Physics for Scientists and Engineers

Raymond A. Serway, John W. Jewett

Chapter 16

Wave Motion - all with Video Answers

Educators

Chapter Questions

Problem 1

At $t=0,$ a transverse pulse in a wire is described by the function $$y=\frac{6}{x^{2}+3}$$ where $x$ and $y$ are in meters. Write the function $y(x, t)$ that describes this pulse if it is traveling in the positive $x$ direction with a speed of $4.50 \mathrm{m} / \mathrm{s}$.

Geoffrey Brandt

Geoffrey Brandt

Numerade Educator

Problem 2

Ocean waves with a crest-to-crest distance of $10.0 \mathrm{m}$ can be described by the wave function $$y(x, t)=(0.800 \mathrm{m}) \sin [0.628(x-v t)]$$ where $v=1.20 \mathrm{m} / \mathrm{s} .$ (a) Sketch $y(x, t)$ at $t=0 .$ (b) Sketch $y(x, t)$ at $t=2.00 \mathrm{s} .$ Note that the entire wave form has shifted $2.40 \mathrm{m}$ in the positive $x$ direction in this time interval.

Neil Pandya

Numerade Educator

Problem 3

A pulse moving along the $x$ axis is described by $$y(x, t)=5.00 e^{-(x+5.00 t)^{2}}$$ where $x$ is in meters and $t$ is in seconds. Determine (a) the direction of the wave motion, and (b) the speed of the pulse.

Neil Pandya

Numerade Educator

Problem 4

Two points $A$ and $B$ on the surface of the Earth are at the same longitude and $60.0^{\circ}$ apart in latitude. Suppose that an earthquake at point $A$ creates a $P$ wave that reaches point $B$ by traveling straight through the body of the Earth at a constant speed of $7.80 \mathrm{km} / \mathrm{s} .$ The earthquake also radiates a Rayleigh wave, which travels across the surface of the Earth in an analogous way to a surface wave on water, at $4.50 \mathrm{km} / \mathrm{s}$.
(a) Which of these two seismic waves arrives at $B$ first?
(b) What is the time difference between the arrivals of the two waves at $B ?$ Take the radius of the Earth to be $6370 \mathrm{km}$.

Sandeep Desai

Numerade Educator

Problem 5

S and P waves, simultaneously radiated from the hypocenter of an earthquake, are received at a seismographic station $17.3 \mathrm{s}$ apart. Assume the waves have traveled over the same path at speeds of $4.50 \mathrm{km} / \mathrm{s}$ and $7.80 \mathrm{km} / \mathrm{s} .$ Find the distance from the seismograph to the hypocenter of the quake.

Neil Pandya

Numerade Educator

Problem 6

For a certain transverse wave, the distance between two successive crests is $1.20 \mathrm{m},$ and eight crests pass a given point along the direction of travel every 12.0 s. Calculate the wave speed.

Geoffrey Brandt

Geoffrey Brandt

Numerade Educator

Problem 7

A sinusoidal wave is traveling along a rope. The oscillator that generates the wave completes 40.0 vibrations in $30.0 \mathrm{s}$ Also, a given maximum travels $425 \mathrm{cm}$ along the rope in $10.0 \mathrm{s} .$ What is the wavelength?

Neil Pandya

Numerade Educator

Problem 8

When a particular wire is vibrating with a frequency of $4.00 \mathrm{Hz},$ a transverse wave of wavelength $60.0 \mathrm{cm}$ is produced. Determine the speed of waves along the wire.

Geoffrey Brandt

Geoffrey Brandt

Numerade Educator

Problem 9

A wave is described by $y=(2.00 \mathrm{cm}) \sin (k x-\omega t),$ where $k=2.11 \mathrm{rad} / \mathrm{m}, \omega=3.62 \mathrm{rad} / \mathrm{s}, x$ is in meters, and $t$ is in seconds. Determine the amplitude, wavelength, frequency, and speed of the wave.

Neil Pandya

Numerade Educator

Problem 10

A sinusoidal wave on a string is described by $$y=(0.51 \mathrm{cm}) \sin (k x-\omega t)$$ where $k=3.10 \mathrm{rad} / \mathrm{cm}$ and $\omega=9.30 \mathrm{rad} / \mathrm{s} .$ How far does a wave crest move in 10.0 s? Does it move in the positive or negative $x$ direction?

Neil Pandya

Numerade Educator

Problem 11

Consider further the string shown in Figure 16.10 and treated in Example $16.3 .$ Calculate (a) the maximum transverse speed and (b) the maximum transverse acceleration of a point on the string.

