University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 15

Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

Fans at a local football stadium are so excited that their team is winning that they start "the wave" in celebration. Which of the following four statements is (are) true?
I. This wave is a traveling wave.
II. This wave is a transverse wave.
III. This wave is a longitudinal wave.
IV. This wave is a combination of a longitudinal wave and a transverse wave.
a) I and II
c) III only
e) I and III
b) II only
d) I and IV

Prabhu Ramji

Prabhu Ramji

Numerade Educator

Problem 2

You wish to decrease the speed of a wave traveling on a string to half its current value by changing the tension in the string. By what factor must you decrease the tension in the string?
a) 1
c) 2
e) none of the above
b) $\sqrt{2}$
d) 4

Prabhu Ramji

Numerade Educator

Problem 3

Suppose that the tension is doubled for a string on which a wave is propagated. How will the velocity of the wave change?
a) It will double.
c) It will be multiplied by $\sqrt{2}$.
b) It will quadruple.
d) It will be multiplied by $\frac{1}{2}$.

Prabhu Ramji

Numerade Educator

Problem 4

Which one of the following transverse waves has the greatest power?
a) a wave with velocity $v$, amplitude $A$, and frequency $f$
b) a wave with velocity $v$, amplitude $2 A$, and frequency $f / 2$
c) a wave with velocity $2 v$, amplitude $A / 2,$ and frequency $f$
d) a wave with velocity $2 v$, amplitude $A$, and frequency $f / 2$
e) a wave with velocity $2 v$, amplitude $A / 2,$ and frequency $2 f$

Prabhu Ramji

Numerade Educator

Problem 5

The speed of light waves in air is greater than the speed of sound waves in air by about a factor of a million. Given a sound wave and a light wave of the same wavelength, both traveling through air, which statement about their frequencies is true?
a) The frequency of the sound wave will be about a million times greater than that of the light wave.
b) The frequency of the sound wave will be about a thousand times greater than that of the light wave.
c) The frequency of the light wave will be about a thousand times greater than that of the sound wave
d) The frequency of the light wave will be about a million times greater than that of the sound wave.
e) There is insufficient information to determine the relationship between the two frequencies.

Prabhu Ramji

Numerade Educator

Problem 7

The different colors of light we perceive are a result of the varying frequencies (and wavelengths) of the electromagnetic radiation. Infrared radiation has lower frequencies than does visible light, and ultraviolet radiation has higher frequencies than visible light does. The primary colors are red (R), yellow (Y), and blue (B). Order these colors by their wavelength, shortest to longest.
a) $\mathrm{B}, \mathrm{Y}, \mathrm{R}$
b) $B, R, Y$
c) $R, Y, B$
d) $R, B, Y$

Prabhu Ramji

Numerade Educator

Problem 8

If transverse waves on a string travel with a velocity of $50 \mathrm{~m} / \mathrm{s}$ when the string is under a tension of $20 \mathrm{~N}$, what tension on the string is required for the waves to travel with a velocity of $30 \mathrm{~m} / \mathrm{s} ?$
a) $7.2 \mathrm{~N}$
c) $33 \mathrm{~N}$
e) $45 \mathrm{~N}$
b) $12 \mathrm{~N}$
d) $40 \mathrm{~N}$
f) $56 \mathrm{~N}$

Prabhu Ramji

Numerade Educator

Problem 9

The intensity of electromagnetic waves from the Sun at the Earth is $1400 \mathrm{~W} / \mathrm{m}^{2}$. How much power does the Sun generate? The distance from the Earth to the Sun is $149.6 \cdot 10^{6} \mathrm{~km}$.
a) $1.21 \cdot 10^{20} \mathrm{~W}$
b) $2.43 \cdot 10^{24} \mathrm{~W}$
c) $3.94 \cdot 10^{26} \mathrm{~W}$
d) $2.11 \cdot 10^{28} \mathrm{~W}$
e) $9.11 \cdot 10^{30} \mathrm{~W}$

Prabhu Ramji

Numerade Educator

Problem 10

A guitar string is $0.750 \mathrm{~m}$ long and has a mass of $5.00 \mathrm{~g}$. The string is tuned to $\mathrm{E}(660 \mathrm{~Hz})$ when it vibrates at its fundamental frequency. What is the required tension on the string?
a) $2.90 \cdot 10^{3} \mathrm{~N}$
b) $4.84 \cdot 10^{3} \mathrm{~N}$
c) $6.53 \cdot 10^{3} \mathrm{~N}$
d) $8.11 \cdot 10^{3} \mathrm{~N}$
e) $1.23 \cdot 10^{4} \mathrm{~N}$

Prabhu Ramji

Numerade Educator

Problem 11

You and a friend are holding the two ends of a Slinky stretched out between you. How would you move your end of the Slinky to create
(a) transverse waves or (b) longitudinal waves?

