College Physics

Eugenia Etkina, Michael Gentle, Alan Van Heuvelen

Chapter 20

Mechanical Waves - all with Video Answers

Educators

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Chapter Questions

Problem 1

Imagine that you and your friend pull tight on the ends of a rope lying on a smooth floor. You shake your end of the rope, causing a transverse pulse. (a) Draw a displacement-versus-time graph representing the motion of your end of the rope. (b) Draw two displacement-versus-position graphs representing all of the points on the rope at two different clock readings.

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Problem 2

Design an experiment to determine whether longi- tudinal or transverse pulses have a higher speed on a Slinky. Describe the assumptions you make when determining the speed from the collected data. Discuss how experimental uncertainties will affect your decision.

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Problem 3

Imagine that you are standing in a swimming pool with a beach ball. You push the ball up and down several times. Draw a picture representing wave fronts in the pool. Explain what you mean by the words wave front.

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Problem 4

Explain the meaning of each symbol in the equation
$$y=A \cos \left[\frac{2 \pi}{T}\left(t-\frac{x}{v}\right)\right]$$
and summarize how we devised this equation.

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Problem 5

Tell all You have a sinusoidal wave that is described by the function
$$y=(0.2 \mathrm{m}) \sin \left[\frac{\pi}{2}\left(t+\frac{x}{20 \mathrm{m} / \mathrm{s}}\right)\right]$$
(a) Say everything you can about this wave. Pay attention to the positive sign in the equation. (b) Draw a $y$ -versus-t graph and a $y$ -versus-x graph for the wave.

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Problem 6

(a) Draw a position-versus-time graph for one coil of an infi- nitely long Slinky when a longitudinal wave of frequency 2.0 Hz and speed 3.0 m/s propagates on the Slinky. (b) Draw a displace-ment-versus-position graph for one particular time for a piece of the Slinky. (c) What is the wavelength of the wave?

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Problem 7

A longitudinal wave of amplitude 3.0 cm, frequency 2.0 Hz, and speed 3.0 m/s travels on an infinitely long Slinky. How far apart are the two nearest points on the Slinky that at one particular time both have the maximum displacements from their equilibrium positions? Explain your reasoning.

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Problem 8

A boat is moving up and down in the ocean with a period of 1.7 s caused by a wave traveling at a speed of 4.0 m/s. What other physical quantities relevant to this wave can you determine using this information? Determine them.

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Problem 9

A large goose lands in a lake and bobs up and down for a short time. A fisherman notices that the first wave created by the goose reaches the shore in 8.0 s. The distance between two wave crests is 80 cm, and in 2.0s he sees four waves hit the shore. How can the fisherman use these observations to determine how far from the shore the goose landed?

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Problem 10

Equation Jeopardy The equation below describes the varia- tion of pressure at different positions and times (relative to atmospheric pressure) caused by a sound wave:
$$\Delta P=\left(2.0 \mathrm{N} / \mathrm{m}^{2}\right) \cos \left[2 \pi\left(\frac{t}{0.010 \mathrm{s}}-\frac{x}{3.4 \mathrm{m}}\right)\right]$$
Say everything you can about this wave.

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Problem 11

Hearing People can hear sounds ranging in frequency from about 20 Hz to 20,000 Hz. Determine the wavelengths of the sounds at these two extremes. What assumptions are you making? If these assumptions are not correct, how will your answer change?

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Problem 12

A dolphin has a sonar system that emits sounds with a frequency of $2.0 \times 10^{5} \mathrm{Hz}$ . Determine everything you can about this sound wave. Remember that the sounds emitted by the dolphin travel in water.

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Problem 13

Antarctic ice Radio waves travel at a speed of $1.7 \times 10^{8} \mathrm{m} / \mathrm{s}$ through ice. A radio wave pulse sent into the Antarctic ice re- flects off the rock at the bottom and returns to the surface in $32.9 \times 10^{-6} \mathrm{s} .$ How deep is the ice? What assumptions did you make to solve the problem?

