Question

When the wavelength of sound changes from 1 m to 1.01 m, the number of beats heard are 4. The velocity of sound is

• A. 404 m/s
• B. 4.04 m/s
• C. 414 m/s
• D. 400 m/s

Hint:

When the wavelength of sound changes from 1 m to 1.01 m, the number of beats heard are 4. The velocity of sound is

Answer 1

The Correct Option is A

The problem involves determining the velocity of sound using the concept of beats and the change in the wavelength of sound. Given that the wavelength of sound changes from 1 m to 1.01 m, and the number of beats heard is 4, we can find the velocity of sound using the relationship between frequency, velocity, and wavelength.

Let's denote:

• \(\lambda_1 = 1 \, \text{m}\)
• \(\lambda_2 = 1.01 \, \text{m}\)
• \(v\) as the velocity of sound (which we need to find).
• The frequency corresponding to a wavelength \(\lambda\) is given by \(f = \frac{v}{\lambda}\).

For the given wavelengths:

• \(f_1 = \frac{v}{1} = v\)
• \(f_2 = \frac{v}{1.01}\)

The number of beats is given by the difference in frequencies:

\(|f_1 - f_2| = 4 \, \text{beats} = 4 \, \text{Hz}\)

Substituting for \(f_1\) and \(f_2\), we have:

\(|v - \frac{v}{1.01}| = 4\)

Simplifying the equation:

\(v - \frac{v}{1.01} = 4\)

\(\Rightarrow v \left(1 - \frac{1}{1.01}\right) = 4\)

\(\Rightarrow v \left(\frac{0.01}{1.01}\right) = 4\)

\(\Rightarrow v = \frac{4 \times 1.01}{0.01} = 404 \, \text{m/s}\)

Therefore, the velocity of sound is 404 m/s, which corresponds to option:

404 m/s