University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 15

Mechanical Waves - all with Video Answers

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Chapter Questions

Problem 1

The speed of sound in air at $20^{\circ} \mathrm{C}$ is $344 \mathrm{~m} / \mathrm{s}$. (a) What is the wavelength of a sound wave with a frequency of $784 \mathrm{~Hz}$, corresponding to the note $\mathrm{G}_{5}$ on a piano, and how many milliseconds does each vibration take? (b) What is the wavelength of a sound wave one octave higher (twice the frequency) than the note in part (a)?

Anonymous User

Anonymous User

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Problem 2

Ultrasound Imaging. Sound having frequencies above the range of human hearing (about $20,000 \mathrm{~Hz}$ ) is called ultrasound. Waves above this frequency can be used to penetrate the body and to produce images by reflecting from surfaces. In a typical ultrasound scan, the waves travel through body tissue with a speed of $1500 \mathrm{~m} / \mathrm{s}$ For a good, detailed image, the wavelength should be no more than 1.0 $\mathrm{mm} .$ What frequency sound is required for a good scan?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 3

Tsunami! On December $26,2004,$ a great earthquake occurred off the coast of Sumatra and triggered immense waves (tsunami) that killed more than 200,000 people. Satellites observing these waves from space measured $800 \mathrm{~km}$ from one wave crest to the next and a period between waves of 1.0 hour. What was the speed of these waves in $\mathrm{m} / \mathrm{s}$ and in $\mathrm{km} / \mathrm{h}$ ? Does your answer help you understand why the waves caused such devastation?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 4

A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes $2.5 \mathrm{~s}$ for the boat to travel from its highest point to its lowest, a total distance of $0.53 \mathrm{~m} .$ The fisherman sees that the wave crests are spaced $4.8 \mathrm{~m}$ apart. (a) How fast are the waves traveling? (b) What is the amplitude of each wave? (c) If the total vertical distance traveled by the boat were $0.30 \mathrm{~m}$ but the other data remained the same, how would the answers to parts (a) and (b) change?

Averell Hause

Carnegie Mellon University

Problem 5

(a) Audible wavelengths. The range of audible frequencies is from about $20 \mathrm{~Hz}$ to $20,000 \mathrm{~Hz}$. What is the range of the wavelengths of audible sound in air? (b) Visible light. The range of visible light extends from $380 \mathrm{nm}$ to $750 \mathrm{nm}$. What is the range of visible frequencies of light? (c) Brain surgery. Surgeons can remove brain tumors by using a cavitron ultrasonic surgical aspirator, which produces sound waves of frequency $23 \mathrm{kHz}$. What is the wavelength of these waves in air? (d) Sound in the body. What would be the wavelength of the sound in part (c) in bodily fluids in which the speed of sound is $1480 \mathrm{~m} / \mathrm{s}$ but the frequency is unchanged?

Anonymous User

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Problem 6

A small bead of mass $4.00 \mathrm{~g}$ is attached to a horizontal string. Transverse waves of amplitude $A=0.800 \mathrm{~cm}$ and frequency $f=20.0 \mathrm{~Hz}$ are set up on the string. Assume the mass of the bead is small enough that the bead doesn't alter the wave motion. During the wave motion, what is the maximum vertical force that the string exerts on the bead?

Bradley Abell

Numerade Educator

Problem 7

Transverse waves on a string have wave speed $8.00 \mathrm{~m} / \mathrm{s},$ amplitude $0.0700 \mathrm{~m},$ and wavelength $0.320 \mathrm{~m} .$ The waves travel in the $-x$ -direction, and at $t=0$ the $x=0$ end of the string has its maximum upward displacement. (a) Find the frequency, period, and wave number of these waves. (b) Write a wave function describing the wave.
(c) Find the transverse displacement of a particle at $x=0.360 \mathrm{~m}$ at time $t=0.150 \mathrm{~s}$. (d) How much time must elapse from the instant in part (c) until the particle at $x=0.360 \mathrm{~m}$ next has maximum upward displacement?

Anonymous User

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Problem 8

A certain transverse wave is described by
$$
y(x, t)=(6.50 \mathrm{~mm}) \cos 2 \pi\left(\frac{x}{28.0 \mathrm{~cm}}-\frac{t}{0.0360 \mathrm{~s}}\right)
$$
Determine the wave's (a) amplitude; (b) wavelength; (c) frequency; (d) speed of propagation; (e) direction of propagation.

Averell Hause

Carnegie Mellon University

Problem 9

Which of the following wave functions satisfies the wave equation, Eq. (15.12)? (a) $y(x, t)=A \cos (k x+\omega t)$; (b) $y(x, t)=A \sin (k x+\omega t) ;$ (c) $y(x, t)=A(\cos k x+\cos \omega t)$. (d) For the wave of part (b), write the equations for the transverse velocity and transverse acceleration of a particle at point $x$.

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 10

A water wave traveling in a straight line on a lake is described by the equation
$$
y(x, t)=(2.75 \mathrm{~cm}) \cos (0.410 \mathrm{rad} / \mathrm{cm} x+6.20 \mathrm{rad} / \mathrm{s} t)
$$
where $y$ is the displacement perpendicular to the undisturbed surface of the lake. (a) How much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor, and what horizontal distance does the wave crest travel in that time? (b) What are the wave number and the number of waves per second that pass the fisherman? (c) How fast does a wave crest travel past the fisherman, and what is the maximum speed of his cork floater as the wave causes it to bob up and down?

Averell Hause

Carnegie Mellon University

Problem 11

A sinusoidal wave is propagating along a stretched string that lies along the $x$ -axis. The displacement of the string as a function of time is graphed in Fig. E15.11 for particles at $x=0$ and at $x=$ $0.0900 \mathrm{~m}$. (a) What is the amplitude of the wave?
(b) What is the period of the wave?
(c) You are told that the two points $x=0$ and $x=0.0900 \mathrm{~m}$ are within one wavelength of each other. If the wave is moving in the $+x$ -direction, determine the wavelength and the wave speed.
(d) If instead the wave is moving in the $-x$ -direction, determine the wavelength and the wave speed. (e) Would it be possible to determine definitively the wavelengths in parts (c) and (d) if you were not told that the two points were within one wavelength of each other? Why or why not?

