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 understand the relationship between the harmonics of a closed organ pipe and its fundamental frequency.

### Step-by-Step Solution:

1. **Understanding Harmonics in a Closed Organ Pipe**:
A closed organ pipe supports only odd harmonics. The frequencies of the harmonics can be expressed in terms of the fundamental frequency (f₁). The frequencies of the harmonics are given by:
- 1st Harmonic (Fundamental): f₁
- 3rd Harmonic: f₃ = 3f₁
- 5th Harmonic: f₅ = 5f₁

2. **Finding the Difference Between the Frequencies**:
According to the problem, the difference between the frequencies of the 3rd and 5th harmonics is given as 100 Hz. We can express this mathematically:
\[ f_5 - f_3 = 100 \text{ Hz} \]
Substituting the expressions for f₅ and f₃:
\[ 5f_1 - 3f_1 = 100 \text{ Hz} \]

3. **Simplifying the Equation**:
Simplifying the left side gives:
\[ (5f_1 - 3f_1) = 2f_1 \]
Thus, we have:
\[ 2f_1 = 100 \text{ Hz} \]

4. **Solving for the Fundamental Frequency**:
Now, we can solve for f₁:
\[ f_1 = \frac{100 \text{ Hz}}{2} = 50 \text{ Hz} \]

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

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