If the velocity of sound in air is 336 m/s. The maximum length of a closed pipe that would produce a just audible sound will be

Answer

To find the maximum length of a closed pipe that would produce a just audible sound, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Closed Pipe**:
In a closed pipe, one end is closed (node) and the other end is open (antinode). The fundamental frequency of a closed pipe corresponds to a quarter of the wavelength (λ) fitting into the length (L) of the pipe. Therefore, we can express this relationship as:
\[ L = \frac{\lambda}{4} \]

2. **Relating Wavelength to Length**:
From the above equation, we can rearrange it to find the wavelength:
\[ \lambda = 4L \]

3. **Using the Wave Equation**:
The velocity (v) of sound is related to frequency (f) and wavelength (λ) by the equation:
\[ v = f \cdot \lambda \]
Substituting the expression for wavelength from step 2, we have:
\[ v = f \cdot (4L) \]

4. **Substituting the Given Values**:
We know the velocity of sound in air is given as \( v = 336 \, \text{m/s} \). Therefore, we can write:
\[ 336 = f \cdot (4L) \]

5. **Finding the Maximum Length**:
Rearranging the equation to solve for L gives:
\[ L = \frac{336}{4f} \]

6. **Choosing the Frequency**:
To find the maximum length of the pipe that produces just audible sound, we should use the minimum frequency of human hearing, which is \( f = 20 \, \text{Hz} \):
\[ L = \frac{336}{4 \cdot 20} \]

7. **Calculating the Length**:
Now, we can calculate the length:
\[ L = \frac{336}{80} = 4.2 \, \text{m} \]

### Final Answer:
The maximum length of a closed pipe that would produce a just audible sound is **4.2 meters**.

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