College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 12

Sound - all with Video Answers

Educators

Chapter Questions

Problem 1

Bats emit ultrasonic waves with a frequency as high as $1.0 \times 10^{5} \mathrm{~Hz}$. What is the wavelength of such a wave in air of temperature $15^{\circ} \mathrm{C} ?$

Kayla Day

Kayla Day

Numerade Educator

Problem 2

Dolphins emit ultrasonic waves with a frequency as high as $2.5 \times 10^{5} \mathrm{~Hz}$. What is the wavelength of such a wave in seawater at $25^{\circ} \mathrm{C}$ ?

Kemuel Roberts

Numerade Educator

Problem 3

At a baseball game, a spectator is $60.0 \mathrm{~m}$ away from the batter. How long does it take the sound of the bat connecting with the ball to travel to the spectator's ears? The air temperature is $27.0^{\circ} \mathrm{C}$.

Kayla Day

Numerade Educator

Problem 4

A lightning flash is seen in the sky and $8.2 \mathrm{~s}$ later the boom of the thunder is heard. The temperature of the air is $12^{\circ} \mathrm{C}$.
(a) What is the speed of sound at that temperature? [Hint:
Light is an electromagnetic wave that travels at a speed of $3.00 \times 10^{8} \mathrm{~m} / \mathrm{s} .7(\mathrm{~b})$ How far away is the lightning strike?

Kemuel Roberts

Numerade Educator

Problem 5

During a thunderstorm, you can easily estimate your distance from a lightning strike. Count the number of seconds that elapse from when you see the flash of lightning to when you hear the thunder. The rule of thumb is that 5 s elapse for each mile of distance. Verify that this rule of thumb is (approximately) correct. (One mile is $1.6 \mathrm{~km}$ and light travels at a speed of $3 \times 10^{8} \mathrm{~m} / \mathrm{s}$.)

Kayla Day

Numerade Educator

Problem 6

A copper alloy has a Young's modulus of $1.1 \times 10^{11} \mathrm{~Pa}$ and a density of $8.92 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$. What is the speed of sound in a thin rod made from this alloy? Compare your result with that given in Table $12.1$

Kemuel Roberts

Numerade Educator

Problem 7

Find the speed of sound in mercury, which has a bulk modulus of $2.8 \times 10^{10} \mathrm{~Pa}$ and a density of $1.36 \times$ $10^{4} \mathrm{~kg} / \mathrm{m}^{3}$

Kayla Day

Numerade Educator

Problem 8

Derive Eq. (12-4) as: (a) Starting with Eq. (12-3), substitute $T=T_{\mathrm{C}}+273.15 .$ (b) Apply the binomial approximation to the square root (see Appendix A.5) and simplify.

Mayukh Banik

Numerade Educator

Problem 9

(a) Show that since the bulk modulus has SI units $\mathrm{N} / \mathrm{m}^{2}$ and mass density has SI units $\mathrm{kg} / \mathrm{m}^{3}$, Eq. (12-1) gives the speed of sound in $\mathrm{m} / \mathrm{s}$. Thus, the equation is dimensionally consistent. (b) Show that no other combination of $B$ and $\rho$ can give dimensions of speed. Thus, Eq. (12-
1) must be correct except for the possibility of a dimensionless constant.

Kayla Day

Numerade Educator

Problem 10

Stan and Ollie are standing next to a train track. Stan puts his ear to the steel track to hear the train coming. He hears the sound of the train whistle through the track $2.1$ s before Ollie hears it through the air. How far away is the train?

Kemuel Roberts

Numerade Educator

Problem 11

A sound wave with an intensity level of $80.0 \mathrm{~dB}$ is incident on an eardrum of area $0.600 \times 10^{-4} \mathrm{~m}^{2}$. How much
energy is absorbed by the eardrum in $3.0$ min?

Kayla Day

Numerade Educator

Problem 12

The sound level $25 \mathrm{~m}$ from a loudspeaker is $71 \mathrm{~dB}$. What is the rate at which sound energy is produced by the loudspeaker, assuming it to be an isotropic source?

