Two tuning forks `A` and `B` lying on opposite sides of observer 'O' and of natural frequency `80 Hz` move with velocity `10 m//s` relative to stationary observer `O`. Fork `A` moves away from the observer while the fork `B` moves towards him. A wind with a speed `10 m//s` is blowing in the direction of motion of fork `A`. Find the beat frequency measured by the observer in Hz. [Take speed of sound in air as `340 m//s`].

Options

A. 5
B. 6
C. 7
D. 8

Explanation

To find the beat frequency measured by the observer due to the two tuning forks A and B, we will follow these steps:

### Step 1: Understand the Problem
We have two tuning forks, A and B, both with a natural frequency of 80 Hz. Fork A is moving away from the observer O, while fork B is moving towards the observer. There is also wind blowing from B to A at a speed of 10 m/s. We need to calculate the frequencies heard by the observer from both tuning forks and then find the beat frequency.

### Step 2: Calculate the Effective Speed of Sound
The speed of sound in air is given as 340 m/s. However, since there is wind blowing in the direction of fork A, we need to adjust the effective speed of sound for fork A.

- For fork A moving away from the observer:
\[ v_{A} = v_{sound} - v_{wind} = 340 \, \text{m/s} - 10 \, \text{m/s} = 330 \, \text{m/s} \]

### Step 3: Calculate the Frequency Heard from Fork A
Using the Doppler effect formula for a source moving away from the observer:
\[ f_A = \left( \frac{v_{sound} - v_{wind}}{v_{sound} - v_{source}} \right) f_0 \]
Substituting the values:
- \( v_{sound} = 340 \, \text{m/s} \)
- \( v_{wind} = 10 \, \text{m/s} \)
- \( v_{source} = 10 \, \text{m/s} \)
- \( f_0 = 80 \, \text{Hz} \)

\[ f_A = \left( \frac{330}{340 - 10} \right) \times 80 = \left( \frac{330}{330} \right) \times 80 = 80 \, \text{Hz} \]

### Step 4: Calculate the Frequency Heard from Fork B
For fork B moving towards the observer:
\[ f_B = \left( \frac{v_{sound} + v_{wind}}{v_{sound} + v_{source}} \right) f_0 \]
Substituting the values:
- \( v_{sound} = 340 \, \text{m/s} \)
- \( v_{wind} = 10 \, \text{m/s} \)
- \( v_{source} = 10 \, \text{m/s} \)

\[ f_B = \left( \frac{340 + 10}{340 + 10} \right) \times 80 = \left( \frac{350}{350} \right) \times 80 = 80 \, \text{Hz} \]

### Step 5: Calculate the Beat Frequency
The beat frequency is given by the absolute difference between the two frequencies:
\[ f_{beat} = |f_A - f_B| \]
Substituting the values:
\[ f_{beat} = |80 - 80| = 0 \, \text{Hz} \]

### Conclusion
The beat frequency measured by the observer is **0 Hz**.

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