A small source of sound vibrating frequency 500 Hz is rotated in a circle of radius `(100)/(pi)` cm at a constant angular speed of `5.0` revolutions per second. The speed of sound in air is `330(m)/(s)`.
Q. For an observer who is at rest at a great distance from the centre of the circle but nearly in the same plane, the minimum `f_(min)` and the maximum `f_(max)` of the range of values of the apparent frequency heard by him will be

Options

A. `f_(min)=455Hz`,`f_(max)=535Hz`
B. `f_(min)=484Hz`,`f_(max)=515Hz`
C. `f_(min)=484Hz`,`f_(max)=500Hz`
D. `f_(min)=500Hz`,`f_(max)=515Hz`

Explanation

To solve the problem, we need to determine the minimum and maximum apparent frequencies (f_min and f_max) heard by an observer due to the Doppler effect, as the sound source is rotating in a circle.

### Step 1: Calculate the linear speed of the sound source
The linear speed (u) of the sound source can be calculated using the formula:
\[ u = r \cdot \omega \]
where:
- \( r \) is the radius of the circle,
- \( \omega \) is the angular speed in radians per second.

Given:
- Radius \( r = \frac{100}{\pi} \) cm = \( \frac{100}{\pi} \times \frac{1}{100} \) m = \( \frac{1}{\pi} \) m,
- Angular speed \( \omega = 5 \) revolutions/second = \( 5 \times 2\pi \) radians/second.

Calculating \( u \):
\[ u = \frac{1}{\pi} \cdot (5 \times 2\pi) = 10 \text{ m/s} \]

### Step 2: Use the Doppler effect formula for minimum frequency (f_min)
The formula for the apparent frequency (f') using the Doppler effect is:
\[ f' = f \cdot \frac{v + v_o}{v + v_s} \]
where:
- \( f \) is the source frequency,
- \( v \) is the speed of sound,
- \( v_o \) is the speed of the observer (0 m/s since the observer is at rest),
- \( v_s \) is the speed of the source.

For minimum frequency, the source is moving away from the observer:
\[ f_{min} = f \cdot \frac{v + 0}{v + u} = 500 \cdot \frac{330}{330 + 10} \]
Calculating \( f_{min} \):
\[ f_{min} = 500 \cdot \frac{330}{340} = 500 \cdot \frac{33}{34} \approx 485 \text{ Hz} \]

### Step 3: Use the Doppler effect formula for maximum frequency (f_max)
For maximum frequency, the source is moving towards the observer:
\[ f_{max} = f \cdot \frac{v + 0}{v - u} = 500 \cdot \frac{330}{330 - 10} \]
Calculating \( f_{max} \):
\[ f_{max} = 500 \cdot \frac{330}{320} = 500 \cdot \frac{33}{32} \approx 515 \text{ Hz} \]

### Final Answer
The minimum and maximum apparent frequencies heard by the observer are:
- \( f_{min} \approx 485 \text{ Hz} \)
- \( f_{max} \approx 515 \text{ Hz} \)