Kemuel Roberts

Numerade Educator

Problem 13

In a factory, three machines produce noise with intensity levels of $85 \mathrm{~dB}, 90 \mathrm{~dB}$, and $93 \mathrm{~dB}$. When all three are running, what is the intensity level? How does this compare to running just the loudest machine?

Kayla Day

Numerade Educator

Problem 14

At the race track, one race car starts its engine with a resulting intensity level of $98.0 \mathrm{~dB}$ at point $P .$ Then seven more cars start their engines. If the other seven cars each produce the same intensity level at point $P$ as the first car, what is the new intensity level with all eight cars running?

Kemuel Roberts

Numerade Educator

Problem 15

(a) What is the pressure amplitude of a sound wave with an intensity level of $120.0 \mathrm{~dB}$ in air? (b) What force does this exert on an eardrum of area $0.550 \times 10^{-4} \mathrm{~m}^{2}$ ?

Kayla Day

Numerade Educator

Problem 16

An intensity level change of $+1.00 \mathrm{~dB}$ corresponds to what percentage change in intensity?

Mayukh Banik

Numerade Educator

Problem 17

(a) Show that if $I_{2}=10.0 I_{1}$, then $\beta_{2}=\beta_{1}+10.0 \mathrm{~dB}$. (A factor of 10 increase in intensity corresponds to a $10.0$ dB increase in intensity level.) (b) Show that if $I_{2}=2.0 I_{1}$, then $\beta_{2}=\beta_{1}+3.0 \mathrm{~dB}$. (A factor of 2 increase in intensity corresponds to a $3.0-\mathrm{dB}$ increase in intensity level. Whe tutorial: decibels)

Narayan Hari

Numerade Educator

Problem 18

At a rock concert, the engineer decides that the music isn't loud enough. He turns up the amplifiers so that the amplitude of the sound, where you're sitting, increases by $50.0 \%$. (a) By what percentage does the intensity increase?
(b) How does the intensity level (in dB) change?

Kemuel Roberts

Numerade Educator

Problem 19

Humans can hear sounds with frequencies up to about $20.0 \mathrm{kHz}$, but dogs can hear frequencies up to about $40.0 \mathrm{kHz}$. Dog whistles are made to emit sounds that dogs can hear but humans cannot. If the part of a dog whistle that actually produces the high frequency is made of a tube open at both ends, what is the longest possible length for the tube?

Kayla Day

Numerade Educator

Problem 20

(a) What should be the length of an organ pipe, closed at one end, if the fundamental frequency is to be $261.5 \mathrm{~Hz}$ ?
(b) What is the fundamental frequency of the organ pipe of part (a) if the temperature drops to $0.0^{\circ} \mathrm{C}$ ?

Kemuel Roberts

Numerade Educator

Problem 21

Repeat Problem 20 for an organ pipe that is open at both ends.

Kayla Day

Numerade Educator

Problem 22

An organ pipe that is open at both ends has a fundamental frequency of $382 \mathrm{~Hz}$ at $0.0^{\circ} \mathrm{C}$. What is the fundamental frequency for this pipe at $20.0^{\circ} \mathrm{C}$ ?

Kemuel Roberts

Numerade Educator

Problem 23

What is the length of the organ pipe in Problem 22 ?

Kayla Day

Numerade Educator

Problem 24

A certain pipe has resonant frequencies of $234 \mathrm{~Hz}$, $390 \mathrm{~Hz}$, and $546 \mathrm{~Hz}$, with no other resonant frequencies between these values. (a) Is this a pipe open at both ends or closed at one end? (b) What is the fundamental fre-
quency of this pipe?
(c) How long is this pipe?

Shoukat Ali

Other Schools

Problem 25

In an experiment to measure the speed of sound in air, standing waves are set up in a narrow pipe open at both ends using a speaker driven at $702 \mathrm{~Hz}$. The length of the pipe is $2.0 \mathrm{~m}$. What is the air temperature inside the pipe (assumed reasonably near room temperature, $20^{\circ} \mathrm{C}$ to $\left.35^{\circ} \mathrm{C}\right) ?$ [Hint: The standing wave is not necessarily the fundamental.]

