An organ piep has a fundamental frequency of 100 Hz. Its second overtone is 500 Hz. What is the nature of the pipe ?

Options

A. Closed at one end
B. Open at both ends
C. Closed at both ends
D. Sometimes open sometimes closed

Explanation

To determine the nature of the organ pipe based on the given frequencies, we can follow these steps:

### Step 1: Identify the Given Frequencies
- The fundamental frequency (first harmonic) of the organ pipe is given as **100 Hz**.
- The second overtone (which is the third harmonic) is given as **500 Hz**.

### Step 2: Understand the Harmonics in Different Types of Pipes
- For a **closed organ pipe**, the harmonics are given by the formula:
\[ f_n = \frac{n v}{4L} \]
where \( n \) can take odd values (1, 3, 5, ...).

- For an **open organ pipe**, the harmonics are given by:
\[ f_n = \frac{n v}{2L} \]
where \( n \) can take any positive integer values (1, 2, 3, ...).

### Step 3: Analyze the Frequencies
- The first harmonic (fundamental frequency) for a closed pipe is:
\[ f_1 = 100 \text{ Hz} \]

- The second overtone corresponds to the third harmonic:
\[ f_3 = 500 \text{ Hz} \]

### Step 4: Calculate the Relationship Between Frequencies
- For a closed organ pipe:
- The first harmonic (fundamental) is \( f_1 = 100 \text{ Hz} \).
- The second overtone (third harmonic) is \( f_3 = 3f_1 \).

Thus:
\[ f_3 = 3 \times 100 \text{ Hz} = 300 \text{ Hz} \]
This does not match the given second overtone frequency of 500 Hz.

- For an open organ pipe:
- The first harmonic (fundamental) is \( f_1 = 100 \text{ Hz} \).
- The second overtone (third harmonic) is \( f_3 = 3f_1 = 3 \times 100 \text{ Hz} = 300 \text{ Hz} \).

Again, this does not match the given second overtone frequency of 500 Hz.

### Step 5: Check the Second Overtone for a Closed Organ Pipe
- For a closed organ pipe:
- The second overtone is actually the fifth harmonic:
\[ f_5 = 5f_1 = 5 \times 100 \text{ Hz} = 500 \text{ Hz} \]
This matches the given second overtone frequency of 500 Hz.

### Conclusion
Since the second overtone corresponds to the fifth harmonic in a closed organ pipe and matches the given frequency, we conclude that the organ pipe is a **closed organ pipe**.

### Final Answer
The nature of the pipe is **closed organ pipe**.

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