Two waves of wavelengths 52.5 cm and 52 cm produces 5 beats per second.their frequencies are

Options

A. 490Hz,495 Hz
B. 500 Hz ,505 Hz
C. 525 Hz,520 Hz
D. 500 Hz, 495 Hz

Answer

To solve the problem, we need to find the frequencies of two waves given their wavelengths and the number of beats produced.

### Step-by-Step Solution:

1. **Identify the Given Data**:
- Wavelength of the first wave, \( \lambda_1 = 52.5 \, \text{cm} = 0.525 \, \text{m} \)
- Wavelength of the second wave, \( \lambda_2 = 52 \, \text{cm} = 0.52 \, \text{m} \)
- Number of beats per second, \( \Delta f = 5 \, \text{Hz} \)

2. **Use the Wave Speed Formula**:
The relationship between wave speed \( v \), frequency \( f \), and wavelength \( \lambda \) is given by:
\[ v = f \lambda \]
Since both waves are sound waves traveling in the same medium, they have the same speed \( v \).

3. **Express Frequencies in Terms of Wave Speed**:
- For the first wave:
\[ f_1 = \frac{v}{\lambda_1} = \frac{v}{0.525} \]
- For the second wave:
\[ f_2 = \frac{v}{\lambda_2} = \frac{v}{0.52} \]

4. **Calculate the Difference in Frequencies**:
The difference in frequencies (which gives the number of beats) is:
\[ |f_2 - f_1| = 5 \, \text{Hz} \]
Since \( f_2 > f_1 \) (because \( \lambda_2 < \lambda_1 \)):
\[ f_2 - f_1 = 5 \]

5. **Substituting Frequencies**:
Substitute the expressions for \( f_1 \) and \( f_2 \):
\[ \frac{v}{0.52} - \frac{v}{0.525} = 5 \]

6. **Finding a Common Denominator**:
The common denominator for the left side is \( 0.52 \times 0.525 \):
\[ v \left( \frac{0.525 - 0.52}{0.52 \times 0.525} \right) = 5 \]
Simplifying the numerator:
\[ 0.525 - 0.52 = 0.005 \]
Thus, we have:
\[ \frac{0.005v}{0.52 \times 0.525} = 5 \]

7. **Solving for Wave Speed \( v \)**:
Rearranging gives:
\[ v = 5 \times \frac{0.52 \times 0.525}{0.005} \]
Calculating the value:
\[ v = 5 \times 54.6 = 273 \, \text{m/s} \]

8. **Finding Frequencies**:
Now substitute \( v \) back to find \( f_1 \) and \( f_2 \):
- For \( f_1 \):
\[ f_1 = \frac{273}{0.525} \approx 520 \, \text{Hz} \]
- For \( f_2 \):
\[ f_2 = \frac{273}{0.52} \approx 525 \, \text{Hz} \]

### Final Answer:
- The frequencies of the two waves are:
- \( f_1 \approx 520 \, \text{Hz} \)
- \( f_2 \approx 525 \, \text{Hz} \)