Kayla Day

Numerade Educator

Problem 26

When a tuning fork is held over the open end of a very thin tube, as in Fig. 12.7, the smallest value of $L$ that produces resonance is found to be $30.0 \mathrm{~cm}$. (a) What is the wavelength of the sound? [Hint: Assume that the displacement antinode is at the open end of the tube.]
(b) What is the next larger value of $L$ that will produce resonance with the same tuning fork? (c) If the frequency of the tuning fork is $282 \mathrm{~Hz}$, what is the speed of sound in the tube?

Kemuel Roberts

Numerade Educator

Problem 27

Two tuning forks, A and B, excite the next-to-lowest resonant frequency in two air columns of the same length, but A's column is closed at one end and B's column is open at both ends. What is the ratio of A's frequency to B's frequency?

Kayla Day

Numerade Educator

Problem 28

How long a pipe is needed to make a tuba whose lowest note is low $\mathrm{C}$ (frequency $130.8 \mathrm{~Hz}$ )? Assume that a tuba is a long straight pipe open at both ends.

Kemuel Roberts

Numerade Educator

Problem 29

An aluminum rod, $1.0 \mathrm{~m}$ long, is held lightly in the middle. One end is struck head-on with a rubber mallet so that a longitudinal pulse-a sound wave-travels down the rod. The fundamental frequency of the longitudinal vibration is $2.55 \mathrm{kHz}$. (a) Describe the location of the node(s) and antinode(s) for the fundamental mode of vibration. Use either displacement or pressure nodes and antinodes. (b) Calculate the speed of sound in aluminum from the information given in the problem.
(c) The vibration of the rod produces a sound wave in air that can be heard. What is the wavelength of the sound wave in the air? Take the speed of sound in air to be $334 \mathrm{~m} / \mathrm{s}$. (d) Do the two ends of the rod vibrate longitudinally in phase or out of phase with each other? That is, at any given instant, do they move in the same direction or in opposite directions?

Kayla Day

Numerade Educator

Problem 30

A violin is tuned by adjusting the tension in the strings. Brian's A string is tuned to a slightly lower frequency than Jennifer's, which is correctly tuned to $440.0 \mathrm{~Hz}$.
(a) What is the frequency of Brian's string if beats of 2.0 $\mathrm{Hz}$ are heard when the two bow the strings together?
(b) Does Brian need to tighten or loosen his A string to get in tune with Jennifer? Explain.

Narayan Hari

Numerade Educator

Problem 31

A piano tuner sounds two strings simultaneously. One has been previously tuned to vibrate at $293.0 \mathrm{~Hz}$. The tuner hears $3.0$ beats per second. The tuner increases the tension on the as-yet untuned string, and now when they are played together the beat frequency is $1.0 \mathrm{~s}^{-1}$. (a) What was the original frequency of the untuned string? (b) By what percentage did the tuner increase the tension on that string?

Narayan Hari

Numerade Educator

Problem 32

An auditorium has organ pipes at the front and at the rear of the hall. Two identical pipes, one at the front and one at the back, have fundamental frequencies of $264.0 \mathrm{~Hz}$ at $20.0^{\circ} \mathrm{C}$. During a performance, the organ pipes at the back of the hall are at $25.0^{\circ} \mathrm{C}$, while those at the front are still at $20.0^{\circ} \mathrm{C}$. What is the beat frequency when the two pipes sound simultaneously?

Kemuel Roberts

Numerade Educator

Problem 33

A musician plays a string on a guitar that has a fundamental frequency of $330.0 \mathrm{~Hz}$. The string is $65.5 \mathrm{~cm}$ long and has a mass of $0.300 \mathrm{~g}$. (a) What is the tension in the string? (b) At what speed do the waves travel on the string? (c) While the guitar string is still being plucked, another musician plays a slide whistle that is closed at one end and open at the other. He starts at a very high frequency and slowly lowers the frequency until beats, with a frequency of $5 \mathrm{~Hz}$, are heard with the guitar. What is the fundamental frequency of the slide whistle with the slide in this position? (d) How long is the open tube in the slide whistle for this frequency?

