5: Calculate the speed of sound
The speed of sound \(v\) can be calculated using the formula:
\[ v = f \cdot \lambda \]
where \(f\) is the frequency of the tuning fork. Given \(f = 480 \, \text{Hz}\) and \(\lambda = 80 \, \text{cm} = 0.8 \, \text{m}\):
\[ v = 480 \, \text{Hz} \times 0.8 \, \text{m} = 384 \, \text{m/s} \]

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

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