Two open organ pipes of lengths 25 cm and `25.5` produce 10 beats/sec. What is the velocity of sound ?

Options

A. 200 m/s
B. 205 m/s
C. 255 m/s
D. 250 m/s

Explanation

To solve the problem, we need to find the velocity of sound using the information provided about the two open organ pipes. Let's break it down step by step.

### Step 1: Understand the relationship between frequency and length of the pipe
For an open organ pipe, the frequency \( f \) is given by the formula:
\[ f = \frac{V}{2L} \]
where \( V \) is the velocity of sound and \( L \) is the length of the pipe.

### Step 2: Define the frequencies of the two pipes
Let:
- \( L_1 = 25 \, \text{cm} = 0.25 \, \text{m} \)
- \( L_2 = 25.5 \, \text{cm} = 0.255 \, \text{m} \)

The frequencies of the two pipes can be defined as:
\[ f_1 = \frac{V}{2L_1} \quad \text{and} \quad f_2 = \frac{V}{2L_2} \]

### Step 3: Calculate the difference in frequencies
According to the problem, the difference in frequencies (which corresponds to the number of beats per second) is given as:
\[ |f_1 - f_2| = 10 \, \text{beats/sec} \]
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ \left| \frac{V}{2L_1} - \frac{V}{2L_2} \right| = 10 \]

### Step 4: Simplify the equation
Factoring out \( \frac{V}{2} \):
\[ \frac{V}{2} \left| \frac{1}{L_1} - \frac{1}{L_2} \right| = 10 \]

### Step 5: Substitute the lengths
Substituting \( L_1 \) and \( L_2 \):
\[ \frac{V}{2} \left| \frac{1}{0.25} - \frac{1}{0.255} \right| = 10 \]

### Step 6: Calculate the difference in reciprocals
Calculating the reciprocals:
\[ \frac{1}{0.25} = 4 \quad \text{and} \quad \frac{1}{0.255} \approx 3.9216 \]
Thus,
\[ \left| 4 - 3.9216 \right| = 0.0784 \]

### Step 7: Substitute back into the equation
Now substitute back into the equation:
\[ \frac{V}{2} \cdot 0.0784 = 10 \]

### Step 8: Solve for \( V \)
Multiplying both sides by 2:
\[ V \cdot 0.0784 = 20 \]
Now, divide both sides by \( 0.0784 \):
\[ V = \frac{20}{0.0784} \approx 255.1 \, \text{m/s} \]

### Conclusion
The velocity of sound is approximately:
\[ V \approx 255 \, \text{m/s} \]