A closed organ pipe and an open organ pipe of same length produce 2 beats when they are set into vibrations simultaneously in their fundamental mode. The length of open organ pipe is now halved and of closed organ pipe is doubled, the number of beats produced wil be
(A)8 (B)7 (C)4 (D)2

Options

A. 8
B. 7
C. 4
D. 2

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the fundamental frequencies of the pipes
For a closed organ pipe, the fundamental frequency \( f_c \) is given by:
\[ f_c = \frac{V}{4L} \]
For an open organ pipe, the fundamental frequency \( f_o \) is given by:
\[ f_o = \frac{V}{2L} \]

### Step 2: Set up the initial conditions
Initially, both pipes have the same length \( L \). According to the problem, they produce 2 beats when vibrating simultaneously:
\[ |f_o - f_c| = 2 \]
Substituting the expressions for \( f_o \) and \( f_c \):
\[ \left| \frac{V}{2L} - \frac{V}{4L} \right| = 2 \]

### Step 3: Simplify the equation
Simplifying the left side:
\[ \left| \frac{2V}{4L} - \frac{V}{4L} \right| = \left| \frac{V}{4L} \right| = 2 \]
This gives us:
\[ \frac{V}{4L} = 2 \implies V = 8L \]

### Step 4: Analyze the new lengths of the pipes
Now, the length of the open organ pipe is halved, so its new length \( L' = \frac{L}{2} \). The closed organ pipe's length is doubled, so its new length \( L'' = 2L \).

### Step 5: Calculate the new fundamental frequencies
For the new open organ pipe:
\[ f_o' = \frac{V}{2L'} = \frac{V}{2 \cdot \frac{L}{2}} = \frac{V}{L} \]
For the new closed organ pipe:
\[ f_c' = \frac{V}{4L''} = \frac{V}{4 \cdot 2L} = \frac{V}{8L} \]

### Step 6: Calculate the new number of beats
Now, we find the number of beats produced:
\[ |f_o' - f_c'| = \left| \frac{V}{L} - \frac{V}{8L} \right| \]
Simplifying this:
\[ = \left| \frac{8V}{8L} - \frac{V}{8L} \right| = \left| \frac{7V}{8L} \right| \]

### Step 7: Substitute the value of \( V \)
From our earlier calculation, we found \( V = 8L \):
\[ |f_o' - f_c'| = \left| \frac{7 \cdot 8L}{8L} \right| = 7 \]

### Final Answer
Thus, the number of beats produced when the lengths are changed is:
\[ \boxed{7} \]