Two closed organ pipes of length 100 cm and 101 cm product 16 beats is 20 sec, When each pipe is sounded in its fundamental mode calculate the velocity of sound .

Options

A. `303 ms^-1`
B. `332 ms ^-1`
C. `323.2 ms ^-1`
D. `300 ms^(-1)`

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Frequencies of Closed Organ Pipes
For a closed organ pipe, the frequency of the fundamental mode is given by the formula:
\[ f = \frac{V}{4L} \]
where \( V \) is the velocity of sound and \( L \) is the length of the pipe.

### Step 2: Define the Frequencies of Both Pipes
Let:
- \( L_1 = 100 \, \text{cm} = 1.00 \, \text{m} \)
- \( L_2 = 101 \, \text{cm} = 1.01 \, \text{m} \)

The frequencies of the two pipes can be expressed as:
- For the first pipe:
\[ f_1 = \frac{V}{4 \times 1.00} = \frac{V}{4} \]
- For the second pipe:
\[ f_2 = \frac{V}{4 \times 1.01} = \frac{V}{4.04} \]

### Step 3: Calculate the Beat Frequency
The beat frequency \( f_b \) is given by the absolute difference of the two frequencies:
\[ f_b = |f_1 - f_2| = \left| \frac{V}{4} - \frac{V}{4.04} \right| \]

### Step 4: Set Up the Equation for Beats
According to the problem, the two pipes produce 16 beats in 20 seconds. Therefore, the beat frequency is:
\[ f_b = \frac{16}{20} = 0.8 \, \text{Hz} \]

### Step 5: Set the Equation Equal to the Beat Frequency
Now, we can equate the two expressions for the beat frequency:
\[ \left| \frac{V}{4} - \frac{V}{4.04} \right| = 0.8 \]

### Step 6: Simplify the Equation
To simplify:
\[ \frac{V}{4} - \frac{V}{4.04} = 0.8 \]
Finding a common denominator:
\[ \frac{V \cdot 4.04 - V \cdot 4}{4 \cdot 4.04} = 0.8 \]
\[ \frac{V(4.04 - 4)}{16.16} = 0.8 \]
\[ \frac{V(0.04)}{16.16} = 0.8 \]

### Step 7: Solve for V
Now, multiply both sides by \( 16.16 \):
\[ V(0.04) = 0.8 \times 16.16 \]
\[ V(0.04) = 12.928 \]
Now, divide both sides by \( 0.04 \):
\[ V = \frac{12.928}{0.04} \]
\[ V = 323.2 \, \text{m/s} \]

### Final Answer
The velocity of sound is approximately:
\[ V \approx 323.2 \, \text{m/s} \]