Kayla Day

Numerade Educator

Problem 34

A cello string has a fundamental frequency of $65.40 \mathrm{~Hz}$. What beat frequency is heard when this cello string is bowed at the same time as a violin string with frequency of $196.0 \mathrm{~Hz}$ ? [Hint: The beats occur between the third harmonic of the cello string and the fundamental of the violin.]

Kemuel Roberts

Numerade Educator

Problem 35

An ambulance traveling at $44 \mathrm{~m} / \mathrm{s}$ approaches a car heading in the same direction at a speed of $28 \mathrm{~m} / \mathrm{s}$. The ambulance driver has a siren sounding at $550 \mathrm{~Hz}$. At what frequency does the driver of the car hear the siren?

Kayla Day

Numerade Educator

Problem 36

At a factory, a noon whistle is sounding with a frequency of $500 \mathrm{~Hz}$. As a car traveling at $85 \mathrm{~km} / \mathrm{h}$ approaches the factory, the driver hears the whistle at frequency $f_{i}$. After driving past the factory, the driver hears frequency $f_{\mathrm{f}}$. What is the change in frequency $f_{\mathrm{f}}-f_{\mathrm{i}}$ heard by the driver?

Kemuel Roberts

Numerade Educator

Problem 37

In parts of the midwestern United States, sirens sound when a severe storm that may produce a tornado is approaching. Mandy is walking at a speed of $1.56 \mathrm{~m} / \mathrm{s}$ directly toward one siren and directly away from another siren when they both begin to sound with a frequency of $698 \mathrm{~Hz}$. What beat frequency does Mandy hear? ( W tutorial: Doppler effect)

Kayla Day

Numerade Educator

Problem 38

A source of sound waves of frequency $1.0 \mathrm{kHz}$ is traveling through the air at $0.50$ times the speed of sound.
(a) Find the frequency of the sound received by a stationary observer if the source moves toward her.
(b) Repeat if the source moves away from her instead.

Kemuel Roberts

Numerade Educator

Problem 39

A source of sound waves of frequency $1.0 \mathrm{kHz}$ is stationary. An observer is traveling at $0.50$ times the speed of sound. (a) What is the observed frequency if the observer moves toward the source? (b) Repeat if the observer moves away from the source instead.

Kayla Day

Numerade Educator

Problem 40

A child swinging on a swing set hears the sound of a whistle that is being blown directly in front of her. At the bottom of her swing when she is moving toward the whistle, she hears a higher pitch, and at the bottom of her swing when she is moving away from the swing she hears a lower pitch. The higher pitch has a frequency that is $5.0 \%$ higher than the lower pitch. What is the speed of the child at the bottom of the swing?

Mayukh Banik

Numerade Educator

Problem 41

A source and an observer are each traveling at $0.50$ times the speed of sound. The source emits sound waves at $1.0 \mathrm{kHz}$. Find the observed frequency if (a) the source and observer are moving toward each other; (b) the source and observer are moving away from each other; (c) the source and observer are moving in the same direction.

Shoukat Ali

Other Schools

Problem 42

Blood flow rates can be found by measuring the Doppler shift in frequency of ultrasound reflected by red blood cells (known as angiodynography). If the speed of the red blood cells is $v$, the speed of sound in blood is $u$, the ultrasound source emits waves of frequency $f$, and we assume that the blood cells are moving directly toward the ultrasound source, show that the frequency $f_{\mathrm{r}}$ of reflected waves detected by the apparatus is given by
$$
f_{\mathrm{r}}=f \frac{1+v / u}{1-v / u}
$$
[Hint: There are two Doppler shifts. A red blood cell first acts as a moving observer; then it acts as a moving source when it reradiates the reflected sound at the same frequency that it received.]

