An open pipe produces fundamental note. All of a sudden one of its ends is closed. If again fundamental note is emitted, the frequency of the note will be :

Options

A. Double
B. Half
C. Same
D. None of these

Answer

To solve the problem, we need to analyze the situation when an open pipe is converted into a closed pipe and how that affects the frequency of the fundamental note produced.

### Step-by-Step Solution:

1. **Understanding the Frequency in an Open Pipe:**
- For an open pipe, the fundamental frequency (first harmonic) is given by the formula:
\[ f = \frac{v}{2L} \]
where \( v \) is the speed of sound in air and \( L \) is the length of the pipe.

2. **Understanding the Frequency in a Closed Pipe:**
- When one end of the pipe is closed, it becomes a closed pipe. The fundamental frequency for a closed pipe is given by:
\[ f' = \frac{v}{4L} \]
Here, the length \( L \) remains the same, but the formula changes because a closed pipe supports a different harmonic series.

3. **Setting Up the Ratio of Frequencies:**
- To find the relationship between the frequencies before and after closing one end of the pipe, we can set up the ratio:
\[ \frac{f}{f'} = \frac{\frac{v}{2L}}{\frac{v}{4L}} \]
- Simplifying this ratio:
\[ \frac{f}{f'} = \frac{v}{2L} \cdot \frac{4L}{v} = \frac{4}{2} = 2 \]

4. **Finding the New Frequency:**
- From the ratio \( \frac{f}{f'} = 2 \), we can express the new frequency \( f' \) in terms of the original frequency \( f \):
\[ f' = \frac{f}{2} \]
- This indicates that the frequency of the fundamental note emitted after one end of the pipe is closed is half of the original frequency.

5. **Conclusion:**
- Therefore, the frequency of the note after one end of the open pipe is closed will be:
\[ \text{Frequency after closing one end} = \frac{f}{2} \]
- The correct answer is that the frequency will be **half** of the original frequency.

### Final Answer:
The frequency of the note will be **half** of the original frequency.

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