A table is revolving on its axis at `5` revolutions per second. A sound source of frequency `1000 Hz `is fixed on the table at `70 cm` from the axis. The minimum frequency heard by a listener standing at a distance from the table will be (speed of sound `=352m//s`).

Options

A. 1000 Hz
B. 1066 Hz
C. 941 Hz
D. 352 Hz

Answer

To solve the problem, we need to find the minimum frequency heard by a listener when a sound source is moving away from them. We will use the Doppler effect formula for sound.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Frequency of the sound source, \( f = 1000 \, \text{Hz} \)
- Speed of sound, \( v = 352 \, \text{m/s} \)
- Radius from the axis to the sound source, \( r = 70 \, \text{cm} = 0.7 \, \text{m} \)
- Revolutions per second, \( n = 5 \, \text{rev/s} \)

2. **Calculate Angular Velocity (ω):**
- The angular velocity \( \omega \) can be calculated using the formula:
\[ \omega = 2\pi n \]
- Substituting \( n = 5 \):
\[ \omega = 2\pi \times 5 = 10\pi \, \text{rad/s} \]

3. **Calculate Linear Velocity (v_s) of the Source:**
- The linear velocity of the source moving in a circular path is given by:
\[ v_s = r \omega \]
- Substituting \( r = 0.7 \, \text{m} \) and \( \omega = 10\pi \):
\[ v_s = 0.7 \times 10\pi \approx 22 \, \text{m/s} \]

4. **Apply the Doppler Effect Formula:**
- The minimum frequency heard by the listener when the source is moving away is given by:
\[ f' = f \frac{v}{v + v_s} \]
- Substituting the values:
\[ f' = 1000 \times \frac{352}{352 + 22} \]

5. **Calculate the Denominator:**
- Calculate \( 352 + 22 = 374 \).

6. **Calculate the Minimum Frequency (f'):**
- Now substitute back into the formula:
\[ f' = 1000 \times \frac{352}{374} \]
- Performing the calculation:
\[ f' \approx 1000 \times 0.941 = 941 \, \text{Hz} \]

7. **Final Answer:**
- The minimum frequency heard by the listener is approximately \( 941 \, \text{Hz} \).