Textbook Question
A factory whistle emits sound of frequency 770 Hz. The wind velocity is 15.0 m/s from the north (heading south). What frequency will observers hear who are located, at rest, due south, of the whistle?
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18. Waves & Sound
The Doppler Effect
8:30 minutesProblem 75
Textbook Question
A physics professor demonstrates the Doppler effect by tying a 600 Hz sound generator to a 1.0-m-long rope and whirling it around her head in a horizontal circle at 100 rpm. What are the highest and lowest frequencies heard by a student in the classroom?
Verified step by step guidance1
Determine the angular velocity of the sound generator. Convert the given rotational speed from revolutions per minute (rpm) to radians per second using the formula: \( \omega = \frac{2\pi \cdot \text{rpm}}{60} \).
Calculate the tangential speed of the sound generator. Use the formula \( v = r \cdot \omega \), where \( r \) is the radius of the circular motion (1.0 m) and \( \omega \) is the angular velocity calculated in the previous step.
Understand the Doppler effect. The highest frequency is heard when the sound source is moving directly toward the observer, and the lowest frequency is heard when the source is moving directly away. Use the Doppler effect formula for sound: \( f' = f \cdot \frac{v_{\text{sound}}}{v_{\text{sound}} \mp v_{\text{source}}} \), where \( f \) is the source frequency (600 Hz), \( v_{\text{sound}} \) is the speed of sound in air (assume 343 m/s unless otherwise stated), and \( v_{\text{source}} \) is the tangential speed of the source.
Substitute the values into the Doppler effect formula. For the highest frequency, use the minus sign in the denominator (approaching observer), and for the lowest frequency, use the plus sign (receding observer).
Simplify the expressions for the highest and lowest frequencies. These will give the maximum and minimum frequencies heard by the student in the classroom.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Doppler Effect
The Doppler effect refers to the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. When the source of sound moves towards an observer, the frequency increases, resulting in a higher pitch. Conversely, when the source moves away, the frequency decreases, leading to a lower pitch. This phenomenon is crucial for understanding how the motion of the sound generator affects the perceived frequency by the student.
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Centripetal Motion
Centripetal motion describes the motion of an object moving in a circular path, where a force acts towards the center of the circle. In this scenario, the sound generator is being whirled in a horizontal circle, which means it experiences centripetal acceleration. The speed of the generator and the radius of the circular path influence how the sound waves are emitted and perceived, impacting the frequencies heard by the observer.
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Frequency Shift Calculation
To determine the highest and lowest frequencies heard by the student, one must calculate the frequency shift due to the Doppler effect. This involves using the formula for the observed frequency, which accounts for the speed of sound, the speed of the source, and the direction of motion. By calculating the maximum and minimum speeds of the sound generator as it moves towards and away from the observer, one can find the corresponding highest and lowest frequencies.
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