Sheh Lit Chang

University of Washington

Problem 12

(a) Show that Eq. (15.3) may be written as
$$
y(x, t)=A \cos \left[\frac{2 \pi}{\lambda}(x-v t)\right]
$$
(b) Use $y(x, t)$ to find an expression for the transverse velocity $v_{y}$ of a particle in the string on which the wave travels. (c) Find the maximum speed of a particle of the string. Under what circumstances is this equal to the propagation speed $v ?$ Less than $v ?$ Greater than $v ?$

Inder Jeet

Numerade Educator

Problem 13

A transverse wave on a string has amplitude $0.300 \mathrm{~cm}$, wavelength $12.0 \mathrm{~cm},$ and speed $6.00 \mathrm{~cm} / \mathrm{s} .$ It is represented by $y(x, t)$ as given in Exercise 15.12 . (a) At time $t=0$, compute $y$ at $1.5 \mathrm{~cm}$ intervals of $x$ (that is, at $x=0, x=1.5 \mathrm{~cm}, x=3.0 \mathrm{~cm},$ and so on ) from $x=0$ to $x=12.0 \mathrm{~cm} .$ Graph the results. This is the shape of the string at time $t=0 .$ (b) Repeat the calculations for the same values of $x$ at times $t=0.400 \mathrm{~s}$ and $t=0.800 \mathrm{~s}$. Graph the shape of the string at these instants. In what direction is the wave traveling?

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Problem 14

A musical novice learns that doubling the fundamental increases the pitch by one octave and decides to do that to a guitar string. What factor increase in tension would be necessary? Do you think this would be a good idea?

David González Cornejo

David González Cornejo

Numerade Educator

Problem 15

One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at $120 \mathrm{~Hz}$. The other end passes over a pulley and supports a $1.50 \mathrm{~kg}$ mass. The linear mass density of the rope is $0.0480 \mathrm{~kg} / \mathrm{m} .$ (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would your answers to parts (a) and (b) change if the mass were increased to $3.00 \mathrm{~kg} ?$

Anonymous User

Numerade Educator

Problem 16

With what tension must a rope with length $2.50 \mathrm{~m}$ and mass $0.120 \mathrm{~kg}$ be stretched for transverse waves of frequency $40.0 \mathrm{~Hz}$ to have a wavelength of $0.750 \mathrm{~m} ?$

Averell Hause

Carnegie Mellon University

Problem 17

The upper end of a 3.8 -m-long steel wire is fastened to the ceiling, and a $54 \mathrm{~kg}$ object is suspended from the lower end of the wire. You observe that it takes a transverse pulse $0.049 \mathrm{~s}$ to travel from the bottom to the top of the wire. What is the mass of the wire?

Inder Jeet

Numerade Educator

Problem 18

A $1.50 \mathrm{~m}$ string of weight $0.0125 \mathrm{~N}$ is tied to the ceiling at its upper end, and the lower end supports a weight $W .$ Ignore the very small variation in tension along the length of the string that is produced by the weight of the string. When you pluck the string slightly, the waves traveling up the string obey the equation
$$
y(x, t)=(8.50 \mathrm{~mm}) \cos (172 \mathrm{rad} / \mathrm{m} x-4830 \mathrm{rad} / \mathrm{s} t)
$$
Assume that the tension of the string is constant and equal to $W$.
(a) How much time does it take a pulse to travel the full length of the string? (b) What is the weight $W ?$ (c) How many wavelengths are on the string at any instant of time?
(d) What is the equation for waves traveling down the string?

Averell Hause

Carnegie Mellon University

Problem 19

A thin, $75.0 \mathrm{~cm}$ wire has a mass of $16.5 \mathrm{~g}$. One end is tied to a nail, and the other end is attached to a screw that can be adjusted to vary the tension in the wire. (a) To what tension (in newtons) must you adjust the screw so that a transverse wave of wavelength $3.33 \mathrm{~cm}$ makes 625 vibrations per second? (b) How fast would this wave travel?

Anonymous User

Numerade Educator

Problem 20

A heavy rope $6.00 \mathrm{~m}$ long and weighing $29.4 \mathrm{~N}$ is attached at one end to a ceiling and hangs vertically. A $0.500 \mathrm{~kg}$ mass is suspended from the lower end of the rope. What is the speed of transverse waves on the rope at the (a) bottom of the rope, (b) middle of the rope, and (c) top of the rope? (d) Is the tension in the middle of the rope the average of the tensions at the top and bottom of the rope? Is the wave speed at the middle of the rope the average of the wave speeds at the top and bottom? Explain.

Averell Hause

Carnegie Mellon University

Problem 21

In Example 15.4 the average power that Throckmorton puts into the clothesline is small, about 1 W. How much better can you do? Assume a rope that has linear mass density $0.500 \mathrm{~kg} / \mathrm{m},$ so $1 \mathrm{~m}$ of the rope weighs about 1 lb. The rope is long and is attached to a post at one end. You hold the other end of the rope in your hand and supply sinusoidal wave pulses by moving your arm up and down. Estimate the amplitude of the pulses to be the length of your arm. Estimate the maximum tension you can supply to the rope by pulling on it horizontally, and estimate the time for you to complete each pulse.
(a) Ignoring any effects from reflection of the pulses from the other end of the rope, what average power can you supply to the rope?
(b) You try to increase your power output by halving the amplitude so you can double the frequency of the pulses. What change in $P_{\text {av }}$ does this produce?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

A piano wire with mass $3.00 \mathrm{~g}$ and length $80.0 \mathrm{~cm}$ is stretched with a tension of $25.0 \mathrm{~N}$. A wave with frequency $120.0 \mathrm{~Hz}$ and amplitude $1.6 \mathrm{~mm}$ travels along the wire. (a) Calculate the average power carried by the wave. (b) What happens to the average power if the wave amplitude is halved?

