A whistle of frequency 500 Hz tied to the end of a string of length 1.2 m revolves at 400 rev / min . A listener standing some distance away in the plane of rotation of whistle hears frequencies in the range (speed of sound = 340 m / s )

Options

A. 436 to 586
B. 426 to 574
C. 426 to 584
D. 436 to 674

Answer

To solve the problem step by step, we will follow the principles of the Doppler effect and the calculations involving the movement of the whistle.

### Step 1: Understand the given data
- Frequency of the whistle (\(f_0\)) = 500 Hz
- Length of the string (which is the radius of the circular path, \(r\)) = 1.2 m
- Revolutions per minute (RPM) = 400 rev/min
- Speed of sound (\(v\)) = 340 m/s

### Step 2: Convert RPM to radians per second
To find the angular speed (\(\omega\)), we convert revolutions per minute to radians per second:
\[ \omega = \text{RPM} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} \]
Substituting the values:
\[ \omega = 400 \times \frac{2\pi}{60} \approx 41.89 \text{ rad/s} \]

### Step 3: Calculate the tangential speed of the whistle
The tangential speed (\(V_s\)) is given by the formula:
\[ V_s = r \cdot \omega \]
Substituting the values:
\[ V_s = 1.2 \times 41.89 \approx 50.27 \text{ m/s} \]

### Step 4: Calculate the maximum frequency heard by the listener
When the source is moving towards the observer, the apparent frequency (\(f'\)) can be calculated using the formula:
\[ f' = \frac{v}{v - V_s} \cdot f_0 \]
Substituting the values:
\[ f' = \frac{340}{340 - 50.27} \cdot 500 \]
Calculating the denominator:
\[ 340 - 50.27 \approx 289.73 \]
Now substituting back:
\[ f' \approx \frac{340}{289.73} \cdot 500 \approx 586.07 \text{ Hz} \]

### Step 5: Calculate the minimum frequency heard by the listener
When the source is moving away from the observer, the apparent frequency (\(f'\)) can be calculated using the formula:
\[ f' = \frac{v}{v + V_s} \cdot f_0 \]
Substituting the values:
\[ f' = \frac{340}{340 + 50.27} \cdot 500 \]
Calculating the denominator:
\[ 340 + 50.27 \approx 390.27 \]
Now substituting back:
\[ f' \approx \frac{340}{390.27} \cdot 500 \approx 436.07 \text{ Hz} \]

### Step 6: State the range of frequencies heard by the listener
The listener hears frequencies in the range of approximately 436 Hz to 586 Hz.

### Final Answer
The frequencies heard by the listener range from approximately **436 Hz to 586 Hz**.

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