Prabhu Ramji

Numerade Educator

Problem 12

Noise results from the superposition of a very large number of sound waves of various frequencies (usually in a continuous spectrum), amplitudes, and phases. Can interference arise with noise produced by two sources?

Ajay Singhal

Numerade Educator

Problem 13

Noise results from the superposition of a very large number of sound waves of various frequencies (usually in a continuous spectrum), amplitudes, and phases. Can interference arise with noise produced by two sources??

Ajay Singhal

Numerade Educator

Problem 14

The $1 / R^{2}$ dependency for intensity can be thought of as being due to the fact that the same power is being spread out over the surface of a larger and larger sphere. What happens to the intensity of a sound wave inside an enclosed space, such as a long hallway?

Prabhu Ramji

Numerade Educator

Problem 15

If two traveling waves have the same wavelength, frequency, and amplitude and are added appropriately, the result is a standing wave. Is it possible to combine two standing waves in some way to give a traveling wave?

Eduard Sanchez

Numerade Educator

Problem 16

A string is made to oscillate, and a standing wave with three antinodes is created. If the tension in the string is increased by a factor of 4 , the number of antinodes
a) increases.
b) remains the same.
c) decreases.
d) will equal the number of nodes.

Prabhu Ramji

Numerade Educator

Problem 16

A ping-pong ball is floating in the middle of a lake and waves begin to propagate on the surface. Can you think of a situation in which the ball remains stationary? Can you think of a situation involving a single wave on the lake in which the ball remains stationary?

Ajay Singhal

Numerade Educator

Problem 17

Why do circular water waves on the surface of a pond decrease in amplitude as they travel away from the source?

Prabhu Ramji

Numerade Educator

Problem 18

Consider a wave on a string, with amplitude $A$ and wavelength $\lambda$, traveling in one direction. Find the relationship between the maximum speed of any portion of string, $v_{\text {max }}$, and the wave speed, $v$

Prabhu Ramji

Numerade Educator

Problem 19

Suppose a tightrope walker is standing in the middle of a rope mile long. If one end of the rope is cut, how long will it be before the tightrope walker begins to fall? What would you have to know to answer this question?

Prabhu Ramji

Numerade Educator

Problem 20

Suppose a long line of cars waits at an intersection for the light to turn green. Once the light turns, the cars in the back cannot move right away, but have to wait until the cars ahead of them move. Explain this effect (called a rarefaction wave) in terms of the wave physics that we have studied in this chapter.

Supratim Pal

Numerade Educator

Problem 21

Derive the relationship for the speed of a wave on a string $v=\sqrt{F / \mu}$ where $F$ is the tension force on the string and $\mu$ is the linear mass density of the string. Start with Derivation 15.1 and substitute the restoring force provided by the springs with a restoring force resulting from the tension of the string.

Prabhu Ramji

Numerade Educator

Problem 22

Derive the expression for the energy of a wave $E=\frac{1}{2} m \omega^{2} A^{2},$ where $m$ is the mass of the substance that is oscillating, $\omega$ is the angular velocity, and $A(r)$ is the amplitude. Start by expressing the kinetic energy and potential energy of the wave, and then add these to get the energy of the wave.

Prabhu Ramji

Numerade Educator

Problem 23

One of the main things allowing humans to determine whether a sound is coming from the left or the right is the fact that the sound will reach one ear before the other. Given that the speed of sound in air is $343 \mathrm{~m} / \mathrm{s}$ and that human ears are typically $20.0 \mathrm{~cm}$ apart, what is the maximum time resolution for human hearing that allows sounds coming from the left to be distinguished from sounds coming from the right? Why is it impossible for a diver to be able to tell from which direction the sound of a motorboat is coming? The speed of sound in water is $1.50 \cdot 10^{3} \mathrm{~m} / \mathrm{s}$.

Prabhu Ramji

Numerade Educator

Problem 24

Hiking in the mountains, you shout "hey," wait $2.00 \mathrm{~s}$ and shout again. What is the distance between the sound waves you cause? If you hear the first echo after $5.00 \mathrm{~s}$, what is the distance between you and the point where your voice hit a mountain?