Averell Hause

Carnegie Mellon University

Problem 23

A horizontal wire is stretched with a tension of $94.0 \mathrm{~N},$ and the speed of transverse waves for the wire is $406 \mathrm{~m} / \mathrm{s}$. What must the amplitude of a traveling wave of frequency $69.0 \mathrm{~Hz}$ be for the average power carried by the wave to be $0.365 \mathrm{~W} ?$

Anonymous User

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Problem 24

Threshold of Pain. You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are $7.5 \mathrm{~m}$ from it, you measure its intensity to be $0.11 \mathrm{~W} / \mathrm{m}^{2}$. An intensity of $1.0 \mathrm{~W} / \mathrm{m}^{2}$ is often used as the "threshold of pain." How much closer to the source can you move before the sound intensity reaches this threshold?

Noor Aldeen Almusleh

Noor Aldeen Almusleh

Numerade Educator

Problem 25

$\mathrm{A}$ jet plane at takeoff can produce sound of intensity $10.0 \mathrm{~W} / \mathrm{m}^{2}$ at $30.0 \mathrm{~m}$ away. But you prefer the tranquil sound of normal conversation, which is $1.0 \mu \mathrm{W} / \mathrm{m}^{2}$. Assume that the plane behaves like a point source of sound. (a) What is the closest distance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at takeoff?

Anonymous User

Numerade Educator

Problem 26

$\mathrm{A}$ fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y(x, t)=(2.30 \mathrm{~mm}) \cos [(6.98 \mathrm{rad} / \mathrm{m}) x+(742 \mathrm{rad} / \mathrm{s}) t] .$ Being more practical, you measure the rope to have a length of $1.35 \mathrm{~m}$ and a mass of $0.00338 \mathrm{~kg}$. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 27

Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is $0.026 \mathrm{~W} / \mathrm{m}^{2}$ at a distance of $4.3 \mathrm{~m}$ from the source. (a) What is the intensity at a distance of $3.1 \mathrm{~m}$ from the source? (b) How much sound energy does the source emit in one hour if its power output remains constant?

Anonymous User

Numerade Educator

Problem 28

Reflection. A wave pulse on a string has the dimensions shown in Fig. E15.28 at $t=0 .$ The wave speed is $40 \mathrm{~cm} / \mathrm{s}$ (a) If point $O$ is a fixed end, draw the total wave on the string at $t=15 \mathrm{~ms}, 20 \mathrm{~ms}, 25 \mathrm{~ms}, 30 \mathrm{~ms}$ $35 \mathrm{~ms}, 40 \mathrm{~ms},$ and $45 \mathrm{~ms}$. (b) Repeat part (a) for the case in which point $O$ is a free end.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

Reflection. A wave pulse on a string has the dimensions shown in Fig. $\mathbf{E} 15.29$ at $t=0 .$ The wave speed is $5.0 \mathrm{~m} / \mathrm{s}$ (a) If point $O$ is a fixed end, draw the total wave on the string at $t=1.0 \mathrm{~ms}, 2.0 \mathrm{~ms}, 3.0 \mathrm{~ms}$ $4.0 \mathrm{~ms}, 5.0 \mathrm{~ms}, 6.0 \mathrm{~ms},$ and $7.0 \mathrm{~ms}$ (b) Repeat part (a) for the case in which point $O$ is a free end.

David González Cornejo

David González Cornejo

Numerade Educator

Problem 30

Interference of Triangular Pulses. Two triangular wave pulses are traveling toward each other on a stretched string as shown in Fig. $\mathbf{E} 15.30 .$ Each pulse is identical to the other and travels at $2.00 \mathrm{~cm} / \mathrm{s}$. The leading edges of the pulses are $1.00 \mathrm{~cm}$ apart at $t=0 .$ Sketch the shape of the string at $t=0.250 \mathrm{~s}, t=0.500 \mathrm{~s}, t=0.750 \mathrm{~s}, t=1.000 \mathrm{~s},$ and $t=1.250 \mathrm{~s}$

Mihajlo Grcic

Numerade Educator

Problem 31

Suppose that the left-traveling pulse in Exercise 15.30 is below the level of the unstretched string instead of above it. Make the same sketches that you did in that exercise.

Anonymous User

Numerade Educator

Problem 32

Interference of Rectangular Pulses. Figure $\mathrm{E} 15.32$ shows two rectangular wave pulses on a stretched string traveling toward each other. Each pulse is traveling with a speed of $1.00 \mathrm{~mm} / \mathrm{s}$ and has the height and width shown in the figure. If the leading edges of the pulses are $8.00 \mathrm{~mm}$ apart at $t=0,$ sketch the shape of the string at $t=4.00 \mathrm{~s}, t=6.00 \mathrm{~s},$ and $t=10.0 \mathrm{~s}$

Brandy Heflin

Numerade Educator

Problem 33

For a violin, estimate the length of the portions of the strings that are free to vibrate. (a) The frequency of the note played by the open E5 string vibrating in its fundamental standing wave is 659 Hz. Use your estimate of the length to calculate the wave speed for the transverse waves on the string. (b) The vibrating string produces sound waves in air with the same frequency as that of the string. Use $344 \mathrm{~m} / \mathrm{s}$ for the speed of sound in air and calculate the wavelength of the E5 note in air. Which is larger: the wavelength on the string or the wavelength in air? (c) Repeat parts (a) and (b) for a bass viol, which is typically played by a person standing up. Start your calculation by estimating the length of the bass viol string that is free to vibrate. The G2 string produces a note with frequency $98 \mathrm{~Hz}$ when vibrating in its fundamental standing wave.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 34

Adjacent antinodes of a standing wave on a string are $15.0 \mathrm{~cm}$ apart. A particle at an antinode oscillates in simple harmonic motion with amplitude $0.850 \mathrm{~cm}$ and period $0.0750 \mathrm{~s}$. The string lies along the $+x$ -axis and is fixed at $x=0 .$ (a) How far apart are the adjacent nodes? (b) What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern? (c) Find the maximum and minimum transverse speeds of a point at an antinode. (d) What is the shortest distance along the string between a node and an antinode?