Kemuel Roberts

Numerade Educator

Problem 43

Show that for a moving source, the fractional shift in observed frequency is equal to $v_{s} / v$, the source's speed as a fraction of the speed of sound. [Hint: Use the binomial approximation from Appendix A.5.]

Kayla Day

Numerade Educator

Problem 44

The pitch of the sound from a race car engine drops the musical interval of a fourth when it passes the spectators. This means the frequency of the sound after passing is $0.75$ times what it was before. How fast is the race car moving?

Mayukh Banik

Numerade Educator

Problem 45

A ship is lost in a dense fog in a Norwegian fjord that is $1.80 \mathrm{~km}$ wide. The air temperature is $5.0^{\circ} \mathrm{C}$. The captain fires a pistol and hears the first echo after $4.0 \mathrm{~s}$. (a) How

Narayan Hari

Numerade Educator

Problem 46

A ship mapping the depth of the ocean emits a sound of $38 \mathrm{kHz}$. The sound travels to the ocean floor and returns $0.68$ s later. (a) How deep is the water at that location?
(b) What is the wavelength of the wave in water? (c) What is the wavelength of the reflected wave as it travels into the air, where the speed of sound is $350 \mathrm{~m} / \mathrm{s}$ ?

Kemuel Roberts

Numerade Educator

Problem 47

A boat is using sonar to detect the bottom of a freshwater lake. If the echo from a sonar signal is heard $0.540 \mathrm{~s}$ after it is emitted, how deep is the lake? Assume the temperature of the lake is uniform and at $25^{\circ} \mathrm{C}$.

Kayla Day

Numerade Educator

Problem 48

A geological survey ship mapping the floor of the ocean sends sound pulses down from the surface and measures the time taken for the echo to return. How deep is the ocean at a point where the echo time (down and back) is $7.07 \mathrm{~s}$ ? The temperature of the seawater is $25^{\circ} \mathrm{C}$.

Mayukh Banik

Numerade Educator

Problem 49

A bat emits chirping sounds of frequency $82.0 \mathrm{kHz}$ while hunting for moths to eat. If the bat is flying toward the moth at a speed of $4.40 \mathrm{~m} / \mathrm{s}$ and the moth is flying away from the bat at $1.20 \mathrm{~m} / \mathrm{s}$, what is the frequency of the sound wave reflected from the moth as observed by the bat? Assume it is a cool night with a temperature of $10.0^{\circ} \mathrm{C}$. [Hint: There are two Doppler shifts. Think of the moth as a receiver, which then becomes a source as it "retransmits" the reflected wave.]

Kayla Day

Numerade Educator

Problem 50

The bat of Problem 49 emits a chirp that lasts for $2.0 \mathrm{~ms}$ and then is silent while it listens for the echo. If the beginning of the echo returns just after the outgoing chirp is finished, how close to the moth is the bat? [Hint:
Is the change in distance between the two significant during a 2.0-ms time interval?]

Mayukh Banik

Numerade Educator

Problem 51

Doppler ultrasound is used to measure the speed of blood flow (see Problem 42). The reflected sound interferes with the emitted sound, producing beats. If the speed of red blood cells is $0.10 \mathrm{~m} / \mathrm{s}$, the ultrasound frequency used is $5.0 \mathrm{MHz}$, and the speed of sound in blood is $1570 \mathrm{~m} / \mathrm{s}$, what is the beat frequency?

Kayla Day

Numerade Educator

Problem 52

(a) In Problem 42, find the beat frequency between the outgoing and reflected sound waves. (b) Show that the beat frequency is proportional to the speed of the blood cell if $v \ll u .$ [Hint: Use the binomial approximation from Appendix A.5.]

Mayukh Banik

Numerade Educator

Problem 53

A $30.0$ -cm-long string has a mass of $0.230 \mathrm{~g}$ and is vibrating at its next-to-lowest natural frequency $f_{2}$. The tension in the string is $7.00 \mathrm{~N}$. (a) What is $f_{2}$ ? (b) What are the frequency and wavelength of the sound in the surrounding air if the speed of sound is $350 \mathrm{~m} / \mathrm{s}$ ?