Prabhu Ramji

Numerade Educator

Problem 25

The displacement from equilibrium caused by a wave on a string is given by $y(x, t)=(-0.00200 \mathrm{~m}) \sin \left[\left(40.0 \mathrm{~m}^{-1}\right) x-\left(800 . \mathrm{s}^{-1}\right) t\right] .$ For this wave,
what are the (a) amplitude, (b) number of waves in $1.00 \mathrm{~m},$ (c) number of complete cycles in $1.00 \mathrm{~s},$ (d) wavelength, and (e) speed?

Prabhu Ramji

Numerade Educator

Problem 26

A traveling wave propagating on a string is described by the following equation:
$$
y(x, t)=(5.00 \mathrm{~mm}) \sin \left(\left(157.08 \mathrm{~m}^{-1}\right) x-\left(314.16 \mathrm{~s}^{-1}\right) t+0.7854\right)
$$
a) Determine the minimum separation, $\Delta x_{\text {min }}$, between two points on the string that oscillate in perfect opposition of phases (move in opposite directions at all times).
b) Determine the minimum separation, $\Delta x_{A B}$, between two points $A$ and $B$ on the string, if point $B$ oscillates with a phase difference of $0.7854 \mathrm{rad}$ compared to point $A$.
c) Find the number of crests of the wave that pass through point $A$ in a time interval $\Delta t=10.0 \mathrm{~s}$ and the number of troughs that pass through point $B$ in the same interval.

Ajay Singhal

Numerade Educator

Problem 27

A traveling wave propagating on a string is described by the following equation:
$$
y(x, t)=(5.00 \mathrm{~mm}) \sin \left(\left(157.08 \mathrm{~m}^{-1}\right) x-\left(314.16 \mathrm{~s}^{-1}\right) t+0.7854\right)
$$
a) Determine the minimum separation, $\Delta x_{\min }$, between two points on the string that oscillate in perfect opposition of phases (move in opposite directions at all times).
b) Determine the minimum separation, $\Delta x_{A B}$, between two points $A$ and $B$ on the string, if point $B$ oscillates with a phase difference of $0.7854 \mathrm{rad}$ compared to point $A$.
c) Find the number of crests of the wave that pass through point $A$ in a time interval $\Delta t=10.0 \mathrm{~s}$ and the number of troughs that pass through point $B$ in the same interval.

Ajay Singhal

Numerade Educator

Problem 28

Show that the function $D=A \ln (x+v t)$ is a solution of the wave equation (equation 15.9).

Prabhu Ramji

Numerade Educator

Problem 29

A wave travels along a string in the positive $x$ -direction at $30.0 \mathrm{~m} / \mathrm{s}$. The frequency of the wave is $50.0 \mathrm{~Hz}$. At $x=0$ and $t=0$, the wave velocity is $2.50 \mathrm{~m} / \mathrm{s}$ and the vertical displacement is $y=4.00 \mathrm{~mm} .$ Write the function $y(x, t)$ for the wave.

Prabhu Ramji

Numerade Educator

Problem 30

A wave on a string has a wave function given by $y(x, t)=(0.0200 \mathrm{~m}) \sin \left[\left(6.35 \mathrm{~m}^{-1}\right) x+\left(2.63 \mathrm{~s}^{-1}\right) t\right]$
a) What is the amplitude of the wave?
b) What is the period of the wave?
c) What is the wavelength of the wave?
d) What is the speed of the wave?
e) In which direction does the wave travel?

Prabhu Ramji

Numerade Educator

Problem 31

A sinusoidal wave traveling in the positive $x$ -direction has a wavelength of $12.0 \mathrm{~cm},$ a frequency of $10.0 \mathrm{~Hz},$ and an amplitude of $10.0 \mathrm{~cm} .$ The part of the wave that is at the origin at $t=0$ has a vertical displacement of $5.00 \mathrm{~cm} .$ For this wave, determine the
a) wave number,
d) speed,
b) period,
e) phase angle, and
c) angular frequency,
f) equation of motion.

Prabhu Ramji

Numerade Educator

Problem 32

A mass $m$ hangs on a string that is connected to the ceiling. You pluck the string just above the mass, and a wave pulse travels up to the ceiling, reflects off the ceiling, and travels back to the mass. Compare the round-trip time for this wave pulse to that of a similar wave pulse on the same string if the attached mass is increased to $3.00 \mathrm{~m}$. (Assume that the string does not stretch in either case and the contribution of the mass of the string to the tension is negligible.