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Problem 14

Lightning and thunder You see a flash of lightning, and 2.4 s later you hear thunder coming from the same location. (a) Why is there this long delay? (b) How far away did the lightning flash occur? Justify any assumptions you make in your calculations.

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Problem 15

A pulse travels at speed v on a stretched rope. By what factor must you increase the force you exert on the rope to cause the speed to increase by a factor of 1.30?

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Problem 16

Two ropes have equal length and are stretched the same way. The speed of a pulse on rope 1 is 1.4 times the speed on rope 2. Determine the ratio of the masses of the two ropes $\left(m_{1} / m_{2}\right)$

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Problem 17

A telephone lineman is told to stretch the wire between two poles so the poles exert an 800-N force on the wire. As the lineman does not have a scale to measure forces, he decides to measure the speed of a pulse created in the wire when he hits it with a wrench. The pulse travels 60 m from one pole to the other and back again in 2.6 s. The 60-m wire has a mass of 15 kg. Should the wire be tightened or loosened? Explain. What assumptions did you make?

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Problem 18

The speed of a wave on a violin A string is 288 m/s and on the G string is 128 m/s. Use this information to determine the ratio of mass per unit length of the strings. What assumptions did you make?

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Problem 19

Describe two experiments to determine the speed of propagation of a transverse wave on a rope. You have the following tools to use: a stopwatch, a meter stick, a mass-measuring scale, and a force-measuring device. Use whatever other items you need for your experiments.

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Problem 20

Show using a sketch and mathematics that the intensity of a two-dimensional wave is inversely proportional to the distance from a source. Give three examples that will explain the meaning of “inversely proportional to the distance” in this case.

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Problem 21

Show using a sketch and mathematics that the intensity of a three-dimensional wave is inversely proportional to the distance squared from the source. Give three examples that will explain what it means to be inversely proportional to the distance squared in this case.

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Problem 22

The Sun radiates energy at a rate of about $4 \times 10^{26} \mathrm{W}$ . Estimate how much of the Sun's energy 1 $\mathrm{m}^{2}$ of Earth's surface facing the Sun receives in 1 $\mathrm{h}$ . Earth is about $150 \times 10^{6} \mathrm{km}$ from the Sun. What assumptions did you make?

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Problem 23

Use the data from the previous problem to estimate the energy coming to Earth from the Sun each second. The radius of Earth is about 6400 km. State your assumptions.

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Problem 24

Compare the intensity of a 100-W lightbulb while you are reading this book to the intensity of the Sun. What assumptions did you make?

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Problem 25

At what distance from the Sun is its intensity the same as that of a 100-W lightbulb that is 1.0 m from you?

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Problem 26

We can hear airplanes flying. Estimate the smallest power of the sound produced by the airplane engine that we can still hear. Explain your estimation method.

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Problem 27

Explain why the transverse pulse traveling on a rope held by two people reflects in the opposite orientation each time it reaches a person.

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Problem 28

Use your knowledge of waves to explain echoes. Use your explanation to devise a system to measure distances to objects that cannot be reached directly.

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Problem 29

A bat receives a reflected sound wave from a fly. If the reflected wave returns to the bat 0.042 s after it is sent, how far is the fly from the bat? Would you expect the reflected wave pulse to be in or out of phase with the incident wave pulse? Explain.

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Problem 30

A sound wave created by an explosion at Earth’s surface is reflected by a discontinuity of some type under Earth’s surface (Figure P20.30). Determine the distance from the surface to the discontinuity. Does the discontinuity have a greater or lesser impedance to sound than the surface above it? Explain. Sound travels at about 3000 m/s through the top layer of Earth.

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Problem 31

A 5.0-kg rope that is 20 m long is woven to an 8.0-kg rope that is 16 m long. The ropes are pulled taut and a pulse initiated in one is reflected at their interface. Draw a picture of what happens just after the pulse reaches the interface between the ropes.