Kayla Day

Numerade Educator

Problem 54

Kyle is climbing a sailboat mast and is $5.00 \mathrm{~m}$ above the surface of the ocean, while his friend Rob is scuba diving below the boat. Kyle shouts to someone on another boat and Rob hears him shout $0.0210 \mathrm{~s}$ later. The ocean temperature is $25^{\circ} \mathrm{C}$ and the air is at $20^{\circ} \mathrm{C}$. How deep is Rob below the boat?

Narayan Hari

Numerade Educator

Problem 55

What are the four lowest standing wave frequencies for an organ pipe that is $4.80 \mathrm{~m}$ long and closed at one end?

Kayla Day

Numerade Educator

Problem 56

The length of the auditory canal in humans averages about $2.5 \mathrm{~cm}$. What are the lowest three standing wave frequencies for a pipe of this length open at one end? What effect might resonance have on the sensitivity of the ear at various frequencies? (Refer to Fig. 12.12. Note that frequencies critical to speech recognition are in the range 2 to $5 \mathrm{kHz}$.

Shoukat Ali

Other Schools

Problem 57

Some bats determine their distance to an object by detecting the difference in intensity between echoes.(a) If intensity falls off at a rate that is inversely proportional to the distance squared, show that the echo intensity is inversely proportional to the fourth power of distance. (b) The bat was originally $0.60 \mathrm{~m}$ from one object and $1.10 \mathrm{~m}$ from another. After flying closer, it is now $0.50 \mathrm{~m}$ from the first and at $1.00 \mathrm{~m}$ from the second object. What is the percentage increase in the intensity of the echo from each object?

Shoukat Ali

Other Schools

Problem 58

The Vespertilionidae family of bats detect the distance to an object by timing how long it takes for an emitted signal to reflect off the object and return. Typically they emit sound pulses $3 \mathrm{~ms}$ long and $70 \mathrm{~ms}$ apart while cruising.
(a) If an echo is heard $60 \mathrm{~ms}$ later $\left(v_{\text {sound }}=331 \mathrm{~m} / \mathrm{s}\right)$, how far away is the object? (b) When an object is only $30 \mathrm{~cm}$ away, how long will it be before the echo is heard?
(c) Will the bat be able to detect this echo?

Mayukh Banik

Numerade Educator

Problem 59

At what frequency $f$ does a sound wave in air have a wavelength of $15 \mathrm{~cm}$, about half the diameter of the human head? Some methods of localization work well only for frequencies below $f$, while others work well only above $f$. (See Conceptual Questions 4 and 5.)

Kayla Day

Numerade Educator

Problem 60

Horseshoe bats use the Doppler effect to determine their location. A Horseshoe bat flies toward a wall at a speed of $15 \mathrm{~m} / \mathrm{s}$ while emitting a sound of frequency $35 \mathrm{kHz}$. What is the beat frequency between the emission frequency and the echo?

Mayukh Banik

Numerade Educator

Problem 61

According to a treasure map, a treasure lies at a depth of $40.0$ fathoms on the ocean floor due east from the lighthouse. The treasure hunters use sonar to find where the depth is $40.0$ fathoms as they head east from the lighthouse. What is the elapsed time between an emitted pulse and the return of its echo at the correct depth if the water temperature is $25^{\circ} \mathrm{C}$ ? [Hint: One fathom is $1.83 \mathrm{~m}$.]

Kayla Day

Numerade Educator

Problem 62

When playing fortissimo (very loudly), a trumpet emits sound energy at a rate of $0.800 \mathrm{~W}$ out of a bell (opening) of diameter $12.7 \mathrm{~cm}$. (a) What is the sound intensity level right in front of the trumpet? (b) If the trumpet radiates sound waves uniformly in all directions, what is the sound intensity level at a distance of $10.0 \mathrm{~m}$ ?