Prabhu Ramji

Numerade Educator

Problem 33

Point $A$ in the figure is a distance $d$ below the ceiling. Determine how much longer it will take for a wave pulse to travel from point $A$ to the ceiling along wire 1 than along wire 2 .

Prabhu Ramji

Numerade Educator

Problem 34

A particular steel guitar string has a mass per unit length of $1.93 \mathrm{~g} / \mathrm{m}$.
a) If the tension on this string is $62.2 \mathrm{~N},$ what is the wave speed on the string?
b) For the wave speed to be increased by $1.00 \%$, how much should the tension be changed?

Prabhu Ramji

Numerade Educator

Problem 35

Bob is talking to Alice using a tin can telephone, which consists of two steel cans connected by a 20.0 - m-long taut steel wire (see the figure). The wire has a linear mass density of $6.13 \mathrm{~g} / \mathrm{m},$ and the tension on the wire is $25.0 \mathrm{~N}$. The sound waves leave Bob's mouth, are collected by the can on the left, and then create vibrations in the wire, which travel to Alice's can and are transformed back into sound waves in air. Alice hears both the sound waves
that have traveled through the wire (wave 1 ) and those that have traveled through the air (wave 2), bypassing the wire. Do these two kinds of waves reach her at the same time? If not, which wave arrives sooner and by how much? The speed of sound in air is $343 \mathrm{~m} / \mathrm{s}$. Assume that the waves on the
string are transverse.

Prabhu Ramji

Numerade Educator

Problem 36

A wire of uniform linear mass density hangs from the ceiling. It takes 1.00 s for a wave pulse to travel the length of the wire. How long is the wire?

Prabhu Ramji

Numerade Educator

Problem 37

a) Starting from the general wave equation (equation 15.9 ), prove through direct derivation that the Gaussian wave packet described by the equation $y(x, t)=(5.00 m) e^{-0.1(x-5 t)^{2}}$ is indeed a traveling wave (that it satisfies the differential wave equation).
b) If $x$ is specified in meters and $t$ in seconds, determine the speed of this wave. On a single graph, plot this wave as a function of $x$ at $t=0, t=1.00 \mathrm{~s}$, $t=2.00 \mathrm{~s},$ and $t=3.00 \mathrm{~s}$
c) More generally, prove that any function $f(x, t)$ that depends on $x$ and $t$ through a combined variable $x \pm v t$ is a solution of the wave equation, irrespective of the specific form of the function $f$.

Dominador Tan

Numerade Educator

Problem 38

Suppose the slope of a beach underneath an ocean is $12.0 \mathrm{~cm}$ of dropoff for every $1.00 \mathrm{~m}$ of horizontal distance. A wave is moving inland, slowing down as it enters shallower water. What is its acceleration when it is $10.0 \mathrm{~m}$ from the shoreline? (Take the direction toward the shoreline as the positive $x$ -direction.)

Prabhu Ramji

Numerade Educator

Problem 39

An earthquake generates three kinds of waves: surface waves (L waves), which are the slowest and weakest; shear (S) waves, which are transverse waves and carry most of the energy; and pressure (P) waves, which are longitudinal waves and travel the fastest. The speed of $\mathrm{P}$ waves is approximately $7.0 \mathrm{~km} / \mathrm{s},$ and that of $\mathrm{S}$ waves is about $4.0 \mathrm{~km} / \mathrm{s}$. Animals seem to feel the P waves. If a dog senses the arrival of P waves and starts barking 30.0 s before an earthquake is felt by humans, approximately how far is the dog from the earthquake's epicenter?

Prabhu Ramji

Numerade Educator

Problem 40

A string with a mass of $30.0 \mathrm{~g}$ and a length of $2.00 \mathrm{~m}$ is stretched under a tension of $70.0 \mathrm{~N}$. How much power must be supplied to the string to generate a traveling wave that has a frequency of $50.0 \mathrm{~Hz}$ and an amplitude of $4.00 \mathrm{~cm} ?$

Prabhu Ramji

Numerade Educator

Problem 41

A string with a linear mass density of $0.100 \mathrm{~kg} / \mathrm{m}$ is under a tension. of $100 .$ N. How much power must be supplied to the string to generate a sinusoidal wave of amplitude $2.00 \mathrm{~cm}$ and frequency $120 . \mathrm{Hz} ?$

Prabhu Ramji

Numerade Educator

Problem 42

A sinusoidal wave on a string is described by the equation $y=(0.100 \mathrm{~m}) \sin (0.750 x-40.0 t),$ where $x$ and $y$ are in meters and $t$ is in seconds. If the linear mass density of the string is $10.0 \mathrm{~g} / \mathrm{m},$ determine
(a) the phase constant, (b) the phase of the wave at $x=2.00 \mathrm{~cm}$ and $t=0.100 \mathrm{~s}$,
(c) the speed of the wave, (d) the wavelength, (e) the frequency, and (f) the power transmitted by the wave.

