Wavelength of two notes in air are 68/176 m and 68/174 m each note produces five beats per second with a third note of fixed frequency. Calculate the velocity of sound in air?

Answer

To solve the problem, we need to calculate the velocity of sound in air using the given wavelengths of two notes and the information about the beats produced.

### Step-by-Step Solution:

1. **Identify the Wavelengths**:
- Wavelength of note 1, \( \lambda_1 = \frac{68}{176} \, \text{m} \)
- Wavelength of note 2, \( \lambda_2 = \frac{68}{174} \, \text{m} \)

2. **Determine the Beat Frequency**:
- The beat frequency is given as 5 beats per second. Since there are two notes producing beats with a third note of fixed frequency, the total beat frequency is \( f_1 - f_2 = 10 \, \text{Hz} \).

3. **Use the Frequency Formula**:
- The frequency \( f \) is related to the velocity \( v \) and wavelength \( \lambda \) by the formula:
\[ f = \frac{v}{\lambda} \]
- For note 1:
\[ f_1 = \frac{v}{\lambda_1} = \frac{v}{\frac{68}{176}} = \frac{176v}{68} \]
- For note 2:
\[ f_2 = \frac{v}{\lambda_2} = \frac{v}{\frac{68}{174}} = \frac{174v}{68} \]

4. **Set Up the Equation for Beat Frequency**:
- From the beat frequency:
\[ f_1 - f_2 = 10 \, \text{Hz} \]
- Substitute \( f_1 \) and \( f_2 \):
\[ \frac{176v}{68} - \frac{174v}{68} = 10 \]
- Simplifying this gives:
\[ \frac{(176 - 174)v}{68} = 10 \]
\[ \frac{2v}{68} = 10 \]

5. **Solve for Velocity \( v \)**:
- Cross-multiply to solve for \( v \):
\[ 2v = 10 \times 68 \]
\[ 2v = 680 \]
\[ v = \frac{680}{2} = 340 \, \text{m/s} \]

6. **Conclusion**:
- The velocity of sound in air is \( v = 340 \, \text{m/s} \).