Averell Hause

Carnegie Mellon University

Problem 35

Standing waves on a wire are described by Eq. (15.28), with $A_{\mathrm{SW}}=2.50 \mathrm{~mm}, \omega=942 \mathrm{rad} / \mathrm{s},$ and $k=0.750 \pi \mathrm{rad} / \mathrm{m} .$ The left end of the wire is at $x=0 .$ At what distances from the left end are (a) the nodes of the standing wave and (b) the antinodes of the standing wave?

Keshav Singh

Numerade Educator

Problem 36

A $1.50-\mathrm{m}$ -long rope is stretched between two supports with a tension that makes the speed of transverse waves $62.0 \mathrm{~m} / \mathrm{s}$. What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

Averell Hause

Carnegie Mellon University

Problem 37

A wire with mass $40.0 \mathrm{~g}$ is stretched so that its ends are tied down at points $80.0 \mathrm{~cm}$ apart. The wire vibrates in its fundamental mode with frequency $60.0 \mathrm{~Hz}$ and with an amplitude at the antinodes of $0.300 \mathrm{~cm}$. (a) What is the speed of propagation of transverse waves in the wire? (b) Compute the tension in the wire. (c) Find the maximum transverse velocity and acceleration of particles in the wire.

Anonymous User

Numerade Educator

Problem 38

A piano tuner stretches a steel piano wire with a tension of $800 \mathrm{~N}$. The steel wire is $0.400 \mathrm{~m}$ long and has a mass of $3.00 \mathrm{~g}$. (a) What is the frequency of its fundamental mode of vibration? (b) What is the number of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to $10,000 \mathrm{~Hz} ?$

Averell Hause

Carnegie Mellon University

Problem 39

A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation $y(x, t)=(5.60 \mathrm{~cm}) \sin [(0.0340 \mathrm{rad} / \mathrm{cm}) x] \sin [(50.0 \mathrm{rad} / \mathrm{s}) t],$ where the origin is at the left end of the string, the $x$ -axis is along the string, and the $y$ -axis is perpendicular to the string. (a) Draw a sketch that shows the standing-wave pattern. (b) Find the amplitude of the two traveling waves that make up this standing wave. (c) What is the length of the string? (d) Find the wavelength, frequency, period, and speed of the traveling waves. (e) Find the maximum transverse speed of a point on the string.
(f) What would be the equation $y(x, t)$ for this string if it were vibrating in its eighth harmonic?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 40

The wave function of a standing wave is $y(x, t)=(4.44 \mathrm{~mm})$ $\sin [(32.5 \mathrm{rad} / \mathrm{m}) x] \sin [(754 \mathrm{rad} / \mathrm{s}) t] .$ For the two traveling waves that make up this standing wave, find the (a) amplitude; (b) wavelength; (c) frequency; (d) wave speed; (e) wave functions. (f) From the information given, can you determine which harmonic this is? Explain.

Averell Hause

Carnegie Mellon University

Problem 41

Standing waves are produced on a string that is held fixed at both ends. The tension in the string is kept constant. (a) For the second overtone standing wave the node-to-node distance is $8.00 \mathrm{~cm} .$ What is the length of the string? (b) What is the node-to-node distance for the fourth harmonic standing wave?

Inder Jeet

Numerade Educator

Problem 42

One string of a certain musical instrument is $75.0 \mathrm{~cm}$ long and has a mass of $8.75 \mathrm{~g}$. It is being played in a room where the speed of sound is $344 \mathrm{~m} / \mathrm{s}$. (a) To what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength $0.765 \mathrm{~m} ?$ (Assume that the breaking stress of the wire is very large and isn't exceeded.) (b) What frequency sound does this string produce in its fundamental mode of vibration?

Averell Hause

Carnegie Mellon University

Problem 43

The portion of the string of a certain musical instrument between the bridge and upper end of the finger board (that part of the string that is free to vibrate) is $60.0 \mathrm{~cm}$ long, and this length of the string has mass $2.00 \mathrm{~g}$. The string sounds an $\mathrm{A}_{4}$ note $(440 \mathrm{~Hz})$ when played. (a) Where must the player put a finger (what distance $x$ from the bridge) to play a $D_{5}$ note $(587 \mathrm{~Hz}) ?$ (See Fig. E15.43.) For both the $\mathrm{A}_{4}$ and $\mathrm{D}_{5}$ notes, the string vibrates in its fundamental mode. (b) Without retuning, is it possible to play a $\mathrm{G}_{4}$ note $(392 \mathrm{~Hz})$ on this string? Why or why not?

Anonymous User

Numerade Educator

Problem 44

(a) A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed $v,$ frequency $f,$ amplitude $A,$ and wavelength $\lambda .$ Calculate the maximum transverse velocity and maximum transverse acceleration of points located at (i) $x=\lambda / 2$, (ii) $x=\lambda / 4,$ and (iii) $x=\lambda / 8,$ from the left-hand end of the string. (b) At each of the points in part (a), what is the amplitude of the motion? (c) At each of the points in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement?

Anonymous User

Numerade Educator

Problem 45

A sinusoidal wave with wavelength 0.400 m travels along a string. The maximum transverse speed of a point on the string is $3.00 \mathrm{~m} / \mathrm{s}$ and the maximum transverse acceleration is $8.50 \times 10^{4} \mathrm{~m} / \mathrm{s}^{2} .$ What are the propagation speed $v$ and the amplitude $A$ of the wave?

Inder Jeet

Numerade Educator

Problem 46

A transverse wave on a rope is given by
$$
y(x, t)=(0.750 \mathrm{~cm}) \cos \pi\left[\left(0.400 \mathrm{~cm}^{-1}\right) x+\left(250 \mathrm{~s}^{-1}\right) t\right]
$$
(a) Find the amplitude, period, frequency, wavelength, and speed of propagation. (b) Sketch the shape of the rope at these values of $t: 0,$ $0.0005 \mathrm{~s}, 0.0010 \mathrm{~s} .$ (c) Is the wave traveling in the $+x$ - or $-x$ -direction?
(d) The mass per unit length of the rope is $0.0500 \mathrm{~kg} / \mathrm{m}$. Find the tension. (e) Find the average power of this wave.

