A closed organ pipe and an open organ pipe of same length produce 4 beats when they are set into vibration simultaneously. If the length of each of them were twice their initial lengths. The number of beats produced will be [Assume same mode of viberation in both cases]

Options

A. 2
B. 4
C. 1
D. 8

Answer

To solve the problem step by step, we will analyze the frequencies of the closed and open organ pipes and how they change when the lengths are doubled.

### Step 1: Understand the initial conditions
We have a closed organ pipe and an open organ pipe of the same length, producing 4 beats when they vibrate simultaneously.

### Step 2: Write the formulas for the frequencies
The frequency of a closed organ pipe (f1) is given by:
\[ f_1 = \frac{v}{4L} \]
where \( v \) is the speed of sound and \( L \) is the length of the pipe.

The frequency of an open organ pipe (f2) is given by:
\[ f_2 = \frac{v}{2L} \]

### Step 3: Set up the equation for beats
The beat frequency is the absolute difference between the two frequencies:
\[ |f_2 - f_1| = 4 \]

Since \( f_2 \) is greater than \( f_1 \), we can write:
\[ f_2 - f_1 = 4 \]

Substituting the formulas for \( f_1 \) and \( f_2 \):
\[ \frac{v}{2L} - \frac{v}{4L} = 4 \]

### Step 4: Simplify the equation
To simplify, we find a common denominator:
\[ \frac{2v}{4L} - \frac{v}{4L} = 4 \]
\[ \frac{v}{4L} = 4 \]

### Step 5: Solve for \( v \) in terms of \( L \)
From the equation:
\[ v = 16L \]

### Step 6: Determine the new frequencies when length is doubled
If the length of each pipe is doubled, the new lengths will be \( 2L \).

For the closed organ pipe:
\[ f_1' = \frac{v}{4(2L)} = \frac{v}{8L} \]

For the open organ pipe:
\[ f_2' = \frac{v}{2(2L)} = \frac{v}{4L} \]

### Step 7: Calculate the new beat frequency
Now, we find the new beat frequency:
\[ |f_2' - f_1'| = \left| \frac{v}{4L} - \frac{v}{8L} \right| \]

Finding a common denominator:
\[ \frac{2v}{8L} - \frac{v}{8L} = \frac{v}{8L} \]

### Step 8: Substitute \( v \) back into the equation
Using \( v = 16L \):
\[ \frac{16L}{8L} = 2 \]

### Conclusion
Thus, the number of beats produced when the lengths are doubled is:
\[ \text{Number of beats} = 2 \]

### Final Answer
The number of beats produced will be **2**.

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