A closed and an open organ pipe have the same length. When they are vibrating simultaneously in their first overtone, they produce three beats. The length of the open pipe is now made one third the original length and one of its ends is closed. On the other hand, the length of the closed pipe is made three times the original length. The number of betats produced when they vibrate with fundamental frequencies will be

Options

A. 8
B. 14
C. 17
D. 8

Answer

To solve the problem step by step, we need to analyze the frequencies of the closed and open pipes at different lengths and then calculate the number of beats produced when they vibrate at their fundamental frequencies.

### Step 1: Understand the frequencies of the pipes
1. **Closed Organ Pipe (Fundamental Frequency)**: The fundamental frequency \( F_c \) of a closed organ pipe is given by the formula:
\[ F_c = \frac{v}{4L} \]
where \( v \) is the speed of sound and \( L \) is the length of the pipe.

2. **Open Organ Pipe (Fundamental Frequency)**: The fundamental frequency \( F_0 \) of an open organ pipe is given by the formula:
\[ F_0 = \frac{v}{2L} \]

### Step 2: Analyze the first overtone frequencies
- The first overtone of a closed pipe is the third harmonic, so:
\[ F_c' = 3F_c = 3 \left( \frac{v}{4L} \right) = \frac{3v}{4L} \]

- The first overtone of an open pipe is the second harmonic, so:
\[ F_0' = 2F_0 = 2 \left( \frac{v}{2L} \right) = \frac{v}{L} \]

### Step 3: Given condition of beats
- When both pipes vibrate simultaneously in their first overtone, they produce 3 beats. This means:
\[ |F_0' - F_c'| = 3 \quad \text{(1)} \]
- Substituting the values:
\[ \left| \frac{v}{L} - \frac{3v}{4L} \right| = 3 \]
Simplifying gives:
\[ \left| \frac{4v - 3v}{4L} \right| = 3 \implies \frac{v}{4L} = 3 \implies v = 12L \]

### Step 4: New lengths of the pipes
- **Open Pipe**: The length is now made one-third the original length:
\[ L' = \frac{L}{3} \]
- **Closed Pipe**: The length is now made three times the original length:
\[ L'' = 3L \]

### Step 5: Calculate new frequencies
1. **New Frequency of Open Pipe**:
\[ F_0' = \frac{v}{2L'} = \frac{v}{2 \cdot \frac{L}{3}} = \frac{3v}{2L} \]
Substituting \( v = 12L \):
\[ F_0' = \frac{3 \cdot 12L}{2L} = 18 \text{ Hz} \]

2. **New Frequency of Closed Pipe**:
\[ F_c' = \frac{v}{4L''} = \frac{v}{4 \cdot 3L} = \frac{v}{12L} \]
Substituting \( v = 12L \):
\[ F_c' = \frac{12L}{12L} = 1 \text{ Hz} \]

### Step 6: Calculate the number of beats produced
- The number of beats produced when they vibrate at their fundamental frequencies is given by:
\[ \text{Beats} = |F_0' - F_c'| = |18 - 1| = 17 \]

### Final Answer
The number of beats produced when they vibrate with fundamental frequencies is **17**.

---