Mihajlo Grcic

Numerade Educator

Problem 47

A transverse sine wave with an amplitude of $2.50 \mathrm{~mm}$ and a wavelength of $1.80 \mathrm{~m}$ travels from left to right along a long, horizontal, stretched string with a speed of $36.0 \mathrm{~m} / \mathrm{s}$. Take the origin at the left end of the undisturbed string. At time $t=0$ the left end of the string has its maximum upward displacement. (a) What are the frequency, angular frequency, and wave number of the wave? (b) What is the function $y(x, t)$ that describes the wave? (c) What is $y(t)$ for a particle at the left end of the string? (d) What is $y(t)$ for a particle $1.35 \mathrm{~m}$ to the right of the origin? (e) What is the maximum magnitude of transverse velocity of any particle of the string? (f) Find the transverse displacement and the transverse velocity of a particle $1.35 \mathrm{~m}$ to the right of the origin at time $t=0.0625 \mathrm{~s}$

Anonymous User

Numerade Educator

Problem 48

A $1750 \mathrm{~N}$ irregular beam is hanging horizontally by its ends from the ceiling by two vertical wires $(A$ and $B),$ each $1.25 \mathrm{~m}$ long and weighing $0.290 \mathrm{~N}$. The center of gravity of this beam is one-third of the way along the beam from the end where wire $A$ is attached. If you pluck both strings at the same time at the beam, what is the time delay between the arrival of the two pulses at the ceiling? Which pulse arrives first? (Ignore the effect of the weight of the wires on the tension in the wires.)

Averell Hause

Carnegie Mellon University

Problem 49

One end of a light uniform rod is attached to a wall by a frictionless hinge. The rod is held in a horizontal position by a wire that runs from the other end of the rod to the wall. The wire has length $2.00 \mathrm{~m}$ and makes an angle of $30.0^{\circ}$ with the rod. A block with mass $m$ is suspended by a light rope attached to the middle of the rod. The transverse fundamental standing wave on the wire has frequency $f$. The mass $m$ is varied and for each value the frequency is measured. You plot $f^{2}$ versus $m$ and find that your data lie close to a straight line with slope $20.4 \mathrm{~kg}^{-1} \cdot \mathrm{s}^{-2}$ What is the mass of the wire? Assume that the change in the length of the wire when the tension changes is small enough to neglect.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

An ant with mass $m$ is standing peacefully on top of a horizontal, stretched rope. The rope has mass per unit length $\mu$ and is under tension $F$. Without warning, Cousin Throckmorton starts a sinusoidal transverse wave of wavelength $\lambda$ propagating along the rope. The motion of the rope is in a vertical plane. What minimum wave amplitude will make the ant become momentarily weightless? Assume that $m$ is so small that the presence of the ant has no effect on the propagation of the wave.

Averell Hause

Carnegie Mellon University

Problem 51

You must determine the length of a long, thin wire that is suspended from the ceiling in the atrium of a tall building. A $2.00-\mathrm{cm}$ -long piece of the wire is left over from its installation. Using an analytical balance, you determine that the mass of the spare piece is $14.5 \mu \mathrm{g}$. You then hang a $0.400 \mathrm{~kg}$ mass from the lower end of the long, suspended wire. When a small-amplitude transverse wave pulse is sent up that wire, sensors at both ends measure that it takes the wave pulse $26.7 \mathrm{~ms}$ to travel the length of the wire. (a) Use these measurements to calculate the length of the wire. Assume that the weight of the wire has a negligible effect on the speed of the transverse waves. (b) Discuss the accuracy of the approximation made in part (a).

Anonymous User

Numerade Educator

Problem 52

You are designing a two-string instrument with metal strings $35.0 \mathrm{~cm}$ long, as shown in Fig. $\mathrm{P} 15.52 .$ Both strings are under the same tension. String $S_{1}$ has a mass of $8.00 \mathrm{~g}$ and produces the note middle $\mathrm{C}$ (frequency $262 \mathrm{~Hz}$ ) in its fundamental mode. (a) What should be the tension in the string? (b) What should be the mass of string $S_{2}$ so that it will produce A-sharp (frequency $466 \mathrm{~Hz}$ ) as its fundamental? (c) To extend the range of your instrument, you include a fret located just under the strings but not normally touching them. How far from the upper end should you put this fret so that when you press $S_{1}$ tightly against it, this string will produce $\mathrm{C}$ -sharp (frequency $277 \mathrm{~Hz}$ ) in its fundamental? That is, what is $x$ in the figure? (d) If you press $S_{2}$ against the fret, what frequency of sound will it produce in its fundamental?

Averell Hause

Carnegie Mellon University

Problem 53

A $5.00 \mathrm{~m}, 0.732 \mathrm{~kg}$ wire is used to support two uniform $235 \mathrm{~N}$ posts of equal length (Fig. P15.53). Assume that the wire is essentially horizontal and that the speed of sound is $344 \mathrm{~m} / \mathrm{s}$ A strong wind is blowing, causing the wire to vibrate in its 5th overtone. What are the frequency and wavelength of the sound this wire produces?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 54

You are exploring a newly discovered planet. The radius of the planet is $7.20 \times 10^{7} \mathrm{~m}$. You suspend a lead weight from the lower end of a light string that is $4.00 \mathrm{~m}$ long and has mass $0.0280 \mathrm{~kg}$. You measure that it takes $0.0685 \mathrm{~s}$ for a transverse pulse to travel from the lower end to the upper end of the string. On the earth, for the same string and lead weight, it takes $0.0390 \mathrm{~s}$ for a transverse pulse to travel the length of the string. The weight of the string is small enough that you ignore its effect on the tension in the string. Assuming that the mass of the planet is distributed with spherical symmetry, what is its mass?

Averell Hause

Carnegie Mellon University

Problem 55

For a string stretched between two supports, two successive standing-wave frequencies are $525 \mathrm{~Hz}$ and $630 \mathrm{~Hz}$. There are other standing-wave frequencies lower than $525 \mathrm{~Hz}$ and higher than $630 \mathrm{~Hz}$. If the speed of transverse waves on the string is $384 \mathrm{~m} / \mathrm{s},$ what is the length of the string? Assume that the mass of the wire is small enough for its effect on the tension in the wire to be ignored.