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Problem 32

The pulses shown in Figure P20.32 (shown at time zero) are moving toward each other at speeds of 10 m/s. Draw y1x2 graphs showing the resultant pulse at 0.10 s, 0.20 s, and 0.30 s.

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Problem 33

Repeat the previous problem for the case where the pulse on the right is upright rather than inverted.

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Problem 34

Find a resultant wave for the two waves shown in Figure P20.34 at the instant they are traveling through the same medium.

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Problem 35

Two waves shown in Figure P20.35 at zero clock reading move toward each
other at a speed of 10 m/s. Draw graphs of the resultant wave at times of 0.10 s, 0.20 s, and 0.30 s.

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Problem 36

Use Huygens’ principle to find the shape of the wave fronts of a wave generated by the long edge of a flat piece of plastic floating horizontally and vibrating up and down in a swimming pool.

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Problem 37

Use Huygens’ principle and a wave front representation of waves to show that if you place a screen with a small circular hole in the path of a wave with flat wave fronts, the wave fronts beyond the screen will be circular.

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Problem 38

Use Huygens’ principle and a wave front representation of waves to find the locations in the swimming pool where no vibrations occur if two identical point-like objects synchronously vibrate in the pool. Assume the pool is infinitely large.

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Problem 39

You have two vibrating objects in an infinitely large pool. The distance between them is 6.0 m. Their frequency of vibration is 2.0 Hz and the wave speed is 4.0 m>s. The vibrations are sinusoidal. Find a location between them where the water does not vibrate and another location between them where the water vibrates with the largest amplitude.

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Problem 40

Describe an experiment to convince a friend that sound is a wave.

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Problem 41

The speed of sound in an ideal gas is given by the relationship
$$v=\sqrt{\frac{\gamma R T}{M}}$$
where $R=$ the universal gas constant $=8.314 \mathrm{J} / \mathrm{mol} \mathrm{K} ; T=$ the absolute temperature; $M=$ the molar mass of the gas in $\mathrm{kg} / \mathrm{mol} ;$ and $\gamma$ is a characteristic of the specific gas. For air, $\gamma=1.4$ and the average molar mass for dry air is 28.95 $\mathrm{g} /$ mol. (a) Show that the equation gives you correct units. (b) Give reasons why the temperature of the gas is in the numerator and the molar mass of the gas is in the denominator.

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Problem 42

Using the information from problem 41, calculate the speed of sound in the air. What assumptions are you making?

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Problem 43

Using the information in problem 41, estimate how much faster sound travels in summer than in winter. Explain how you arrived at your answer and the assumptions that you made.

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Problem 44

The energy of a sound wave is proportional to its amplitude squared. Why is it difficult to hear poolside sounds when swimming underwater?

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Problem 45

The sound intensity at a gasoline station next to a freeway averages $10^{-3} \mathrm{W} / \mathrm{m}^{2}$ The owner decides to collect this energy and convert it to thermal energy for heating his building. Assuming that this conversion is 100% efficient, what is the length of one side of a square sound collector that is needed to provide thermal energy at a rate of 500 W? Is this a practical idea for the owner? Explain.

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Problem 46

The sound intensity 5 km from the place where a supersonic jet takes off is 0.60 $\mathrm{W} / \mathrm{m}^{2} .$ Determine the area of a sound collector you would need to run a 40-W light-bulb from the energy collected. What might require you to create a larger collector?

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Problem 47

In music a very soft sound called “pianississimo” (ppp) might have an intensity of about $10^{-8} \mathrm{W} / \mathrm{m}^{2} .$ A very loud sound called fortississimo (fff) might have an intensity of about $10^{-2} \mathrm{W} / \mathrm{m}^{2}$ . Convert these intensities to intensity levels (units of $\mathrm{dB} )$

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Problem 48

Two sounds differ by 1 dB. What is the difference in their intensities?

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Problem 49

Calculate the change in intensity level when a sound intensity is increased by a factor of 8, by a factor of 80, and by a factor of 800.

