To solve the problem step by step, we will use the Doppler effect formula for sound. The scenario involves a whistle revolving in a circle, and we need to calculate the minimum frequency heard by an observer when the whistle is moving away from the observer.
### Step 1: Understand the Given Data
- Angular speed of the whistle, \( \omega = 20 \, \text{rad/s} \)
- Length of the string (radius of the circular path), \( r = 50 \, \text{cm} = 0.5 \, \text{m} \)
- Frequency of the whistle, \( f = 385 \, \text{Hz} \)
- Speed of sound in air, \( v = 340 \, \text{m/s} \)
### Step 2: Calculate the Linear Velocity of the Whistle
The linear velocity \( v_s \) of the whistle can be calculated using the formula:
\[
v_s = r \cdot \omega
\]
Substituting the values:
\[
v_s = 0.5 \, \text{m} \cdot 20 \, \text{rad/s} = 10 \, \text{m/s}
\]
### Step 3: Identify the Doppler Effect Formula
The apparent frequency \( f' \) heard by the observer when the source is moving away from the observer is given by:
\[
f' = \frac{v - v_0}{v + v_s} \cdot f
\]
Where:
- \( v \) = speed of sound
- \( v_0 \) = speed of the observer (0 since the observer is stationary)
- \( v_s \) = speed of the source (whistle)
### Step 4: Substitute the Values into the Doppler Effect Formula
Since the observer is stationary, \( v_0 = 0 \). Therefore, the formula simplifies to:
\[
f' = \frac{v}{v + v_s} \cdot f
\]
Substituting the known values:
\[
f' = \frac{340 \, \text{m/s}}{340 \, \text{m/s} + 10 \, \text{m/s}} \cdot 385 \, \text{Hz}
\]
### Step 5: Calculate the Apparent Frequency
Calculating the denominator:
\[
340 + 10 = 350 \, \text{m/s}
\]
Now substituting this back into the formula:
\[
f' = \frac{340}{350} \cdot 385
\]
Calculating the fraction:
\[
\frac{340}{350} = 0.9714
\]
Now, multiplying by the actual frequency:
\[
f' = 0.9714 \cdot 385 \approx 374 \, \text{Hz}
\]
### Final Answer
The minimum frequency heard by the observer is approximately \( 374 \, \text{Hz} \).
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