Anonymous User

Numerade Educator

Problem 56

Transverse standing waves are produced on a string that has length $0.800 \mathrm{~m}$ and is held fixed at each end. Each standing-wave pattern has a node at the fixed ends plus additional nodes along the length of the string. You measure the frequencies $f_{n}$ for standing waves that have $n$ of these nodes along their length. The tension in the string is kept constant. You plot $n$ versus $f_{n}$ and find that your data lie close to a straight line that has slope $7.30 \times 10^{-3} \mathrm{~s}$. What is the speed of transverse waves on the string?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

A 1.80 -m-long uniform bar that weighs $638 \mathrm{~N}$ is suspended in a horizontal position by two vertical wires that are attached to the ceiling. One wire is aluminum and the other is copper. The aluminum wire is attached to the left-hand end of the bar, and the copper wire is attached $0.40 \mathrm{~m}$ to the left of the right-hand end. Each wire has length $0.600 \mathrm{~m}$ and a circular cross section with radius $0.280 \mathrm{~mm} .$ What is the fundamental frequency of transverse standing waves for each wire?

Anonymous User

Numerade Educator

Problem 58

A transverse standing wave is set up on a string that is held fixed at both ends. The amplitude of the standing wave at an antinode is $1.80 \mathrm{~mm}$ and the speed of propagation of transverse waves on the string is $260 \mathrm{~m} / \mathrm{s}$. The string extends along the $x$ -axis, with one of the fixed ends at $x=0,$ so that there is a node at $x=0 .$ The smallest value of $x$ where there is an antinode is $x=0.150 \mathrm{~m}$. (a) What is the maximum transverse speed of a point on the string at an antinode? (b) What is the maximum transverse speed of a point on the string at $x=0.075 \mathrm{~m} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

A horizontal wire is tied to supports at each end and vibrates in its second-overtone standing wave. The tension in the wire is $5.00 \mathrm{~N}$, and the node-to-node distance in the standing wave is $6.28 \mathrm{~cm}$.
(a) What is the length of the wire? (b) A point at an antinode of the standing wave on the wire travels from its maximum upward displacement to its maximum downward displacement in $8.40 \mathrm{~ms}$. What is the wire's mass?

Anonymous User

Numerade Educator

Problem 60

A vertical, $1.20 \mathrm{~m}$ length of 18 gauge (diameter of $1.024 \mathrm{~mm}$ ) copper wire has a $100.0 \mathrm{~N}$ ball hanging from it. (a) What is the wavelength of the third harmonic for this wire? (b) A $500.0 \mathrm{~N}$ ball now replaces the original ball. What is the change in the wavelength of the third harmonic caused by replacing the light ball with the heavy one? (Hint: See Table 11.1 for Young's modulus.)

Averell Hause

Carnegie Mellon University

Problem 61

A sinusoidal transverse wave travels on a string. The string has length $8.00 \mathrm{~m}$ and mass $6.00 \mathrm{~g}$. The wave speed is $30.0 \mathrm{~m} / \mathrm{s},$ and the wavelength is $0.200 \mathrm{~m}$. (a) If the wave is to have an average power of $50.0 \mathrm{~W}$, what must be the amplitude of the wave? (b) For this same string, if the amplitude and wavelength are the same as in part (a), what is the average power for the wave if the tension is increased such that the wave speed is doubled?

Anonymous User

Numerade Educator

Problem 62

A vibrating string $50.0 \mathrm{~cm}$ long is under a tension of $1.00 \mathrm{~N}$. The results from five successive stroboscopic pictures are shown in Fig. $P 15.62$. The strobe rate is set at 5000 flashes per minute, and observations reveal that the maximum displacement occurred at flashes 1 and 5 with no other maxima in between. (a) Find the period, frequency, and wavelength for the traveling waves on this string. (b) In what normal mode (harmonic) is the string vibrating? (c) What is the speed of the traveling waves on the string? (d) How fast is point $P$ moving when the string is in (i) position 1 and (ii) position $3 ?$ (e) What is the mass of this string?

Averell Hause

Carnegie Mellon University

Problem 63

A $1.005 \mathrm{~m}$ chain consists of small spherical beads, each with a mass of $1.00 \mathrm{~g}$ and a diameter of $5.00 \mathrm{~mm},$ threaded on an elastic strand with negligible mass such that adjacent beads are separated by a center-to-center distance of $10.0 \mathrm{~mm}$. There are beads at each end of the chain. The strand has a spring constant of $28.8 \mathrm{~N} / \mathrm{m}$. The chain is stretched horizontally on a frictionless tabletop to a length of $1.50 \mathrm{~m}$, and the beads at both ends are fixed in place. (a) What is the linear mass density of the chain? (b) What is the tension in the chain? (c) With what speed would a pulse travel down the chain? (d) The chain is set vibrating and exhibits a standing-wave pattern with four antinodes. What is the frequency of this motion? (e) If the beads are numbered sequentially from 1 to $101,$ what are the numbers of the five beads that remain motionless? (f) The 13th bead has a maximum speed of $7.54 \mathrm{~m} / \mathrm{s}$. What is the amplitude of that bead's motion? (g) If $x_{0}=0$ corresponds to the center of the 1 st bead and $x_{101}=1.50 \mathrm{~m}$ corresponds to the center of the 101 st bead, what is the position $x_{n}$ of the $n$ th bead? (h) What is the maximum speed of the 30 th bead?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 64

A strong string of mass $3.00 \mathrm{~g}$ and length $2.20 \mathrm{~m}$ is tied to supports at each end and is vibrating in its fundamental mode. The maximum transverse speed of a point at the middle of the string is $9.00 \mathrm{~m} / \mathrm{s}$ The tension in the string is $330 \mathrm{~N}$. (a) What is the amplitude of the standing wave at its antinode? (b) What is the magnitude of the maximum transverse acceleration of a point at the antinode?

Averell Hause

Carnegie Mellon University

Problem 65

A thin string $2.50 \mathrm{~m}$ in length is stretched with a tension of $90.0 \mathrm{~N}$ between two supports. When the string vibrates in its first overtone, a point at an antinode of the standing wave on the string has an amplitude of $3.50 \mathrm{~cm}$ and a maximum transverse speed of $28.0 \mathrm{~m} / \mathrm{s}$ (a) What is the string's mass? (b) What is the magnitude of the maximum transverse acceleration of this point on the string?

