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

Options

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

Explanation

To solve the problem of finding the velocity of sound using the information provided about two closed organ pipes, follow these steps:

### Step 1: Understand the relationship between frequency, length, and velocity
For a closed organ pipe, the fundamental frequency \( f \) 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: Calculate the frequencies of both pipes
Let the lengths of the two pipes be:
- \( L_1 = 100 \, \text{cm} = 1.00 \, \text{m} \)
- \( L_2 = 101 \, \text{cm} = 1.01 \, \text{m} \)

Using the formula for frequency, we can express the frequencies of the two pipes as:
\[ f_1 = \frac{V}{4L_1} = \frac{V}{4 \times 1.00} = \frac{V}{4} \]
\[ f_2 = \frac{V}{4L_2} = \frac{V}{4 \times 1.01} = \frac{V}{4.04} \]

### Step 3: Determine the beat frequency
The beat frequency \( n \) is given as 16 beats in 20 seconds, which means:
\[ n = \frac{16}{20} = 0.8 \, \text{Hz} \]
The beat frequency is also given by the difference in frequencies of the two pipes:
\[ n = |f_1 - f_2| = \left| \frac{V}{4} - \frac{V}{4.04} \right| \]

### Step 4: Set up the equation for beat frequency
We can express the beat frequency in terms of \( V \):
\[ 0.8 = \left| \frac{V}{4} - \frac{V}{4.04} \right| \]
This simplifies to:
\[ 0.8 = \frac{V}{4} - \frac{V}{4.04} \]

### Step 5: Solve for \( V \)
To solve for \( V \), we need a common denominator:
\[ 0.8 = \frac{V \cdot 4.04 - V \cdot 4}{4 \cdot 4.04} \]
This simplifies to:
\[ 0.8 = \frac{V(4.04 - 4)}{16.16} \]
\[ 0.8 = \frac{0.04V}{16.16} \]
Now, multiply both sides by 16.16:
\[ 0.8 \cdot 16.16 = 0.04V \]
\[ 12.928 = 0.04V \]
Now, divide both sides by 0.04:
\[ V = \frac{12.928}{0.04} = 323.2 \, \text{m/s} \]

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