Prabhu Ramji

Numerade Educator

Problem 43

In an acoustics experiment, a piano string with a mass of $5.00 \mathrm{~g}$ and a length of $70.0 \mathrm{~cm}$ is held under tension by running the string over a frictionless pulley and hanging a $250 .-\mathrm{kg}$ weight from it. The whole system is placed in an elevator.
a) What is the fundamental frequency of oscillation for the string when the elevator is at rest?
b) With what acceleration and in what direction (up or down) should the elevator move for the string to produce the proper frequency of $440 . \mathrm{Hz}$ corresponding to middle A?

Prabhu Ramji

Numerade Educator

Problem 44

A string is $35.0 \mathrm{~cm}$ long and has a mass per unit length of $5.51 \cdot 10^{-4} \mathrm{~kg} / \mathrm{m} .$ What tension must be applied to the string so that it vibrates at the fundamental frequency of $660 . \mathrm{Hz?}$

Prabhu Ramji

Numerade Educator

Problem 45

A $2.00-\mathrm{m}$ -long string of mass $10.0 \mathrm{~g}$ is clamped at both ends. The tension in the string is $150 .$ N.
a) What is the speed of a wave on this string?
b) The string is plucked so that it oscillates. What is the wavelength and frequency of the resulting wave if it produces a standing wave with two antinodes?

Prabhu Ramji

Numerade Educator

Problem 46

Write the equation for a standing wave that has three antinodes of amplitude $2.00 \mathrm{~cm}$ on a $3.00-\mathrm{m}$ -long string that is fixed at both ends and vibrates 15.0 times a second. The time $t=0$ is chosen to be an instant when the string is flat. If a wave pulse were propagated along this string, how fast would it travel?

Prabhu Ramji

Numerade Educator

Problem 47

A $3.00-\mathrm{m}$ -long string, fixed at both ends, has a mass of $6.00 \mathrm{~g}$. If you want to set up a standing wave in this string having a frequency of $300 . \mathrm{Hz}$ and three antinodes, what tension should you put the string under?

Ajay Singhal

Numerade Educator

Problem 48

A cowboy walks at a pace of about two steps per second, holding a glass of diameter $10.0 \mathrm{~cm}$ that contains milk. The milk sloshes higher and higher in the glass until it eventually starts to spill over the top. Determine the maximum speed of the waves in the milk.

Ajay Singhal

Numerade Educator

Problem 49

Students in a lab produce standing waves on stretched strings connected to vibration generators. One such wave is described by the wave function $y(x, t)=(2.00 \mathrm{~cm}) \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) x\right] \cos \left[\left(150 . \mathrm{s}^{-1}\right) t\right],$ where $y$ is the
transverse displacement of the string, $x$ is the position along the string, and $t$ is time. Rewrite this wave function in the form for a wave moving in the positive $x$ -direction and a wave moving in the negative $x$ -direction:
$y(x, t)=f(x-v t)+g(x+v t) ;$ that is, find the functions $f$ and $g$ and the speed, $v$

Suzanne W.

Numerade Educator

Problem 50

An array of wave emitters, as shown in the figure, emits a wave of wavelength $\lambda$ that is to be detected at a distance $L$ directly above the rightmost emitter. The distance between adjacent wave emitters is $d$.
a) Show that when $L \gg d$, the wave from the $n$ th emitter (counting from right to left with $n=0$ being the rightmost emitter) has to travel an extra distance of $\Delta s=n^{2}\left(d^{2} / 2 L\right)$.
b) If $\lambda=d^{2} / 2 L$, will the interference at the detector be constructive or destructive?
c) If $\lambda=d^{2} / 2 L=10^{-3} \mathrm{~m}$ and $L=1.00 \cdot 10^{3} \mathrm{~m},$ what is $d$, the distance between adjacent emitters?

