Two notes of wavelength 2.08 m and 2.12 m produce 180 beats per minute in a gas, Find the velocity of sound in the gas.

Answer

To find the velocity of sound in the gas given the wavelengths and the beat frequency, we can follow these steps:

### Step 1: Identify the given values
- Wavelength 1 (\( \lambda_1 \)) = 2.08 m
- Wavelength 2 (\( \lambda_2 \)) = 2.12 m
- Beat frequency = 180 beats per minute

### Step 2: Convert the beat frequency to Hz
To convert beats per minute to beats per second (Hz), we divide by 60:
\[ \text{Beat frequency} = \frac{180 \text{ beats/min}}{60} = 3 \text{ Hz} \]

### Step 3: Use the formula for beat frequency
The beat frequency (\( f_b \)) is given by the difference in frequencies of the two waves:
\[ f_b = |f_1 - f_2| \]
Where:
- \( f_1 = \frac{v}{\lambda_1} \)
- \( f_2 = \frac{v}{\lambda_2} \)

### Step 4: Substitute the frequencies into the beat frequency formula
Substituting the expressions for \( f_1 \) and \( f_2 \):
\[ 3 = \left| \frac{v}{\lambda_1} - \frac{v}{\lambda_2} \right| \]

### Step 5: Factor out \( v \)
Factoring out \( v \) gives:
\[ 3 = v \left( \frac{1}{\lambda_1} - \frac{1}{\lambda_2} \right) \]

### Step 6: Substitute the values of \( \lambda_1 \) and \( \lambda_2 \)
Substituting \( \lambda_1 = 2.08 \) m and \( \lambda_2 = 2.12 \) m:
\[ 3 = v \left( \frac{1}{2.08} - \frac{1}{2.12} \right) \]

### Step 7: Calculate \( \frac{1}{\lambda_1} - \frac{1}{\lambda_2} \)
Calculating the difference:
\[ \frac{1}{2.08} - \frac{1}{2.12} = \frac{2.12 - 2.08}{2.08 \times 2.12} = \frac{0.04}{2.08 \times 2.12} \]

### Step 8: Calculate \( 2.08 \times 2.12 \)
Calculating the product:
\[ 2.08 \times 2.12 = 4.4096 \]

### Step 9: Substitute back into the equation
Now substituting back:
\[ 3 = v \left( \frac{0.04}{4.4096} \right) \]

### Step 10: Solve for \( v \)
Rearranging gives:
\[ v = 3 \times \frac{4.4096}{0.04} \]
Calculating:
\[ v = 3 \times 110.24 = 330.72 \text{ m/s} \]

### Conclusion
The velocity of sound in the gas is approximately:
\[ v \approx 330 \text{ m/s} \]