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Problem 50

A banjo D string is 0.69 m long and has a fundamental frequency of 294 Hz. (a) Determine the speed of a wave or pulse on the string. (b) Identify three other frequencies at which the string can vibrate.

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Problem 51

How far from the end of the banjo string discussed in the previous problem must a fret be so that the string’s fundamental frequency is 330 Hz when you hold it down at the fret?

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Problem 52

A 0.33-m-long violin string has a mass of 0.89 g. The peg exerts a 45 N force on it. What can you determine about the sound produced by that string?

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Problem 53

A person secures a 5.0-m-long rope of mass 0.40 kg at one end and pulls on the rope, exerting a 120-N force. The rope vibrates in three segments with nodes separating each segment. List the physical quantities you can determine using this information and determine three of them.

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Problem 54

A canary sits 10 m from the edge of a 30-m-long clothesline, and a grackle sits 5 m from the other end. The rope is pulled by two poles that each exerts a 200-N force on it. The mass per unit length is 0.10 kg/m. At what frequency must you vibrate the line in order to dislodge the grackle while allowing the canary to sit undisturbed?

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Problem 55

Estimate the fundamental frequency of vibration of a telephone line between adjacent poles near where you live. Explain how you arrived at your answer.

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Problem 56

Two wires on a piano are the same length and are pulled by pegs that exert the same force on the wires. Wire 1 vibrates at a fundamental frequency 1.5 times that of wire $2 .$ Determine the ratio of their masses $\left(m_{1} / m_{2}\right)$

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Problem 57

By what percent does the frequency of a piano string change if the force that the peg exerts on it increases by 10%?

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Problem 58

(a) Determine the first three standing wave frequencies of a 40-cm-long open pipe. (b) Do the same for a 40-cm-long closed pipe.

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Problem 59

The 2779-m Brooklyn-Battery Tunnel, connecting Brooklyn and Manhattan, is one of the world’s longest underwater vehicular tunnels. (a) Determine its fundamental frequency of vibration. (b) What harmonic must be excited so that it resonates in the audio region at 20 Hz or greater?

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Problem 60

A wooden flute, open at both ends, is 0.48 m long. (a) Determine its fundamental vibration frequency. (b) How far from one end should a finger hole be placed to produce a sound whose frequency is four-thirds that calculated in part (a)? Be sure to justify how you arrive at your answer.

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Problem 61

A wooden flute, open at both ends, is 0.48 m long. (a) Determine its fundamental vibration frequency. (b) How far from one end should a finger hole be placed to produce a sound whose frequency is four-thirds that calculated in part (a)? Be sure to justify how you arrive at your answer.

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Problem 62

(a) Use the dimensions of a small soft drink bottle to estimate its fundamental resonant frequency when empty. (b) Determine the depth of water that must be added to increase its frequency by a factor of 4/3.

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Problem 63

The speed of sound can be measured using the apparatus shown in Figure P20.63. A 440-Hz tuning fork vibrating above a tube partially filled with water initiates sound waves in the tube. The air inside the tube vibrates at the same frequency as the tuning fork when the water in the tube is lowered 0.20 m and 0.60 m from the top of the tube. Use this information to determine the speed of sound in air.

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Problem 64

The fundamental frequency of a closed pipe, such as your vocal tract, is 240 Hz when filled with air (a mix of different molecules) that has molar mass of 29 g/mole. Using ideas from the kinetic theory of gases, estimate the vibration frequency when the vocal tract is filled with helium that has a molar mass of 4 g/mole.

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Problem 65

A source of ultrasound emits waves at a frequency of $2.00 \times 10^{6} \mathrm{Hz}$. The waves are reflected by red blood cells moving toward the source at a speed of 0.30 m/s. Determine the frequency of sound detected at a receiver next to the source. The speed of sound in the blood is 1500 m/s.

