A small source of sound vibrating at frequency 500 Hz is rotated in a circle of radius `100/2pi` cm at a constant angular speed of 5-0 revolutions per second. A listener situates himself in the plane of the circle. Find the minimum and the maximum frequency of the sound observed. Speed of sound in air `= 332 m s^-1`.

Explanation

To solve the problem, we need to find the minimum and maximum frequency of sound observed by a listener when a sound source is rotating in a circle. The given parameters are:

- Frequency of the sound source, \( f_0 = 500 \, \text{Hz} \)
- Radius of the circle, \( r = \frac{100}{2\pi} \, \text{cm} = \frac{1}{\pi} \, \text{m} \)
- Angular speed, \( \omega = 5 \, \text{revolutions/second} = 5 \times 2\pi \, \text{radians/second} = 10\pi \, \text{rad/s} \)
- Speed of sound in air, \( v = 332 \, \text{m/s} \)

### Step 1: Calculate the linear speed of the sound source
The linear speed \( v_s \) of the sound source can be calculated using the formula:
\[ v_s = \omega \cdot r \]
Substituting the values:
\[ v_s = 10\pi \cdot \frac{1}{\pi} = 10 \, \text{m/s} \]

### Step 2: Calculate the minimum frequency observed
The minimum frequency occurs when the sound source is moving directly away from the listener. The formula for the observed frequency \( f \) in this case is given by:
\[ f_{\text{min}} = \frac{v + v_s}{v} \cdot f_0 \]
Substituting the values:
\[ f_{\text{min}} = \frac{332 + 10}{332} \cdot 500 \]
Calculating this:
\[ f_{\text{min}} = \frac{342}{332} \cdot 500 \approx 485.38 \, \text{Hz} \]

### Step 3: Calculate the maximum frequency observed
The maximum frequency occurs when the sound source is moving directly towards the listener. The formula for the observed frequency \( f \) in this case is given by:
\[ f_{\text{max}} = \frac{v - v_s}{v} \cdot f_0 \]
Substituting the values:
\[ f_{\text{max}} = \frac{332 - 10}{332} \cdot 500 \]
Calculating this:
\[ f_{\text{max}} = \frac{322}{332} \cdot 500 \approx 485.53 \, \text{Hz} \]

### Final Results
Thus, the minimum and maximum frequencies observed by the listener are:
- Minimum frequency \( f_{\text{min}} \approx 485.38 \, \text{Hz} \)
- Maximum frequency \( f_{\text{max}} \approx 505.53 \, \text{Hz} \)