A closed organ pipe and an open organ pipe of same length produce `4` beats when they are set into vibrations simultaneously. If the length of each of them were twice their initial lengths, the number of beats produced will be

Options

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

Explanation

To solve the problem of how many beats will be produced when the lengths of a closed organ pipe and an open organ pipe are doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Frequencies**:
- The fundamental frequency \( f_c \) of a closed organ pipe is given by the formula:
\[ f_c = \frac{v}{4L} \]
- The fundamental frequency \( f_o \) of an open organ pipe is given by the formula:
\[ f_o = \frac{v}{2L} \]
Where \( v \) is the speed of sound in air, and \( L \) is the length of the pipe.

2. **Initial Frequencies**:
- Since both pipes have the same length \( L \), we can express their frequencies:
\[ f_c = \frac{v}{4L} \]
\[ f_o = \frac{v}{2L} \]

3. **Calculating the Beat Frequency**:
- The beat frequency \( f_b \) is the absolute difference between the two frequencies:
\[ f_b = |f_o - f_c| = \left| \frac{v}{2L} - \frac{v}{4L} \right| \]
- Simplifying this expression:
\[ f_b = \left| \frac{2v}{4L} - \frac{v}{4L} \right| = \left| \frac{v}{4L} \right| = \frac{v}{4L} \]
- We know from the problem statement that this beat frequency is given as 4 beats per second:
\[ \frac{v}{4L} = 4 \quad \Rightarrow \quad v = 16L \]

4. **Doubling the Length**:
- Now, if the lengths of both pipes are doubled, the new lengths become \( 2L \).
- The new frequencies will be:
\[ f_c' = \frac{v}{4(2L)} = \frac{v}{8L} \]
\[ f_o' = \frac{v}{2(2L)} = \frac{v}{4L} \]

5. **Calculating the New Beat Frequency**:
- The new beat frequency \( f_b' \) is:
\[ f_b' = |f_o' - f_c'| = \left| \frac{v}{4L} - \frac{v}{8L} \right| \]
- Simplifying this:
\[ f_b' = \left| \frac{2v}{8L} - \frac{v}{8L} \right| = \left| \frac{v}{8L} \right| \]
- Substituting \( v = 16L \):
\[ f_b' = \frac{16L}{8L} = 2 \text{ beats per second} \]

### Final Answer:
The number of beats produced when the lengths of both pipes are doubled is **2 beats per second**.

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