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Problem 66

A car horn vibrates at a frequency of 250 Hz. Determine the frequency a stationary observer hears as the car (a) approaches at a speed of 20 m/s and (b) departs at 20 m/s. If the car is stationary, what frequency is heard (c) by an observer approaching the car at 20 m/s and (d) by an observer departing from the car at 20 m/s?

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Problem 67

A car drives at a speed of 25 m/s along a road parallel to a railroad track. A train traveling at 15 m/s sounds a horn that vibrates at 300 Hz. (a) If the train and car are moving toward each other, what frequency of sound is heard by a person in the car? (b) If the train and car are moving away from each other, what frequency of sound is heard in the car?

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Problem 68

A whistle with frequency 400 Hz moves at speed 20 m/s in a horizontal circle at the end of a rotating stick. Determine the highest and lowest frequencies heard by a person riding a bicycle at speed 10 m/s toward the whistle.

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Problem 69

A bat emits short pulses of sound at a frequency of $1.60 \times 10^{5} \mathrm{Hz}$ . As the bat swoops toward a flat wall at speed 30 m/s, this sound is reflected from the wall back to the bat. (a) What is the frequency of sound incident on the wall? (b) Consider the wall as a sound source at the frequency calculated in part (a). What frequency of sound does the bat hear coming from the wall?

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Problem 70

A hungry student working in a cafeteria decides to eat from plates of food that pass on a conveyor belt. The plates are separated by 3.0 m, and the belt moves at a speed of 9.0 m/min. (a) How many plates of food does the student eat per minute? (b) As the student’s hunger is appeased, he moves with the belt at a speed of 6.0 m/min. Determine the number of plates of food that now reach the student each minute. (This change is similar to the Doppler shift for a moving listener.)

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Problem 71

A Doppler speed meter operating at exactly 1.02 * 105 Hz emits sound waves and detects the same waves after they are reflected from a baseball thrown by the pitcher. The receiver “mixes” the reflected wave with a small amount of the emitted wave and measures a beat frequency of 0.30 * 105 Hz. How fast is the ball moving?

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Problem 72

Use Huygens’ principle and a wave front representation of waves to show that a plane wave incident on a barrier traveling at an angle $\theta$ relative to a line perpendicular to the barrier (the so-called normal line) reflects, forming a wave front that travels at angle $\theta$ on the other side of the normal line.

Khoobchandra Agrawal

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Problem 73

Use Huygens’ principle and a wave front representation of waves to show that when a plane wave traveling in one medium hits a border with another different-speed medium that is not perpendicular to the direction of wave propagation, the wave changes its direction of propagation.

Khoobchandra Agrawal

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Problem 74

Describe an experiment that you can perform to measure the speed of sound in air using a graduated cylinder and a tuning fork that produces sound of a known frequency. Draw a picture. Carefully outline your experimental and mathematical procedure.

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Problem 75

A basketball player’s teammate shouts at her to catch a ball. Estimate the time required for the sound to travel a distance of 10 m, be detected by the ear, travel as a nerve impulse to the brain, be processed, then travel as a nerve signal for muscle action back to an arm, and finally cause a muscle contraction. Nerve impulses travel at a speed of about 120 m/s in humans. You will have to make reasonable assumptions about quantities not stated in the problem.

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Problem 76

While camping, you record a thunderclap whose intensity is $10^{-2} \mathrm{W} / \mathrm{m}^{2}$The clap reaches you 3.0 s after a flash of lightning. Estimate the total acoustical power generated by the bolt of lightning. Clearly state all assumptions that you made.

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Problem 77

A red blood cell travels at speed 0.40 m/s in a large artery. A sound of frequency $2.00 \times 10^{6} \mathrm{Hz}$ enters the blood opposite the direction of flow. (a) Determine the frequency of sound reflected from the cell and detected by a receiver. (b) If the emitted and received sounds are combined in the receiver, what beat frequency is measured?