Mayukh Banik

Numerade Educator

Problem 63

One cold and windy winter day, Zach notices a humming sound coming from his chimney when the chimney is open at the top and closed at the bottom. He opens the chimney at the bottom and notices that the sound changes. He goes over to the piano to try to match the note that the chimney is producing with the bottom open. He finds that the "C" three octaves below middle " $\mathrm{C}^{\prime \prime}$ matches the chimney's fundamental frequency. Zach knows that the frequency of middle "C" is $261.6 \mathrm{~Hz}$, and each lower octave is $\frac{1}{2}$ of the frequency of the octave above. From this information, Zach finds the height of the chimney and the fundamental frequency of the note that was produced when the chimney was closed at the bottom. Assuming that the speed of sound in the cold air is $330 \mathrm{~m} / \mathrm{s}$, reproduce Zach's calculations to find (a) the height of the chimney and (b) the fundamental frequency of the chimney when it is closed at the bottom.

Kayla Day

Numerade Educator

Problem 64

A periodic wave is composed of the superposition of three sine waves whose frequencies are 36,60, and $84 \mathrm{~Hz}$. The speed of the wave is $180 \mathrm{~m} / \mathrm{s}$. What is the wavelength of the wave? [Hint: The $36 \mathrm{~Hz}$ is not necessarily the fundamental frequency.]

Mayukh Banik

Numerade Educator

Problem 65

Analysis of the periodic sound wave produced by a violin's G string includes three frequencies: 392,588 , and $980 \mathrm{~Hz}$. What is the fundamental frequency? [Hint: The wave on the string is the superposition of several different standing wave patterns.]

Kayla Day

Numerade Educator

Problem 66

Your friend needs advice on her newest "acoustic sculpture." She attaches one end of a steel wire, of diameter $4.00 \mathrm{~mm}$ and density $7860 \mathrm{~kg} / \mathrm{m}^{3}$, to a wall. After passing over a pulley, located $1.00 \mathrm{~m}$ from the wall, the other end of the wire is attached to a hanging weight. Below the horizontal length of wire she places a $1.50$ -m-long hollow tube, open at one end and closed at the other. Once the sculpture is in place, air will blow through the tube, creating a sound. Your friend wants this sound to cause the steel wire to oscillate at the same resonant frequency as the tube. What weight (in newtons) should she hang from the wire if the temperature is $18.0^{\circ} \mathrm{C}$ ?

Mayukh Banik

Numerade Educator

Problem 67

A sound wave arriving at your ear is transferred to the fluid in the cochlea. If the intensity in the fluid is $0.80$ times that in air and the frequency is the same as for the wave in air, what will be the ratio of the pressure amplitude of the wave in air to that in the fluid? Approximate the fluid as having the same values of density and speed of sound as water.

Kayla Day

Numerade Educator

Problem 68

In this problem, you will estimate the smallest kinetic energy of vibration that the human ear can detect. Suppose that a harmonic sound wave at the threshold of hearing $\left(I=1.0 \times 10^{-12} \mathrm{~W} / \mathrm{m}^{2}\right)$ is incident on the eardrum. The speed of sound is $340 \mathrm{~m} / \mathrm{s}$ and the density of air is $1.3 \mathrm{~kg} / \mathrm{m}^{3}$. (a) What is the maximum speed of an element of air in the sound wave? [Hint: See Eq. (10-21).]
(b) Assume the eardrum vibrates with displacement $s_{0}$ at angular frequency $\omega$; its maximum speed is then equal to the maximum speed of an air element. The mass of the eardrum is approximately $0.1 \mathrm{~g}$. What is the average kinetic energy of the eardrum? (c) The average kinetic energy of the eardrum due to collisions with air molecules in the absence of a sound wave is about $10^{-20} \mathrm{~J} .$ Compare your answer with (b) and discuss.

Mayukh Banik

Numerade Educator

Problem 69

During a rehearsal, all eight members of the first violin section of an orchestra play a very soft passage. The sound intensity level at a certain point in the concert hall is $38.0 \mathrm{~dB}$. What is the sound intensity level at the same point if only one of the violinists plays the same passage? [Hint: When playing together, the violins are incoherent sources of sound.

Kayla Day

Numerade Educator