Eduard Sanchez

Numerade Educator

Problem 51

A small ball floats in the center of a circular pool that has a radius of $5.00 \mathrm{~m}$. Three wave generators are placed at the edge of the pool, separated by $120 .^{\circ}$. The first wave generator operates at a frequency of $2.00 \mathrm{~Hz}$. The second wave generator operates at a frequency of $3.00 \mathrm{~Hz}$. The third wave generator operates at a frequency of $4.00 \mathrm{~Hz}$. If the speed of each water wave is $5.00 \mathrm{~m} / \mathrm{s}$, and the amplitude of the waves is the same, sketch the height of the ball as a function of time from $t=0$ to $t=2.00 \mathrm{~s}$, assuming that the water surface is at zero height. Assume that all the wave generators impart a phase shift of zero. How would your answer change if one of the wave generators was moved to a different location at the edge of the pool?

Dominador Tan

Numerade Educator

Problem 52

A string with linear mass density $\mu=0.0250 \mathrm{~kg} / \mathrm{m}$ under a tension of $F=250 . \mathrm{N}$ is oriented in the $x$ -direction. Two transverse waves of equal amplitude and with a phase angle of zero (at $t=0$ ) but with different angular frequencies $(\omega=3000 . \mathrm{rad} / \mathrm{s}$ and $\omega / 3=1000 . \mathrm{rad} / \mathrm{s})$ are created in the string by an oscillator located at $x=0 .$ The resulting waves, which travel in the positive $x$ -direction, are reflected at a distant point where both waves have a node, so there is a similar pair of waves traveling in the negative $x$ -direction. Find the values of $x$ at which the first two nodes in the standing wave are produced by these four waves.

Ankur S

Numerade Educator

Problem 53

The equation for a standing wave on a string with mass density $\mu$ is $y(x, t)=2 A \cos (\omega t) \sin (\kappa x) .$ Show that the average kinetic energy and potential energy over time for this wave per unit length of string are given by $K_{\text {ave }}(x)=\mu \omega^{2} A^{2} \sin ^{2} \kappa x$ and $U_{\text {ave }}(x)=F(\kappa A)^{2}\left(\cos ^{2} \kappa x\right)$

Dominador Tan

Numerade Educator

Problem 54

A sinusoidal wave traveling on a string is moving in the positive $x$ -direction. The wave has a wavelength of $4 \mathrm{~m}$, a frequency of $50.0 \mathrm{~Hz}$, and an amplitude of $3.00 \mathrm{~cm}$. What is the wave function for this wave?

Prabhu Ramji

Numerade Educator

Problem 55

A guitar string with a mass of $10.0 \mathrm{~g}$ is $1.00 \mathrm{~m}$ long and attached to the guitar at two points separated by $65.0 \mathrm{~cm}$.
a) What is the frequency of the first harmonic of this string when it is placed under a tension of $81.0 \mathrm{~N}$ ?
b) If the guitar string is replaced by a heavier one that has a mass of $16.0 \mathrm{~g}$ and is $1.00 \mathrm{~m}$ long, what is the frequency of the replacement string's first harmonic?

Prabhu Ramji

Numerade Educator

Problem 56

Write the equation for a sinusoidal wave propagating in the negative $x$ -direction with a speed of $120 . \mathrm{m} / \mathrm{s}$, if a particle in the medium in which the wave is moving is observed to swing back and forth through a $6.00-\mathrm{cm}$ range in $4.00 \mathrm{~s}$. Assume that $t=0$ is taken to be the instant when the particle is at $y=0$ and that the particle moves in the positive $y$ -direction immediately after $t=0 .$

Prabhu Ramji

Numerade Educator

Problem 57

Shown in the figure is a plot of the displacement, $y$, due to a sinusoidal wave traveling along a string as a function of time, $t$. What are the (a) period, (b) maximum speed, and (c) maximum acceleration of this wave perpendicular to the direction in which it is traveling?

Prabhu Ramji

Numerade Educator

Problem 58

A $50.0-\mathrm{cm}$ -long wire with a mass of $10.0 \mathrm{~g}$ is under a tension of $50.0 \mathrm{~N}$ Both ends of the wire are held rigidly while it is plucked.
a) What is the speed of the waves on the wire?
b) What is the fundamental frequency of the standing wave?
c) What is the frequency of the third harmonic?

Prabhu Ramji

Numerade Educator

Problem 59

What is the wave speed along a brass wire with a radius of $0.500 \mathrm{~mm}$ stretched at a tension of $125 \mathrm{~N}$ ? The density of brass is $8.60 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$

Prabhu Ramji

Numerade Educator

Problem 60

Two steel wires are stretched under the same tension. The first wire has a diameter of $0.500 \mathrm{~mm}$, and the second wire has a diameter of $1.00 \mathrm{~mm}$. If the speed of waves traveling along the first wire is $50.0 \mathrm{~m} / \mathrm{s}$, what is the speed of waves traveling along the second wire?

