Question

The velocity of sound in a gas, in which two wavelengths, 4.08 m and 4.16 m produce 40 beats in 12 s, will be:

• A. 282.8 ms^–1
• B. 175.5 ms^–1
• C. 353.6 ms^–1
• D. 707.2 ms^–1

Answer 1

The Correct Option is D

To find the velocity of sound in the given gas, we start with understanding the provided data and use the formula for frequency and velocity of sound. We are given two wavelengths, 4.08 m and 4.16 m, and the production of 40 beats in 12 seconds.

1. First, calculate the frequencies corresponding to these wavelengths using the velocity formula: v = f \lambda, where v is the velocity of sound, f is the frequency, and \lambda is the wavelength.
2. The beat frequency (number of beats per second) can be calculated as: f_{\text{beat}} = \frac{40}{12} \approx 3.33 \, \text{Hz}.
3. Frequency difference between the two sound waves is given by the beat frequency: |f_1 - f_2| = 3.33 .
4. The velocities for the respective wavelengths can be defined as:
   • v = f_1 \times 4.08 \qquad \text{(1)}
   • v = f_2 \times 4.16 \qquad \text{(2)}
5. Using Equation (1) and rearranging, we find: f_1 = \frac{v}{4.08}.
6. Using Equation (2) and rearranging: f_2 = \frac{v}{4.16}.
7. Since known: |f_1 - f_2| = \frac{v}{4.08} - \frac{v}{4.16} = 3.33 .
8. Find v from: \frac{v}{4.08} - \frac{v}{4.16} = 3.33 : v \left(\frac{1}{4.08} - \frac{1}{4.16}\right) = 3.33 .
9. Calculate: \frac{1}{4.08} - \frac{1}{4.16} = \frac{1}{4.08 \times 4.16} \times (4.16 - 4.08) = \frac{0.08}{16.9728} .
10. Simplifying, \frac{1}{4.08} - \frac{1}{4.16} \approx 0.0047.
11. Substitute back: v \times 0.0047 \approx 3.33 .
12. Solve for v: v = \frac{3.33}{0.0047} \approx 707.2 \, \text{ms}^{-1}.

Therefore, the velocity of sound in the gas is 707.2 ms^–1. This confirms the given correct answer.