If the velocity of sound in air is `320 m//s`, then the maximum and minimum length of a pipe closed at one end , that would produce a just audible sound would be

Options

A. `2.6 m` and `3.6 mm`
B. `4 m` and `4.2 mm`
C. `3 m and 3 mm`
D. `4 m and 4 mm`

Explanation

To solve the problem of finding the maximum and minimum lengths of a pipe closed at one end that would produce a just audible sound, we can follow these steps:

### Step 1: Understand the relationship between frequency, velocity, and length of the pipe
For a pipe closed at one end, the frequency of the sound produced is given by the formula:
\[ f = \frac{V}{4L} (2N - 1) \]
where:
- \( f \) = frequency of the sound,
- \( V \) = velocity of sound in air (given as \( 320 \, \text{m/s} \)),
- \( L \) = length of the pipe,
- \( N \) = harmonic number (1, 2, 3,...).

### Step 2: Identify the audible frequency range
The audible frequency range for humans is from \( 20 \, \text{Hz} \) to \( 20 \, \text{kHz} \).
- For maximum length, we will use the minimum frequency \( f_{\text{min}} = 20 \, \text{Hz} \).
- For minimum length, we will use the maximum frequency \( f_{\text{max}} = 20 \, \text{kHz} = 20 \times 10^3 \, \text{Hz} \).

### Step 3: Calculate the maximum length of the pipe
Using the minimum frequency \( f_{\text{min}} = 20 \, \text{Hz} \):
\[ 20 = \frac{320}{4L_{\text{max}}} (2 \times 1 - 1) \]
This simplifies to:
\[ 20 = \frac{320}{4L_{\text{max}}} \]
Rearranging gives:
\[ 4L_{\text{max}} = \frac{320}{20} \]
\[ 4L_{\text{max}} = 16 \]
\[ L_{\text{max}} = \frac{16}{4} = 4 \, \text{m} \]

### Step 4: Calculate the minimum length of the pipe
Using the maximum frequency \( f_{\text{max}} = 20 \times 10^3 \, \text{Hz} \):
\[ 20 \times 10^3 = \frac{320}{4L_{\text{min}}} (2 \times 1 - 1) \]
This simplifies to:
\[ 20 \times 10^3 = \frac{320}{4L_{\text{min}}} \]
Rearranging gives:
\[ 4L_{\text{min}} = \frac{320}{20 \times 10^3} \]
\[ 4L_{\text{min}} = \frac{320}{20000} = 0.016 \]
\[ L_{\text{min}} = \frac{0.016}{4} = 0.004 \, \text{m} = 4 \, \text{mm} \]

### Final Answer
- Maximum length \( L_{\text{max}} = 4 \, \text{m} \)
- Minimum length \( L_{\text{min}} = 4 \, \text{mm} \)