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 beats produced when they vibrate with fundamental frequencies will be

Options

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

Answer

To solve the problem, we need to analyze the frequencies of the open and closed organ pipes under the given conditions. Let's break it down step by step.

### Step 1: Understanding the Initial Conditions
We have an open organ pipe and a closed organ pipe of the same length \( L \). When they vibrate in their first overtone, they produce 3 beats.

- **Frequency of the first overtone for the open pipe**:
\[ f_1 = \frac{3V}{4L} \]

- **Frequency of the first overtone for the closed pipe**:
\[ f_2 = \frac{5V}{4L} \]

### Step 2: Calculating the Beat Frequency
The beat frequency is given by the absolute difference between the two frequencies:
\[ \text{Beats} = |f_1 - f_2| = \left| \frac{3V}{4L} - \frac{5V}{4L} \right| = \left| \frac{-2V}{4L} \right| = \frac{2V}{4L} = \frac{V}{2L} \]
Given that this produces 3 beats, we have:
\[ \frac{V}{2L} = 3 \implies V = 6L \]

### Step 3: Modifying the Lengths of the Pipes
Now, we modify the lengths of the pipes:
- The length of the open pipe is made one third of the original length: \( L' = \frac{L}{3} \).
- The closed pipe's length is made three times the original length: \( L'' = 3L \).

### Step 4: Finding the New Frequencies
Now we calculate the fundamental frequencies for the modified pipes.

1. **For the new closed pipe (originally open)**:
- Since it is now closed at one end, its fundamental frequency is:
\[ f_{01} = \frac{V}{4L'} = \frac{V}{4 \times \frac{L}{3}} = \frac{3V}{4L} \]

2. **For the new closed pipe (originally closed)**:
- Its fundamental frequency is:
\[ f_{02} = \frac{V}{4L''} = \frac{V}{4 \times 3L} = \frac{V}{12L} \]

### Step 5: Calculating the Beat Frequency for the New Frequencies
Now we find the beat frequency when both pipes vibrate at their fundamental frequencies:
\[ \text{Beats} = |f_{01} - f_{02}| = \left| \frac{3V}{4L} - \frac{V}{12L} \right| \]
To combine these, we need a common denominator (12L):
\[ f_{01} = \frac{9V}{12L}, \quad f_{02} = \frac{V}{12L} \]
Thus,
\[ \text{Beats} = \left| \frac{9V}{12L} - \frac{V}{12L} \right| = \left| \frac{8V}{12L} \right| = \frac{2V}{3L} \]

### Step 6: Relating to the Original Beats
From our earlier calculation, we know:
\[ \frac{V}{4L} = 3 \implies V = 12L \]
Substituting this back into our beat frequency:
\[ \text{Beats} = \frac{2 \times 12L}{3L} = 8 \]

### Final Answer
The number of beats produced when they vibrate with their fundamental frequencies is **8 beats**.

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