Neil Pandya

Numerade Educator

Problem 12

Consider the sinusoidal wave of Example $16.2,$ with the wave function $$y=(15.0 \mathrm{cm}) \cos (0.157 x-50.3 t)$$ At a certain instant, let point $A$ be at the origin and point $B$ be the first point along the $x$ axis where the wave is $60.0^{\circ}$ out of phase with point $A .$ What is the coordinate of point $B ?$

Neil Pandya

Numerade Educator

Problem 13

A sinusoidal wave is described by $$y=(0.25 \mathrm{m}) \sin (0.30 x-40 t)$$ where $x$ and $y$ are in meters and $t$ is in seconds. Determine for this wave the (a) amplitude, (b) angular frequency, (c) angular wave number, (d) wavelength, (e) wave speed, and (f) direction of motion.

Prabhat Tyagi

Numerade Educator

Problem 14

(a) Plot $y$ versus $t$ at $x=0$ for a sinusoidal wave of the form $y=(15.0 \mathrm{cm}) \cos (0.157 x-50.3 t),$ where $x$ and $y$ are in centimeters and $t$ is in seconds. (b) Determine the period of vibration from this plot and compare your result with the value found in Example 16.2.

Neil Pandya

Numerade Educator

Problem 15

(a) Write the expression for $y$ as a function of $x$ and $t$ for a sinusoidal wave traveling along a rope in the negative $x$ direction with the following characteristics: $A=8.00 \mathrm{cm}$ $\lambda=80.0 \mathrm{cm}, \quad f=3.00 \mathrm{Hz}, \quad$ and $\quad y(0, \quad t)=0 \quad$ at $\quad t=0$. (b) What If? Write the expression for $y$ as a function of $x$ and $t$ for the wave in part (a) assuming that $y(x, 0)=0$ at the point $x=10.0 \mathrm{cm}$.

Sandeep Desai

Numerade Educator

Problem 16

A sinusoidal wave traveling in the $-x$ direction (to the left) has an amplitude of $20.0 \mathrm{cm},$ a wavelength of $35.0 \mathrm{cm},$ and a frequency of $12.0 \mathrm{Hz}$. The transverse position of an element of the medium at $t=0, x=0$ is $y=-3.00 \mathrm{cm},$ and the element has a positive velocity here. (a) Sketch the wave at $t=0 .$ (b) Find the angular wave number, period, angular frequency, and wave speed of the wave. (c) Write an expression for the wave function $y(x, t)$.

Neil Pandya

Numerade Educator

Problem 17

A transverse wave on a string is described by the wave function $$y=(0.120 \mathrm{m}) \sin [(\pi x / 8)+4 \pi t]$$ (a) Determine the transverse speed and acceleration at $t=0.200 \mathrm{s}$ for the point on the string located at $x=$ 1.60 $\mathrm{m}$. (b) What are the wavelength, period, and speed of propagation of this wave?

Sandeep Desai

Numerade Educator

Problem 18

A transverse sinusoidal wave on a string has a period $T=25.0 \mathrm{ms}$ and travels in the negative $x$ direction with a speed of $30.0 \mathrm{m} / \mathrm{s} .$ At $t=0,$ a particle on the string at $x=0$ has a transverse position of $2.00 \mathrm{cm}$ and is traveling downward with a speed of $2.00 \mathrm{m} / \mathrm{s} .$ (a) What is the amplitude of the wave? (b) What is the initial phase angle? (c) What is the maximum transverse speed of the string? (d) Write the wave function for the wave.

Sandeep Desai

Numerade Educator

Problem 19

A sinusoidal wave of wavelength $2.00 \mathrm{m}$ and amplitude $0.100 \mathrm{m}$ travels on a string with a speed of $1.00 \mathrm{m} / \mathrm{s}$ to the right. Initially, the left end of the string is at the origin. Find (a) the frequency and angular frequency, (b) the angular wave number, and (c) the wave function for this wave. Determine the equation of motion for (d) the left end of the string and (e) the point on the string at $x=$ $1.50 \mathrm{m}$ to the right of the left end. (f) What is the maximum speed of any point on the string?

Neil Pandya

Numerade Educator

Problem 20

A wave on a string is described by the wave function $y=(0.100 \mathrm{m}) \sin (0.50 x-20 t) .$ (a) Show that a particle in
the string at $x=2.00 \mathrm{m}$ executes simple harmonic motion. (b) Determine the frequency of oscillation of this particular point.

Neil Pandya

Numerade Educator

Problem 21

A telephone cord is $4.00 \mathrm{m}$ long. The cord has a mass of $0.200 \mathrm{kg} .$ A transverse pulse is produced by plucking one end of the taut cord. The pulse makes four trips down and back along the cord in 0.800 s. What is the tension in the cord?

Neil Pandya

Numerade Educator

Problem 22

Transverse waves with a speed of $50.0 \mathrm{m} / \mathrm{s}$ are to be produced in a taut string. A 5.00 -m length of string with a total mass of $0.0600 \mathrm{kg}$ is used. What is the required tension?

