The difference between the frequencies of the third and fifth harmonic of a closed organ pipe is 100 Hz. Its fundamental frequency is

Options

A. 100 Hz
B. 50 Hz
C. 25 Hz
D. 12.5 Hz

Answer

To solve the problem, we need to determine the fundamental frequency of a closed organ pipe given that the difference between the frequencies of the third and fifth harmonics is 100 Hz.

### Step-by-Step Solution:

1. **Understanding Harmonics in a Closed Organ Pipe**:
- In a closed organ pipe, the harmonics are given by the formula:
- \( f_n = n \cdot f_0 \)
- where \( f_n \) is the frequency of the nth harmonic and \( f_0 \) is the fundamental frequency.

2. **Identify the Frequencies of the Third and Fifth Harmonics**:
- The frequency of the third harmonic is:
- \( f_3 = 3 \cdot f_0 \)
- The frequency of the fifth harmonic is:
- \( f_5 = 5 \cdot f_0 \)

3. **Calculate the Difference Between the Frequencies**:
- According to the problem, the difference between the frequencies of the third and fifth harmonics is given as:
- \( f_5 - f_3 = 100 \, \text{Hz} \)

4. **Substituting the Harmonic Frequencies**:
- Substitute the expressions for \( f_3 \) and \( f_5 \):
- \( 5f_0 - 3f_0 = 100 \, \text{Hz} \)

5. **Simplifying the Equation**:
- Simplifying the left side gives:
- \( 2f_0 = 100 \, \text{Hz} \)

6. **Solving for the Fundamental Frequency**:
- Divide both sides by 2 to find \( f_0 \):
- \( f_0 = \frac{100 \, \text{Hz}}{2} = 50 \, \text{Hz} \)

7. **Conclusion**:
- The fundamental frequency \( f_0 \) of the closed organ pipe is:
- \( f_0 = 50 \, \text{Hz} \)

### Final Answer:
The fundamental frequency of the closed organ pipe is **50 Hz**.

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