Two open pipes of length `20cm` and `20.1cm` produces `10` beats/s. The velocity of sound in the gas is

Options

A. `804ms^(-1)`
B. `402ms^(-1)`
C. `420ms^(-1)`
D. `330ms^(-1)`

Explanation

To solve the problem, we need to find the velocity of sound in the gas using the information provided about the two open pipes and the beats produced. Here’s a step-by-step solution:

### Step 1: Understand the concept of beats
When two sound waves of slightly different frequencies interfere, they produce a phenomenon known as beats. The beat frequency is equal to the absolute difference between the two frequencies.

### Step 2: Write the formula for the fundamental frequency of an open pipe
The fundamental frequency \( f \) of an open pipe is given by the formula:
\[ f = \frac{V}{2L} \]
where \( V \) is the velocity of sound in the gas, and \( L \) is the length of the pipe.

### Step 3: Define the frequencies for both pipes
Let:
- \( L_1 = 20 \, \text{cm} = 0.20 \, \text{m} \)
- \( L_2 = 20.1 \, \text{cm} = 0.201 \, \text{m} \)

The frequencies for the two pipes can be expressed as:
\[ f_1 = \frac{V}{2L_1} = \frac{V}{2 \times 0.20} \]
\[ f_2 = \frac{V}{2L_2} = \frac{V}{2 \times 0.201} \]

### Step 4: Calculate the difference in frequencies
The beat frequency is given as \( 10 \, \text{beats/s} \). Thus, we have:
\[ |f_1 - f_2| = 10 \]
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ \left| \frac{V}{2 \times 0.20} - \frac{V}{2 \times 0.201} \right| = 10 \]

### Step 5: Simplify the equation
Taking \( \frac{V}{2} \) common:
\[ \frac{V}{2} \left( \frac{1}{0.20} - \frac{1}{0.201} \right) = 10 \]

### Step 6: Calculate the difference of the fractions
To simplify \( \frac{1}{0.20} - \frac{1}{0.201} \):
\[ \frac{1}{0.20} = 5 \quad \text{and} \quad \frac{1}{0.201} \approx 4.975 \]
Thus:
\[ 5 - 4.975 = 0.025 \]

### Step 7: Substitute back into the equation
Now substituting back:
\[ \frac{V}{2} \times 0.025 = 10 \]

### Step 8: Solve for \( V \)
Multiply both sides by 2:
\[ V \times 0.025 = 20 \]
Now divide by 0.025:
\[ V = \frac{20}{0.025} = 800 \, \text{m/s} \]

### Final Answer
The velocity of sound in the gas is:
\[ V = 800 \, \text{m/s} \]