Anonymous User

Numerade Educator

Problem 66

A guitar string is vibrating in its fundamental mode, with nodes at each end. The length of the segment of the string that is free to vibrate is $0.386 \mathrm{~m}$. The maximum transverse acceleration of a point at the middle of the segment is $8.40 \times 10^{3} \mathrm{~m} / \mathrm{s}^{2}$ and the maximum transverse velocity is $3.80 \mathrm{~m} / \mathrm{s}$. (a) What is the amplitude of this standing wave? (b) What is the wave speed for the transverse traveling waves on this string?

Averell Hause

Carnegie Mellon University

Problem 67

A uniform cylindrical steel wire, $55.0 \mathrm{~cm}$ long and 1.14 $\mathrm{mm}$ in diameter, is fixed at both ends. To what tension must it be adjusted so that, when vibrating in its first overtone, it produces the note D-sharp of frequency 311 Hz? Assume that it stretches an insignificant amount. (Hint: See Table 12.1.)

Nishant Kumar

Numerade Educator

Problem 68

A string with both ends held fixed is vibrating in its third harmonic. The waves have a speed of $192 \mathrm{~m} / \mathrm{s}$ and a frequency of $240 \mathrm{~Hz}$. The amplitude of the standing wave at an antinode is $0.400 \mathrm{~cm}$. (a) Calculate the amplitude at points on the string a distance of (i) $40.0 \mathrm{~cm}$ (ii) $20.0 \mathrm{~cm} ;$ and (iii) $10.0 \mathrm{~cm}$ from the left end of the string. (b) At each point in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement? (c) Calculate the maximum transverse velocity and the maximum transverse acceleration of the string at each of the points in part (a).

Averell Hause

Carnegie Mellon University

Problem 69

A large rock that weighs $164.0 \mathrm{~N}$ is suspended from the lower end of a thin wire that is $3.00 \mathrm{~m}$ long. The density of the rock is $3200 \mathrm{~kg} / \mathrm{m}^{3} .$ The mass of the wire is small enough that its effect on the tension in the wire can be ignored. The upper end of the wire is held fixed. When the rock is in air, the fundamental frequency for transverse standing waves on the wire is $42.0 \mathrm{~Hz}$. When the rock is totally submerged in a liquid, with the top of the rock just below the surface, the fundamental frequency for the wire is $28.0 \mathrm{~Hz}$. What is the density of the liquid?

Anonymous User

Numerade Educator

Problem 70

(a) Estimate the tension you would need to apply to a standard small rubber band to stretch it between your fingers to a doubled length of $10 \mathrm{~cm}$. (b) Such a rubber band has a typical mass of $0.10 \mathrm{~g}$. Use this value to estimate the mass density of the stretched rubber band in SI units. (c) Use your estimated values to determine the expected frequency of vibration when the string is plucked. (d) Is this result realistic?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

A musician tunes the C-string of her instrument to a fundamental frequency of $65.4 \mathrm{~Hz}$. The vibrating portion of the string is $0.600 \mathrm{~m}$ long and has a mass of $14.4 \mathrm{~g}$. (a) With what tension must the musician stretch it? (b) What percent increase in tension is needed to increase the frequency from $65.4 \mathrm{~Hz}$ to $73.4 \mathrm{~Hz}$, corresponding to a rise in pitch from $\mathrm{C}$ to $\mathrm{D}$ ?

Anonymous User

Numerade Educator

Problem 72

A string or rope will break apart if it is placed under too much tensile stress [see Eq. (11.8)]. Thicker ropes can withstand more tension without breaking because the thicker the rope, the greater the cross-sectional area and the smaller the stress. One type of steel has density $7800 \mathrm{~kg} / \mathrm{m}^{3}$ and will break if the tensile stress exceeds $7.0 \times 10^{8} \mathrm{~N} / \mathrm{m}^{2}$. You want to make a guitar string from $4.0 \mathrm{~g}$ of this type of steel. In use, the guitar string must be able to withstand a tension of $900 \mathrm{~N}$ without breaking. Your job is to determine (a) the maximum length and minimum radius the string can have; (b) the highest possible fundamental frequency of standing waves on this string, if the entire length of the string is free to vibrate.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

In your physics lab, an oscillator is attached to one end of a horizontal string. The other end of the string passes over a frictionless pulley. You suspend a mass $M$ from the free end of the string, producing tension $M g$ in the string. The oscillator produces transverse waves of frequency $f$ on the string. You don't vary this frequency during the experiment, but you try strings with three different linear mass densities $\mu .$ You also keep a fixed distance between the end of the string where the oscillator is attached and the point where the string is in contact with the pulley's rim. To produce standing waves on the string, you vary $M ;$ then you measure the node-to-node distance $d$ for each standing-wave pattern and obtain the following data:
$$
\begin{array}{l|lllll}
\text { String } & \text { A } & \text { A } & \text { B } & \text { B } & \text { C } \\
\hline \mu(\mathrm{g} / \mathrm{cm}) & 0.0260 & 0.0260 & 0.0374 & 0.0374 & 0.0482 \\
M(\mathrm{~g}) & 559 & 249 & 365 & 207 & 262 \\
d(\mathrm{~cm}) & 48.1 & 31.9 & 32.0 & 24.2 & 23.8
\end{array}
$$
(a) Explain why you obtain only certain values of $d$. (b) Graph $\mu d^{2}($ in $\mathrm{kg} \cdot \mathrm{m})$ versus $M($ in $\mathrm{kg}) .$ Explain why the data plotted this way should fall close to a straight line. (c) Use the slope of the best straightline fit to the data to determine the frequency $f$ of the waves produced on the string by the oscillator. Take $g=9.80 \mathrm{~m} / \mathrm{s}^{2}$. (d) For string A $(\mu=0.0260 \mathrm{~g} / \mathrm{cm}),$ what value of $M$ (in grams) would be required to produce a standing wave with a node-to-node distance of $24.0 \mathrm{~cm}$ ? Use the value of $f$ that you calculated in part (c).

