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 situated at a great distance on a straight line perpendicular to the plane of the circle, through its centre, the apparent frequency of the source will be

Options

A. greater that 500 Hx
B. smaller than 500 Hz
C. always remain 500 Hz
D. greater for half the circle and smaller during the other half

Explanation

To solve the problem, we need to determine the apparent frequency of a sound source that is rotating in a circle while being observed from a distance. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have a sound source with a frequency of 500 Hz rotating in a circle of radius \( \frac{100}{\pi} \) cm at an angular speed of 5 revolutions per second. The speed of sound in air is given as 330 m/s. An observer is situated at a great distance perpendicular to the plane of the circle.

### Step 2: Convert Units
First, we need to convert the radius from centimeters to meters:
\[ R = \frac{100}{\pi} \text{ cm} = \frac{100}{\pi} \times \frac{1}{100} \text{ m} = \frac{1}{\pi} \text{ m} \]

### Step 3: Calculate the Linear Speed of the Source
The angular speed \( \omega \) in radians per second can be calculated from the revolutions per second:
\[ \omega = 5 \text{ revolutions/second} \times 2\pi \text{ radians/revolution} = 10\pi \text{ radians/second} \]
The linear speed \( v \) of the source is given by:
\[ v = R \cdot \omega = \frac{1}{\pi} \cdot 10\pi = 10 \text{ m/s} \]

### Step 4: Analyze the Observer's Position
The observer is at a great distance \( L \) from the source, and the distance \( L \) is much greater than the radius \( R \). This means that the angle subtended by the radius at the observer's position is very small.

### Step 5: Use the Doppler Effect
For an observer at a great distance, the apparent frequency \( f' \) can be approximated using the Doppler effect formula for a moving source:
\[ f' = f \left( \frac{v + v_o}{v - v_s} \right) \]
Where:
- \( f \) is the source frequency (500 Hz),
- \( v \) is the speed of sound (330 m/s),
- \( v_o \) is the speed of the observer (0 m/s, since the observer is stationary),
- \( v_s \) is the speed of the source (10 m/s).

### Step 6: Substitute Values
Substituting the values into the Doppler effect formula:
\[ f' = 500 \left( \frac{330 + 0}{330 - 10} \right) = 500 \left( \frac{330}{320} \right) \]

### Step 7: Calculate the Apparent Frequency
Calculating the above expression:
\[ f' = 500 \times \frac{330}{320} = 500 \times 1.03125 = 515.625 \text{ Hz} \]

### Step 8: Conclusion
However, since the observer is at a great distance and the source is moving in a circular path, the apparent frequency will average out to the original frequency due to symmetry in the motion. Therefore, the apparent frequency remains approximately:
\[ f' \approx 500 \text{ Hz} \]

### Final Answer
The apparent frequency of the source for the observer will be **500 Hz**.

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