Two open organ pipes of lengths 60 cm and 60.5 cm produce 2 beats per second. Calculate the velocity of sound in air.?

Answer

To solve the problem of finding the velocity of sound in air using the information given about two open organ pipes, we can follow these steps:

### Step 1: Understand the problem and gather the data
We have two open organ pipes with lengths:
- Pipe 1 (L1) = 60 cm = 0.6 m
- Pipe 2 (L2) = 60.5 cm = 0.605 m

The frequency difference that produces beats is given as:
- Beat frequency = f1 - f2 = 2 beats per second

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

For Pipe 1:
\[ f_1 = \frac{V}{2L_1} = \frac{V}{2 \times 0.6} \]

For Pipe 2:
\[ f_2 = \frac{V}{2L_2} = \frac{V}{2 \times 0.605} \]

### Step 3: Set up the equation for beat frequency
From the beat frequency, we have:
\[ f_1 - f_2 = 2 \]

Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ \frac{V}{2 \times 0.6} - \frac{V}{2 \times 0.605} = 2 \]

### Step 4: Simplify the equation
Factor out \( V \):
\[ V \left( \frac{1}{1.2} - \frac{1}{1.21} \right) = 2 \]

### Step 5: Find a common denominator and simplify further
The common denominator for 1.2 and 1.21 is \( 1.2 \times 1.21 \):
\[ \frac{1.21 - 1.2}{1.2 \times 1.21} = \frac{0.01}{1.2 \times 1.21} \]

Thus, we can rewrite the equation as:
\[ V \cdot \frac{0.01}{1.2 \times 1.21} = 2 \]

### Step 6: Solve for V
Rearranging gives:
\[ V = 2 \cdot \frac{1.2 \times 1.21}{0.01} \]

Calculating the values:
1. \( 1.2 \times 1.21 = 1.452 \)
2. \( V = 2 \cdot \frac{1.452}{0.01} = 2 \cdot 145.2 = 290.4 \, \text{m/s} \)

### Final Answer
The velocity of sound in air is approximately:
\[ V \approx 290.4 \, \text{m/s} \]