Two open pipes of length L are vibrated simultaneously. If length of one of the pipes is reduced by y. then the number of beats heats per second will be if the velocity of sound is v and `y lt lt L` :-

Options

A. `(vy)/(2L^(2))`
B. `(vy)/(L^(2))`
C. `(vy)/(2L)`
D. `(2L^(2))/(vy)`

Explanation

To solve the problem step by step, we will follow these procedures:

### Step 1: Understand the Problem
We have two open pipes of length \( L \) that are producing sound simultaneously. One of the pipes has its length reduced by \( y \). We need to find the number of beats per second produced by the two pipes when they vibrate.

### Step 2: Determine the Frequencies of the Pipes
The frequency of a pipe is given by the formula:
\[ f = \frac{v}{\lambda} \]
where \( v \) is the speed of sound and \( \lambda \) is the wavelength.

For the first pipe (length \( L \)):
- The wavelength \( \lambda_1 \) is given by:
\[ \lambda_1 = 2L \quad \text{(since it is an open pipe)} \]
Thus, the frequency \( f_1 \) is:
\[ f_1 = \frac{v}{\lambda_1} = \frac{v}{2L} \]

For the second pipe (length \( L - y \)):
- The wavelength \( \lambda_2 \) is given by:
\[ \lambda_2 = 2(L - y) \]
Thus, the frequency \( f_2 \) is:
\[ f_2 = \frac{v}{\lambda_2} = \frac{v}{2(L - y)} \]

### Step 3: Calculate the Beat Frequency
The number of beats per second is given by the absolute difference between the two frequencies:
\[ \text{Beats per second} = |f_2 - f_1| \]
Substituting the frequencies we found:
\[ \text{Beats per second} = \left| \frac{v}{2(L - y)} - \frac{v}{2L} \right| \]

### Step 4: Simplify the Expression
Factoring out \( \frac{v}{2} \):
\[ \text{Beats per second} = \frac{v}{2} \left| \frac{1}{L - y} - \frac{1}{L} \right| \]
Finding a common denominator:
\[ \text{Beats per second} = \frac{v}{2} \left| \frac{L - (L - y)}{L(L - y)} \right| = \frac{v}{2} \left| \frac{y}{L(L - y)} \right| \]
Since \( y \ll L \), we can approximate \( L - y \approx L \):
\[ \text{Beats per second} \approx \frac{v}{2} \cdot \frac{y}{L^2} \]

### Step 5: Final Result
Thus, the number of beats per second is:
\[ \text{Beats per second} = \frac{v y}{2 L^2} \]