Prabhu Ramji

Numerade Educator

Problem 61

The middle-C key (key 40 ) on a piano corresponds to a fundamental frequency of about $262 \mathrm{~Hz}$, and the soprano-C key (key 64 ) corresponds to a fundamental frequency of $1046.5 \mathrm{~Hz}$. If the strings used for both keys are identical in density and length, determine the ratio of the tensions in the two strings.

Prabhu Ramji

Numerade Educator

Problem 62

A sinusoidal wave travels along a stretched string. A point along the string has a maximum velocity of $1.00 \mathrm{~m} / \mathrm{s}$ and a maximum displacement of $2.00 \mathrm{~cm}$. What is the maximum acceleration at that point?

Prabhu Ramji

Numerade Educator

Problem 63

As shown in the figure, a sinusoidal wave travels to the right at a speed of $v_{1}$ along string 1 , which has linear mass density $\mu_{1}$. This wave has frequency $f_{1}$ and wavelength $\lambda_{1}$. Since string 1 is attached to string 2 (which has linear mass density $\mu_{2}=3 \mu_{1}$ ), the first wave will excite a new wave in string $2,$ which will also move to the right. What is the frequency, $f_{2}$, of the wave produced in string $2 ?$ What is the speed, $v_{2}$, of the wave produced in string 2 ? What is the wavelength, $\lambda_{2}$, of the wave produced in string $2 ?$ Write all answers in terms of $f_{1}, v_{1},$ and $\lambda_{1}$

Ajay Singhal

Numerade Educator

Problem 64

The tension in a $2.70-\mathrm{m}$ -long, $1.00-\mathrm{cm}$ -diameter steel cable $\left(\rho=7800 . \mathrm{kg} / \mathrm{m}^{3}\right)$ is $840 . \mathrm{N}$. What is the fundamental frequency of vibration of the cable?

Ajay Singhal

Numerade Educator

Problem 65

A wave traveling on a string has the equation of motion $y(x, t)=0.0200 \sin (5.00 x-8.00 t),$ where $x$ and $y$ are in meters and $t$ is in seconds.
a) Calculate the wavelength and the frequency of the wave.
b) Calculate its velocity.
c) If the linear mass density of the string is $\mu=0.100 \mathrm{~kg} / \mathrm{m},$ what is the tension on the string?

Ajay Singhal

Numerade Educator

Problem 66

Calvin sloshes back and forth in his bathtub, producing a standing wave. What is the frequency of such a wave if the bathtub is $150 . \mathrm{cm}$ long and $80.0 \mathrm{~cm}$ wide and contains water that is $38.0 \mathrm{~cm}$ deep?

Ajay Singhal

Numerade Educator

Problem 67

Consider a guitar string stretching $80.0 \mathrm{~cm}$ between its anchored ends. The string is tuned to play middle $C$, with a frequency of $261.6 \mathrm{~Hz}$, when oscillating in its fundamental mode, that is, with one antinode between the ends. If the string is displaced $2.00 \mathrm{~mm}$ at its midpoint and released to produce this note, what are the wave speed, $v$, and the maximum speed, $v_{\max }$, of the midpoint of the string?

Ajay Singhal

Numerade Educator

Problem 68

Two waves traveling in opposite directions along a string fixed at both ends create a standing wave described by $y(x, t)=$ $1.00 \cdot 10^{-2} \sin (25.0 x) \cos (1200 . t) .$ The string has a linear mass density of
$0.0100 \mathrm{~kg} / \mathrm{m},$ and the tension in the string is supplied by a mass hanging from one end. If the string vibrates in its third harmonic, calculate (a) the length of the string, (b) the velocity of the waves, and (c) the mass of the hanging mass.

Ajay Singhal

Numerade Educator

Problem 69

A sinusoidal transverse wave of wavelength $20.0 \mathrm{~cm}$ and frequency
500. Hz travels along a string in the positive $z$ -direction. The wave oscillations take place in the $x z$ -plane and have an amplitude of $3.00 \mathrm{~cm}$. At time $t=0,$ the displacement of the string at $x=0$ is $z=3.00 \mathrm{~cm}$
a) A photo of the wave is taken at $t=0 .$ Make a simple sketch (including axes) of the string at this time.
b) Determine the speed of the wave.
c) Determine the wave's wave number.
d) If the linear mass density of the string is $30.0 \mathrm{~g} / \mathrm{m},$ what is the tension in the string?
e) Determine the function $D(z, t)$ that describes the displacement $x$ that is produced in the string by this wave.

