A whistle revolves in a circle with an angular speed of `20 rad//sec` using a string of length `50 cm`. If the frequency of sound from the whistle is `385 Hz`, then what is the minimum frequency heard by an observer which is far away from the centre in the same plane? `v=340m//s`

Options

A. `333 Hz`
B. `374 Hz`
C. `385 Hz`
D. `394 Hz`

Explanation

To solve the problem, we need to determine the minimum frequency heard by an observer when a whistle is revolving in a circle. We will use the Doppler effect formula for sound.

### Given Data:
- Angular speed, \( \omega = 20 \, \text{rad/s} \)
- Length of string (radius), \( r = 50 \, \text{cm} = 0.5 \, \text{m} \)
- Frequency of sound from the whistle, \( f_0 = 385 \, \text{Hz} \)
- Speed of sound, \( v = 340 \, \text{m/s} \)

### Step-by-Step Solution:

1. **Calculate the linear speed of the whistle (source)**:
The linear speed \( v_s \) of the whistle can be calculated using the formula:
\[ v_s = r \cdot \omega \]
Substituting the values:
\[ v_s = 0.5 \, \text{m} \cdot 20 \, \text{rad/s} = 10 \, \text{m/s} \]

2. **Identify the observer's position**:
The observer is positioned far away from the center of the circle, in the same plane as the whistle. Since the whistle is moving away from the observer at the point of minimum frequency, we will use the Doppler effect formula for a source moving away from a stationary observer.

3. **Apply the Doppler effect formula**:
The formula for the frequency heard by the observer when the source is moving away is given by:
\[ f' = f_0 \cdot \frac{v}{v + v_s} \]
Where:
- \( f' \) is the frequency heard by the observer,
- \( f_0 \) is the emitted frequency,
- \( v \) is the speed of sound,
- \( v_s \) is the speed of the source.

4. **Substituting the values**:
Now, substitute the known values into the formula:
\[ f' = 385 \, \text{Hz} \cdot \frac{340 \, \text{m/s}}{340 \, \text{m/s} + 10 \, \text{m/s}} \]
\[ f' = 385 \, \text{Hz} \cdot \frac{340}{350} \]

5. **Calculating the frequency**:
Now calculate the fraction:
\[ f' = 385 \, \text{Hz} \cdot \frac{34}{35} \]
\[ f' = 385 \, \text{Hz} \cdot 0.9714 \approx 374 \, \text{Hz} \]

### Final Answer:
The minimum frequency heard by the observer is approximately \( 374 \, \text{Hz} \).

---