A tuning fork is used to produce resonance in glass tuve. The length of the air column in the tube can be adjusted by a variable piston. At room temperature of `27^@C` two succesive resonance are produced at 20 cm and 73 cm column length. If the frequency of the tuning fork is 320 Hz. the velocity of sound is air at `27^(@)C` is

Options

A. `300 m//s`
B. `330m//s`
C. `350m//s`
D. `339 m//s`

Explanation

To find the velocity of sound in air at \(27^\circ C\) using the information given about the tuning fork and the resonance in the glass tube, we can follow these steps:

### Step 1: Understand the Resonance Condition
In a closed pipe (one end closed, one end open), the resonant wavelengths fit in such a way that the length of the air column corresponds to odd multiples of \(\frac{\lambda}{4}\). The first resonance occurs at \(L_1 = 20 \, \text{cm}\) and the second resonance occurs at \(L_2 = 73 \, \text{cm}\).

### Step 2: Calculate the Difference in Lengths
The difference in lengths of the air column for the two successive resonances is:
\[ \Delta L = L_2 - L_1 = 73 \, \text{cm} - 20 \, \text{cm} = 53 \, \text{cm} \]

### Step 3: Relate the Length Difference to Wavelength
For a closed pipe, the difference in length between two successive resonances corresponds to half the wavelength (\(\frac{\lambda}{2}\)):
\[ \Delta L = \frac{\lambda}{2} \]
Thus, we can write:
\[ \lambda = 2 \Delta L = 2 \times 53 \, \text{cm} = 106 \, \text{cm} = 1.06 \, \text{m} \]

### Step 4: Use the Frequency to Find the Velocity
The velocity of sound \(v\) can be calculated using the formula:
\[ v = f \cdot \lambda \]
where \(f\) is the frequency of the tuning fork. Given \(f = 320 \, \text{Hz}\), we have:
\[ v = 320 \, \text{Hz} \times 1.06 \, \text{m} = 339.2 \, \text{m/s} \]

### Final Answer
The velocity of sound in air at \(27^\circ C\) is approximately \(339.2 \, \text{m/s}\).

---