Ch 16: Traveling Waves

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 16: Traveling Waves

Problem 75

Chapter 16, Problem 75

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 guidance

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.

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.

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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Textbook Question

A communications truck with a 44-cm-diameter dish receiver on the roof starts out 10 km from its base station. It drives directly away from the base station at 50 km/h for 1.0 h, keeping the receiver pointed at the base station. The base station antenna broadcasts continuously with 2.5 kW of power, radiated uniformly in all directions. How much electromagnetic energy does the truck's dish receive during that 1.0 h?

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Textbook Question

An avant-garde composer wants to use the Doppler effect in his new opera. As the soprano sings, he wants a large bat to fly toward her from the back of the stage. The bat will be outfitted with a microphone to pick up the singer's voice and a loudspeaker to rebroadcast the sound toward the audience. The composer wants the sound the audience hears from the bat to be, in musical terms, one half-step higher in frequency than the note they are hearing from the singer. Two notes a half-step apart have a frequency ratio of 2^1/12 = 1.059. With what speed must the bat fly toward the singer?

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Textbook Question

A battery-powered siren emits 0.50 W of sound power at 1000 Hz. It is dropped from 100 m directly over your head on a 20°C day. 4.0 s after it is released, what are (a) the frequency and (b) the sound intensity level you hear?

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Textbook Question

A loudspeaker, mounted on a tall pole, is engineered to emit 75% of its sound energy into the forward hemisphere, 25% toward the back. You measure an 85 dB sound intensity level when standing 3.5 m in front of and 2.5 m below the speaker. What is the speaker's power output?

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Textbook Question

LASIK eye surgery uses pulses of laser light to shave off tissue from the cornea, reshaping it. A typical LASIK laser emits a 1.0-mm-diameter laser beam with a wavelength of 193 nm. Each laser pulse lasts 15 ns and contains 1.0 mJ of light energy. During the very brief time of the pulse, what is the intensity of the light wave?

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Textbook Question

Some modern optical devices are made with glass whose index of refraction changes with distance from the front surface. FIGURE P16.72 shows the index of refraction as a function of the distance into a slab of glass of thickness L. The index of refraction increases linearly from n₁ at the front surface to n₂ at the rear surface. Evaluate your expression for a 1.0-cm-thick piece of glass for which n₁ = 1.50 and n₂ = 1.60.