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A whistle of frequency 500 Hz tied to th...

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 )

A

386 Hz - 436 Hz

B

426 Hz - 474 Hz

C

426 Hz - 586 Hz

D

436 Hz - 586 Hz

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to calculate the range of frequencies that a listener hears from a whistle revolving in a circular motion. Here’s a step-by-step breakdown of the solution: ### Step 1: Convert the rotational speed from revolutions per minute to radians per second. Given: - Rotational speed, \( n = 400 \) revolutions/minute. To convert this to radians per second: \[ \omega = 2\pi \frac{n}{60} = 2\pi \frac{400}{60} \approx 41.89 \text{ radians/second} \] ### Step 2: Calculate the radius of the circular path. Given: - Length of the string, \( r = 1.2 \) m. ### Step 3: Calculate the linear velocity of the whistle. The linear velocity \( V \) of the whistle can be calculated using the formula: \[ V = \omega r \] Substituting the values: \[ V = 41.89 \times 1.2 \approx 50.27 \text{ m/s} \] ### Step 4: Identify the speed of sound. Given: - Speed of sound, \( c = 340 \) m/s. ### Step 5: Calculate the frequency heard by the listener when the whistle is moving towards and away from the listener. The emitted frequency of the whistle is given as: - \( f = 500 \) Hz. #### For the maximum frequency (when the whistle is moving towards the listener): Using the Doppler effect formula: \[ f_{max} = \frac{c + V}{c - V} f \] Substituting the values: \[ f_{max} = \frac{340 + 50.27}{340 - 50.27} \times 500 \] Calculating: \[ f_{max} = \frac{390.27}{289.73} \times 500 \approx 586 \text{ Hz} \] #### For the minimum frequency (when the whistle is moving away from the listener): Using the Doppler effect formula: \[ f_{min} = \frac{c - V}{c + V} f \] Substituting the values: \[ f_{min} = \frac{340 - 50.27}{340 + 50.27} \times 500 \] Calculating: \[ f_{min} = \frac{289.73}{390.27} \times 500 \approx 436 \text{ Hz} \] ### Step 6: Conclusion The range of frequencies that the listener hears is from approximately 436 Hz to 586 Hz. ### Final Answer: The listener hears frequencies in the range of \( 436 \text{ Hz} \) to \( 586 \text{ Hz} \). ---
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