A closed organ pipe and an open organ pie of same length produce four bets in their fundamental mode when sounded together, If length of the open organ pipe is increased, then the number of beats will

Options

A. increase
B. decrease
C. remain constant
D. may increase or decrease

Answer

To solve the problem step by step, we need to analyze the frequencies of the closed and open organ pipes and how they change when the length of the open organ pipe is increased.

### Step 1: Understand the Frequencies of the Pipes
1. **Frequency of the Open Organ Pipe (f₀)**: The frequency of an open organ pipe is given by the formula:
\[ f_0 = \frac{v}{2L} \]
where \( v \) is the speed of sound and \( L \) is the length of the pipe.

2. **Frequency of the Closed Organ Pipe (f_c)**: The frequency of a closed organ pipe is given by the formula:
\[ f_c = \frac{v}{4L} \]

### Step 2: Calculate the Initial Beat Frequency
3. **Given Information**: The problem states that the two pipes produce 4 beats when sounded together. This means:
\[ |f_0 - f_c| = 4 \]

4. **Substituting the Frequencies**:
\[ \left|\frac{v}{2L} - \frac{v}{4L}\right| = 4 \]

5. **Finding a Common Denominator**:
\[ \left|\frac{2v - v}{4L}\right| = 4 \]
\[ \left|\frac{v}{4L}\right| = 4 \]

6. **Solving for v/L**:
\[ \frac{v}{4L} = 4 \implies \frac{v}{L} = 16 \]

### Step 3: Analyze the Effect of Increasing the Length of the Open Pipe
7. **Increasing the Length of the Open Pipe**: Let’s say the length of the open organ pipe is increased to \( kL \) (where \( k > 1 \)). The new frequency of the open organ pipe becomes:
\[ f'_0 = \frac{v}{2(kL)} = \frac{v}{2kL} \]

### Step 4: Calculate the New Beat Frequency
8. **New Beat Frequency**:
\[ |f'_0 - f_c| = \left|\frac{v}{2kL} - \frac{v}{4L}\right| \]

9. **Finding a Common Denominator**:
\[ \left|\frac{2v - vk}{4kL}\right| = \left|\frac{(2 - k)v}{4kL}\right| \]

10. **Substituting \( v/L = 16 \)**:
\[ \left|\frac{(2 - k) \cdot 16}{4k}\right| = \left|\frac{4(2 - k)}{k}\right| \]

### Step 5: Conclusion on the Number of Beats
11. **Behavior of Beats**:
- If \( k < 2 \), then \( 2 - k > 0 \) and the number of beats increases.
- If \( k = 2 \), then \( 2 - k = 0 \) and the number of beats becomes zero.
- If \( k > 2 \), then \( 2 - k < 0 \) and the number of beats decreases.

### Final Answer
The number of beats may increase or decrease depending on how much the length of the open organ pipe is increased.