In a resonance tube experiment to determine the speed of sound in air, a pipe of diameter `5 cm` is used . The column in pipe resonates with a tuning fork of frequency `480 H_(Z)` when the minimum length of the air column is `16 cm` . Find the speed in air column at room temperature.

Explanation

To solve the problem of determining the speed of sound in air using a resonance tube experiment, we will follow these steps:

### Step 1: Identify the Given Values
- Diameter of the pipe (d) = 5 cm = 0.05 m
- Frequency of the tuning fork (f) = 480 Hz
- Minimum length of the air column (L1) = 16 cm = 0.16 m

### Step 2: Calculate the Radius of the Pipe
The radius (R) is half of the diameter:
\[ R = \frac{d}{2} = \frac{0.05 \, \text{m}}{2} = 0.025 \, \text{m} \]

### Step 3: Calculate the End Correction
The end correction (e) can be calculated using the formula:
\[ e = 0.6 \times R \]
Substituting the value of R:
\[ e = 0.6 \times 0.025 \, \text{m} = 0.015 \, \text{m} \]

### Step 4: Calculate the Effective Length of the Air Column (L2)
The effective length (L2) can be calculated using the relationship:
\[ e = L2 - 3L1 / 2 \]
Rearranging this gives:
\[ L2 = e + \frac{3L1}{2} \]
Substituting the values:
\[ L2 = 0.015 + \frac{3 \times 0.16}{2} \]
Calculating:
\[ L2 = 0.015 + 0.24 = 0.255 \, \text{m} \]

### Step 5: Calculate the Speed of Sound in Air
Using the formula for the speed of sound:
\[ V = 2f(L2 - L1) \]
Substituting the known values:
\[ V = 2 \times 480 \, \text{Hz} \times (0.255 \, \text{m} - 0.16 \, \text{m}) \]
Calculating:
\[ V = 2 \times 480 \times 0.095 \]
\[ V = 2 \times 480 \times 0.095 = 91.2 \times 2 = 182.4 \, \text{m/s} \]

### Step 6: Final Answer
Thus, the speed of sound in the air column at room temperature is approximately:
\[ \text{Speed} \approx 182.4 \, \text{m/s} \]