The air column in a pipe closed at one end is made to vibrate in its third over tone by tuning fork of frequency 220Hz. The speed of sound in air is 330m/sec. End correction may be neglected. Let `P_0` denote the mean pressure at any point in the pipe and `Delta P_0` the maximum amplitude of pressure vibration
Find the length of the air column

Options

A. 3.2 m
B. 2.625m
C. 4.23m
D. 1.16 m

Answer

To find the length of the air column in a closed pipe vibrating in its third overtone, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Harmonic**:
- In a closed pipe, only odd harmonics are present. The fundamental frequency (1st harmonic) is the first overtone (3rd harmonic), and the second overtone is the 5th harmonic. Therefore, the third overtone corresponds to the 7th harmonic.

2. **Use the Formula for Frequency**:
- The frequency of the nth harmonic in a closed pipe is given by:
\[ f_n = \frac{n \cdot v}{4L} \]
- For the 7th harmonic (third overtone), we have:
\[ f_7 = \frac{7 \cdot v}{4L} \]

3. **Substitute Known Values**:
- We know the frequency \( f = 220 \, \text{Hz} \) and the speed of sound \( v = 330 \, \text{m/s} \). Substitute these values into the equation:
\[ 220 = \frac{7 \cdot 330}{4L} \]

4. **Rearrange to Solve for L**:
- Rearranging the equation gives:
\[ 4L \cdot 220 = 7 \cdot 330 \]
- Simplifying this:
\[ 880L = 2310 \]
- Now, solve for \( L \):
\[ L = \frac{2310}{880} \]

5. **Calculate the Length**:
- Performing the division:
\[ L = 2.625 \, \text{m} \]

### Final Answer:
The length of the air column in the pipe is approximately \( 2.625 \, \text{m} \).

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