Manik Pulyani

Numerade Educator

Problem 70

A heavy cable of total mass $M$ and length $L=5.00 \mathrm{~m}$ has one end attached to a rigid support and the other end hanging free. A small transverse displacement is initiated at the bottom of the cable. How long does it take for the displacement to travel to the top of the cable?

Ajay Singhal

Numerade Educator

Problem 71

A rubber band of mass $0.3491 \mathrm{~g}$ is stretched between two fingers, putting each side under a tension of $1.777 \mathrm{~N}$. The overall stretched length of the band is $20.27 \mathrm{~cm} .$ One side of the band is plucked, setting up a vibration in $8.725 \mathrm{~cm}$ of the band's stretched length. What is the lowest-frequency vibration that can be set up on this part of the rubber band? Assume that the band stretches uniformly.

Ajay Singhal

Numerade Educator

Problem 72

A rubber band of mass $0.4245 \mathrm{~g}$ is stretched between two fingers. The overall stretched length of the band is $20.91 \mathrm{~cm} .$ One side of the band is plucked, setting up a vibration in $8.117 \mathrm{~cm}$ of the band's stretched length. The lowest-frequency vibration that can be set up on this part of the rubber band is $184.2 \mathrm{~Hz}$. What is the tension in each side of the rubber band? Assume that the band stretches uniformly.

Ajay Singhal

Numerade Educator

Problem 73

A rubber band of mass $0.1701 \mathrm{~g}$ is stretched between two fingers, putting each side under a tension of $1.851 \mathrm{~N}$. The overall stretched length of the band is $21.55 \mathrm{~cm} .$ One side of the band is plucked, setting up a vibration in the band's stretched length. The lowest-frequency vibration that can be set up on this part of the rubber band is $254.6 \mathrm{~Hz}$. What is the length of the vibrating part of the band? Assume that the band stretches uniformly.

Ajay Singhal

Numerade Educator

Problem 74

A satellite in a circular orbit around the Sun uses a square solar panel as a power source. The edges of the panel are $1.459 \mathrm{~m}$ long. The panel's efficiency is $16.57 \% .$ The satellite is $4.949 \cdot 10^{7} \mathrm{~km}$ from the Sun. How much power does the solar panel provide to the satellite? Assume that the total power output of the Sun is $3.937 \cdot 10^{26} \mathrm{~W}$.

Ajay Singhal

Numerade Educator

Problem 75

A satellite in a circular orbit around the Sun uses a square solar panel as a power source. The panel's efficiency is $16.87 \% .$ The satellite is $6.103 \cdot 10^{7} \mathrm{~km}$ from the Sun. The solar panel provides $5.215 \cdot 10^{3} \mathrm{~W}$ to the satellite. How long are the edges of the solar panel? Assume that the total power output of the Sun is $3.937 \cdot 10^{26} \mathrm{~W}$.

Ajay Singhal

Numerade Educator

Problem 76

A satellite in a circular orbit around the Sun uses a square solar panel as a power source. The edges of the panel are $2.375 \mathrm{~m}$ long. The satellite is $7.257 \cdot 10^{7} \mathrm{~km}$ from the Sun. The solar panel provides $5.768 \cdot 10^{3} \mathrm{~W}$ to the satellite. What is the efficiency of the solar panel? Assume that the total power output of the Sun is $3.937 \cdot 10^{26} \mathrm{~W}$.

Ajay Singhal

Numerade Educator

Problem 77

Consider a string with a linear mass density of $0.2833 \mathrm{~g} / \mathrm{cm}$ and a length of $116.7 \mathrm{~cm},$ which is oscillating as shown in the figure. The tension on the string is $18.25 \mathrm{~N}$. At what frequency is the string vibrating?

Ajay Singhal

Numerade Educator

Problem 78

Consider a string with a linear mass density of $0.1291 \mathrm{~g} / \mathrm{cm}$ and a length of $117.5 \mathrm{~cm}$, which is oscillating as shown in the figure. The string is vibrating at a frequency of $93.63 \mathrm{~Hz}$. What is the tension on the string?

Ajay Singhal

Numerade Educator

Problem 79

Consider a string with a linear mass density of $0.1747 \mathrm{~g} / \mathrm{cm},$ which is oscillating as shown in the figure. The string is under a tension of $10.81 \mathrm{~N}$ and is vibrating at a frequency of $59.47 \mathrm{~Hz}$. What is the length of the string?

Ajay Singhal

Numerade Educator