A whistle of frequency 500 Hz is rotated in a circle of radius cne metre with an angular speed of 10 radian/sec. Calculate the lowest and the highest frequency heard by a listener at a long distance away at rest with respect to the centre of the circle ? (Velocity of sound = 340m/s)

Answer

To solve the problem of finding the lowest and highest frequency heard by a listener from a rotating whistle, we can follow these steps:

### Step 1: Identify Given Values
- Frequency of the whistle (f) = 500 Hz
- Angular speed (ω) = 10 rad/s
- Radius of the circle (r) = 1 m
- Velocity of sound (v) = 340 m/s

### Step 2: Calculate Linear Velocity of the Whistle
The linear velocity (Vs) of the whistle can be calculated using the formula:
\[ Vs = \omega \times r \]
Substituting the values:
\[ Vs = 10 \, \text{rad/s} \times 1 \, \text{m} = 10 \, \text{m/s} \]

### Step 3: Calculate Maximum Frequency (fmax)
The maximum frequency occurs when the source is approaching the observer. The formula for maximum frequency is:
\[ f_{max} = f \times \frac{v - v_0}{v - Vs} \]
Where:
- \( v_0 \) (velocity of observer) = 0 (since the observer is at rest)

Substituting the values:
\[ f_{max} = 500 \times \frac{340 - 0}{340 - 10} \]
\[ f_{max} = 500 \times \frac{340}{330} \]
\[ f_{max} = 500 \times 1.0303 \approx 515.15 \, \text{Hz} \]

### Step 4: Calculate Minimum Frequency (fmin)
The minimum frequency occurs when the source is receding from the observer. The formula for minimum frequency is:
\[ f_{min} = f \times \frac{v - v_0}{v + Vs} \]

Substituting the values:
\[ f_{min} = 500 \times \frac{340 - 0}{340 + 10} \]
\[ f_{min} = 500 \times \frac{340}{350} \]
\[ f_{min} = 500 \times 0.9714 \approx 485.71 \, \text{Hz} \]

### Final Answers
- Highest frequency (fmax) = 515.15 Hz
- Lowest frequency (fmin) = 485.71 Hz

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