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Problem 78

You have a bat-like echolocation system on your car, which emits a 20,000-Hz frequency sound that returns to the car in 0.18 s after being reflected by another stationary car. Which answer below is closest to the distance from your car to the other car?
(a) 10 $\mathrm{m}$ (b) 20 $\mathrm{m}$ (c) 30 $\mathrm{m}$
(d) 40 $\mathrm{m}$ (e) 50 $\mathrm{m}$

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Problem 79

If the car from Problem 78 is moving at 20 m/s toward the stationary car, which answer below is closest to the frequency of the reflected wave detected by the moving car from the stationary car?
(a) $18,000 \mathrm{Hz}$ (b) $19,000 \mathrm{Hz}$ (c) $20,000 \mathrm{Hz}$
(d) $21,000 \mathrm{Hz}$ (e) $23,000 \mathrm{Hz}$

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Problem 80

Which answer below is closest to the distance that the moving car in Problem 79 travels during the 0.18 s?
(a) 0.18 $\mathrm{m}$ (b) 0.36 $\mathrm{m}$ (c) 1.8 $\mathrm{m}$
(d) 3.6 $\mathrm{m}$ (e) 5.4 $\mathrm{m}$

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Problem 81

Compare your answers to Problems 78 and 80. If you want a more accurate measure of the object’s distance from you, you must consider the distance you travel during the time delay for the sound to get from you to the object and back to you. By approximately what percent does your change in position affect the distance calculated in Problem 78?
(a) 6$\%$ (b) 12$\%$ (c) 18$\%$
(d) 24$\%$ (e) 30$\%$

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Problem 82

While your car from Problem 78 is stationary, you emit a 20,000-Hz signal and get a 22,000-Hz signal back from a reflecting object. What can you say about the object?
(a) It is moving away from you.
(b) It is stationary.
(c) It is moving toward you.

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Problem 83

Your echolocation system has one transmitter in the middle of the front of the car and two detectors of reflected waves, one by each headlight (like the bat’s ears). You detect a reflected signal in the left detector slightly before you detect the signal from the right detector. Where must the reflecting object be?
(a) To the left of center
(b) Straight in front of you
(c) To the right of center

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Problem 84

The human ear can detect sound waves whose intensity is about $10^{-12} \mathrm{W} / \mathrm{m}^{2}$ . Where should you be with respect to a stereo speaker that produces sound of intensity $10^{-5} \mathrm{W} / \mathrm{m}^{2}$ when 1 $\mathrm{m}$ from the speaker if you do not want to listen to the music?
(a) About 3 m away
(b) About 30 m away
(c) About 3 km away
(d) It depends on the size of the speaker.

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Problem 85

What amplifies the air pressure in the ear?
(a) Pinna (2 times) and ossicles (30 times)
(b) Pinna (2 times), auditory canal (3 times), and ossicles (30 times)
(c) Pinna and auditory canal (2 times), the area change from the eardrum to the oval window (30 times), and the lever action of the ossicles (3 times)

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Problem 86

Where is the mechanism that allows the ear to distinguish between low-frequency and high-frequency sounds located?
(a) The fluid inside the cochlea of the inner ear
(b) In the oval window
(c) In the basilar membrane
(d) both b and c

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Problem 87

What frequency difference can the human ear distinguish near 1000 Hz?
(a) 100 Hz (b) 10 Hz (c) 3 Hz (d) 1 Hz

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Problem 88

The threshold for pressure variation of a barely audible sound is about $2 \times 10^{-5} \mathrm{N} / \mathrm{m}^{2}$ Which answer below is closest to the pressure variation of a barely audible sound in the cochlea of the inner ear?
(a) $1 \times 10^{-7} \mathrm{N} / \mathrm{m}^{2}$
(b) $1 \times 10^{-6} \mathrm{N} / \mathrm{m}^{2}$
(c) $2 \times 10^{-5} \mathrm{N} / \mathrm{m}^{2}$
(d) $4 \times 10^{-3} \mathrm{N} / \mathrm{m}^{2}$
(e) $4 \times 10^{-2} \mathrm{N} / \mathrm{m}^{2}$

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