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. If the observer moves towards the source with a constant speed of `20(m)/(s)`, along the radial line to the centre, the fractional change in the apparent frequency over the frequency that the source will have if considered at reat at the centre will be

Options

A. `6%`
B. `3%`
C. `2%`
D. `9%`

Explanation

To solve the problem step by step, we need to calculate the apparent frequency of the sound as perceived by the observer moving towards the source, and then find the fractional change in frequency.

### Step 1: Identify the given data
- Frequency of the source, \( f_0 = 500 \, \text{Hz} \)
- Speed of sound in air, \( v = 330 \, \text{m/s} \)
- Speed of the observer, \( u = 20 \, \text{m/s} \)

### Step 2: Calculate the apparent frequency when the observer moves towards the source
The formula for the apparent frequency when the source is stationary and the observer is moving towards it is given by:

\[ f' = f_0 \left( \frac{v + u}{v} \right) \]

Substituting the values:

\[ f' = 500 \left( \frac{330 + 20}{330} \right) \]

\[ f' = 500 \left( \frac{350}{330} \right) \]

\[ f' = 500 \times \frac{35}{33} \]

### Step 3: Calculate the change in frequency
Now, we find the change in frequency:

\[ \Delta f = f' - f_0 \]

Substituting the values we calculated:

\[ \Delta f = \left( 500 \times \frac{35}{33} \right) - 500 \]

\[ \Delta f = 500 \left( \frac{35}{33} - 1 \right) \]

\[ \Delta f = 500 \left( \frac{35 - 33}{33} \right) \]

\[ \Delta f = 500 \left( \frac{2}{33} \right) \]

\[ \Delta f = \frac{1000}{33} \, \text{Hz} \]

### Step 4: Calculate the fractional change in frequency
The fractional change in frequency is given by:

\[ \text{Fractional Change} = \frac{\Delta f}{f_0} \]

Substituting the values:

\[ \text{Fractional Change} = \frac{\frac{1000}{33}}{500} \]

\[ \text{Fractional Change} = \frac{1000}{33 \times 500} \]

\[ \text{Fractional Change} = \frac{2}{33} \]

### Final Answer
The fractional change in the apparent frequency is:

\[ \text{Fractional Change} \approx 0.0606 \]