The fundamental frequency of a vibrating organ pipe is 200 Hz.

Options

A. the first overtone is 400 Hz.
B. The first overtone may be 400 Hz
C. The first overtone may be 600 Hz
D. 600 Hz is an overtone

Answer

To solve the problem regarding the fundamental frequency of a vibrating organ pipe, we will analyze both open and closed organ pipes and determine the possible frequencies.

### Step-by-Step Solution:

1. **Understanding the Fundamental Frequency**:
The fundamental frequency (first harmonic) of an organ pipe is the lowest frequency at which it vibrates. For an open organ pipe, the fundamental frequency \( f_0 \) is given by the formula:
\[ f_0 = \frac{v}{2L} \]
where \( v \) is the speed of sound in air and \( L \) is the length of the pipe.

2. **Given Information**:
We are given that the fundamental frequency \( f_0 = 200 \, \text{Hz} \).

3. **Calculating Frequencies for Open Organ Pipe**:
For an open organ pipe, the harmonics are given by:
\[ f_n = n \cdot f_0 \]
where \( n \) is a positive integer (1, 2, 3,...). Therefore, the frequencies will be:
- For \( n = 1 \): \( f_1 = 200 \, \text{Hz} \) (fundamental frequency)
- For \( n = 2 \): \( f_2 = 400 \, \text{Hz} \) (first overtone)
- For \( n = 3 \): \( f_3 = 600 \, \text{Hz} \) (second overtone)
- And so on...

4. **Calculating Frequencies for Closed Organ Pipe**:
For a closed organ pipe, the fundamental frequency is given by:
\[ f_0' = \frac{v}{4L} \]
The harmonics for a closed pipe occur at odd multiples:
\[ f_n' = n \cdot f_0' \quad \text{where } n = 1, 3, 5, ... \]
Thus, for \( n = 1 \):
- \( f_1' = 200 \, \text{Hz} \) (fundamental frequency)
- For \( n = 3 \): \( f_3' = 600 \, \text{Hz} \) (first overtone)
- For \( n = 5 \): \( f_5' = 1000 \, \text{Hz} \) (second overtone)
- And so on...

5. **Identifying Common Frequencies**:
From our calculations:
- The first overtone for the open pipe is \( 400 \, \text{Hz} \) and for the closed pipe is \( 600 \, \text{Hz} \).
- The second overtone for the open pipe is \( 600 \, \text{Hz} \) and for the closed pipe is \( 1000 \, \text{Hz} \).

6. **Conclusion**:
The first overtone for the closed organ pipe can also be \( 600 \, \text{Hz} \). Therefore, the options provided in the question can include \( 400 \, \text{Hz} \) as the first overtone for the open pipe and \( 600 \, \text{Hz} \) as a possible overtone for both pipes.

### Final Answer:
The correct options for the overtone frequencies based on the analysis are \( 400 \, \text{Hz} \) and \( 600 \, \text{Hz} \).

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