Question

The velocity of sound in a gas in which two wavelengths \( 4.08 \, \text{m} \) and \( 4.16 \, \text{m} \) produce 40 beats in 12 seconds will be:

• A. \( 2.828 \, \text{ms}^{-1} \)
• B. \( 175.5 \, \text{ms}^{-1} \)
• C. \( 353.6 \, \text{ms}^{-1} \)
• D. \( 707.2 \, \text{ms}^{-1} \)

Hint:

To find the velocity of sound using beat frequency, use the relation \( f_{\text{beat}} = f_1 - f_2 \) and the wave equation \( f = \frac{v}{\lambda} \).

Answer 1

The Correct Option is D

The beat frequency (\( f_{\text{beat}} \)) is the difference between two frequencies: \[ f_{\text{beat}} = f_1 - f_2, \] where \( f_1 \) and \( f_2 \) are the frequencies corresponding to the wavelengths \( \lambda_1 \) and \( \lambda_2 \), respectively. Step 1: Calculate the beat frequency Given the number of beats and time: \[ f_{\text{beat}} = \frac{\text{Number of beats}}{\text{Time}} = \frac{40}{12} \, \text{Hz}. \] Simplifying: \[ f_{\text{beat}} = \frac{10}{3} \, \text{Hz}. \] Step 2: Relating wavelength to velocity and frequency The frequency of a wave is related to its velocity by the equation: \[ f = \frac{v}{\lambda}. \] Using the given wavelengths \( \lambda_1 = 4.08 \, \text{m} \) and \( \lambda_2 = 4.16 \, \text{m} \), we have: \[ f_1 = \frac{v}{4.08}, \quad f_2 = \frac{v}{4.16}. \] Step 3: Solve for the velocity \( v \) The beat frequency is the difference in frequencies: \[ f_{\text{beat}} = f_1 - f_2 = \frac{v}{4.08} - \frac{v}{4.16}. \] Simplifying: \[ \frac{10}{3} = v \left(\frac{1}{4.08} - \frac{1}{4.16}\right). \] Now, calculate the difference in reciprocals: \[ \frac{1}{4.08} - \frac{1}{4.16} = \frac{4.16 - 4.08}{4.08 \cdot 4.16} = \frac{0.08}{16.9728}. \] Substitute this value: \[ \frac{10}{3} = v \cdot \frac{0.08}{16.9728}. \] Solving for \( v \): \[ v = \frac{\frac{10}{3} \cdot 16.9728}{0.08} = 707.2 \, \text{m/s}. \] Final Answer: \[ \boxed{707.2 \, \text{m/s}}. \]