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 74

Scale length is the length of the part of a guitar string that is free to vibrate. A standard value of scale length for an acoustic guitar is 25.5 in. The frequency of the fundamental standing wave on a string is determined by the string's scale length, tension, and linear mass density. The standard frequencies $f$ to which the strings of a sixstring guitar are tuned are given in the table:
$$
\begin{array}{l|llllll}
\text { String } & \text { E2 } & \text { A2 } & \text { D3 } & \text { G3 } & \text { B3 } & \text { E4 } \\
\hline f(\mathbf{H z}) & 82.4 & 110.0 & 146.8 & 196.0 & 246.9 & 329.6
\end{array}
$$
Assume that a typical value of the tension of a guitar string is $78.0 \mathrm{~N}$ (although tension varies somewhat for different strings). (a) Calculate the linear mass density $\mu$ (in $\mathrm{g} / \mathrm{cm}$ ) for the $\mathrm{E} 2, \mathrm{G} 3$, and $\mathrm{E} 4$ strings. (b) Just before your band is going to perform, your G3 string breaks. The only replacement string you have is an E2. If your strings have the linear mass densities calculated in part (a), what must be the tension in the replacement string to bring its fundamental frequency to the G3 value of $196.0 \mathrm{~Hz}$ ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 75

You are measuring the frequency dependence of the average power $P_{\mathrm{av}}$ transmitted by traveling waves on a wire. In your experiment you use a wire with linear mass density $3.5 \mathrm{~g} / \mathrm{m}$. For a transverse wave on the wire with amplitude $4.0 \mathrm{~mm},$ you measure $P_{\mathrm{av}}$ (in watts) as a function of the frequency $f$ of the wave (in $\mathrm{Hz}$ ). You have chosen to plot $P_{\text {av }}$ as a function of $f^{2}$ (Fig. $\mathbf{P 1 5 . 7 5}$ ). (a) Explain why values of $P_{\text {av }}$ plotted versus $f^{2}$ should be well fit by a straight line. (b) Use the slope of the straight-line fit to the data shown in Fig. $\mathrm{P} 15.75$ to calculate the speed of the waves. (c) What angular frequency $\omega$ would result in $P_{\mathrm{av}}=10.0 \mathrm{~W} ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 76

A rectangular neoprene sheet has width $W=1.00 \mathrm{~m}$ and length $L=4.00 \mathrm{~m}$. The two shorter edges are affixed to rigid steel bars that are used to stretch the sheet taut and horizontal. The force applied to either end of the sheet is $F=81.0 \mathrm{~N}$. The sheet has a total mass $M=4.00 \mathrm{~kg} .$ The left edge of the sheet is wiggled vertically in a uniform sinusoidal motion with amplitude $A=10.0 \mathrm{~cm}$ and frequency $f=1.00 \mathrm{~Hz}$. This sends waves spanning the width of the sheet rippling from left to right. The right side of the sheet moves upward and downward freely as these waves complete their traversal. (a) Use a twodimensional generalization of the discussion in Section 15.4 to derive an expression for the velocity with which the waves move along the sheet in terms of generic values of $W, L, F, M, f,$ and $A .$ What is the value of this speed for the specified choices of these parameters? (b) If the positive $x$ -axis is oriented rightward and the steel bars are parallel to the $y$ -axis, the height of the sheet may be characterized as $z(x, y)=A \sin (k x-\omega t)$ What is the value of the wave number $k ?$ (c) Write down an expression with generic parameters for the rate of rightward energy transfer by the slice of sheet at a given value of $x$ at generic time $t$. (d) The power at $x=0$ is supplied by the agent wiggling the left bar upward and downward. How much energy is supplied each second by that agent? Express your answer in terms of generic parameters and also as a specific energy for the given parameters.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 77

A deep-sea diver is suspended beneath the surface of Loch Ness by a $100-\mathrm{m}$ -long cable that is attached to a boat on the surface (Fig. $\mathbf{P} 15.77$ ). The diver and his suit have a total mass of $120 \mathrm{~kg}$ and a volume of $0.0800 \mathrm{~m}^{3} .$ The cable has a diameter of $2.00 \mathrm{~cm}$ and a linear mass density of $\mu=1.10 \mathrm{~kg} / \mathrm{m} .$ The diver thinks he sees something moving in the murky depths and jerks the end of the cable back and forth to send transverse waves up the cable as a signal to his companions in the boat. (a) What is the tension in the cable at its lower end, where it is attached to the diver? Do not forget to include the buoyant force that the water (density $1000 \mathrm{~kg} / \mathrm{m}^{3}$ ) exerts on him. (b) Calculate the tension in the cable a distance $x$ above the diver. In your calculation, include the buoyant force on the cable. (c) The speed of transverse waves on the cable is given by $v=\sqrt{F / \mu}$ [Eq. (15.14)]. The speed therefore varies along the cable, since the tension is not constant. (This expression ignores the damping force that the water exerts on the moving cable.) Integrate to find the time required for the first signal to reach the surface.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 78

What is the wavelength of the wave that travels on the surface of the vocal folds when they are vibrating at frequency $f ?$
(a) $2.0 \mathrm{~mm} ;$
(b) $3.3 \mathrm{~mm}$
(c) $0.50 \mathrm{~cm}$
(d) $3.0 \mathrm{~cm}$.

Averell Hause

Carnegie Mellon University

Problem 79

Which of these is a possible mathematical description of the wave in Problem $15.78 ?$
(a) $A \sin [2 \pi f(t+z / v)]$
(b) $A \sin [2 \pi f(t-z / v)] ;$
(c) $A \sin (2 \pi f t) \cos (2 \pi z / \lambda) ;$
(d) $A \sin (2 \pi f t)$
$\sin (2 \pi z / \lambda)$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 80

The wave speed is measured for different vibration frequencies. A graph of the wave speed as a function of frequency (Fig. $\mathbf{P} 15.80$ ) indicates that as the frequency increases, the wavelength (a) increases; (b) decreases; (c) doesn't change; (d) becomes undefined.

Averell Hause

Carnegie Mellon University