Neil Pandya

Numerade Educator

Problem 23

A piano string having a mass per unit length equal to $5.00 \times 10^{-3} \mathrm{kg} / \mathrm{m}$ is under a tension of $1350 \mathrm{N} .$ Find the speed of a wave traveling on this string.

Neil Pandya

Numerade Educator

Problem 24

A transverse traveling wave on a taut wire has an amplitude of $0.200 \mathrm{mm}$ and a frequency of $500 \mathrm{Hz}$. It travels with a speed of $196 \mathrm{m} / \mathrm{s}$. (a) Write an equation in SI units of the form $y=A \sin (k x-\omega t)$ for this wave. (b) The mass per unit length of this wire is $4.10 \mathrm{g} / \mathrm{m} .$ Find the tension in the wire.

Geoffrey Brandt

Geoffrey Brandt

Numerade Educator

Problem 25

An astronaut on the Moon wishes to measure the local value of the free-fall acceleration by timing pulses traveling down a wire that has an object of large mass suspended from it. Assume a wire has a mass of $4.00 \mathrm{g}$ and a length of $1.60 \mathrm{m},$ and that a $3.00-\mathrm{kg}$ object is suspended from it. A pulse requires 36.1 ms to traverse the length of the wire. Calculate $g_{\text {Moon }}$ from these data. (You may ignore the mass of the wire when calculating the tension in it.)

Sandeep Desai

Numerade Educator

Problem 26

Transverse pulses travel with a speed of $200 \mathrm{m} / \mathrm{s}$ along a taut copper wire whose diameter is $1.50 \mathrm{mm}$. What is the tension in the wire? (The density of copper is $8.92 \mathrm{g} / \mathrm{cm}^{3} .$)

Geoffrey Brandt

Geoffrey Brandt

Numerade Educator

Problem 27

Transverse waves travel with a speed of $20.0 \mathrm{m} / \mathrm{s}$ in a string under a tension of $6.00 \mathrm{N} .$ What tension is required for a wave speed of $30.0 \mathrm{m} / \mathrm{s}$ in the same string?

Geoffrey Brandt

Geoffrey Brandt

Numerade Educator

Problem 28

A simple pendulum consists of a ball of mass $M$ hanging from a uniform string of mass $m$ and length $I,$ with $m \ll M$ If the period of oscillations for the pendulum is $T$, determine the speed of a transverse wave in the string when the pendulum hangs at rest.

Sandeep Desai

Numerade Educator

Problem 29

The elastic limit of the steel forming a piece of wire is equal to $2.70 \times 10^{8} \mathrm{Pa} .$ What is the maximum speed at which transverse wave pulses can propagate along this wire without exceeding this stress? (The density of steel is $7.86 \times 10^{3}$ $\left.\mathrm{km} / \mathrm{m}^{3} .\right)$

Sandeep Desai

Numerade Educator

Problem 30

Review problem. A light string with a mass per unit length of $8.00 \mathrm{g} / \mathrm{m}$ has its ends tied to two walls separated by a distance equal to three fourths of the length of the string (Fig. P16.30). An object of mass $m$ is suspended from the center of the string, putting a tension in the string. (a) Find an expression for the transverse wave speed in the string as a function of the mass of the hanging object. (b) What should be the mass of the object suspended from the string in order to produce a wave speed of $60.0 \mathrm{m} / \mathrm{s} ?$ (FIGURE CAN'T COPY)

Neil Pandya

Numerade Educator

Problem 31

A 30.0 -m steel wire and a 20.0 -m copper wire, both with 1.00 -mm diameters, are connected end to end and stretched to a tension of $150 \mathrm{N}$. How long does it take a transverse wave to travel the entire length of the two wires?

Neil Pandya

Numerade Educator

Problem 32

A light string of mass $m$ and length $L$ has its ends tied to two walls that are separated by the distance D. Two objects, each of mass $M$, are suspended from the string as in Figure $\mathrm{P} 16.32 .$ If a wave pulse is sent from point $A$, how long does it take to travel to point $B ?$ (FIGURE CAN'T COPY)

Sandeep Desai

Numerade Educator

Problem 33

A student taking a quiz finds on a reference sheet the two equations $$f=1 / T \text { and } v=\sqrt{T / \mu}$$ She has forgotten what $T$ represents in each equation. (a) Use dimensional analysis to determine the units required for $T$ in each equation. (b) Identify the physical quantity each $T$ represents.

Sandeep Desai

Numerade Educator

Problem 34

A taut rope has a mass of $0.180 \mathrm{kg}$ and a length of $3.60 \mathrm{m}$ What power must be supplied to the rope in order to generate sinusoidal waves having an amplitude of $0.100 \mathrm{m}$ and a wavelength of $0.500 \mathrm{m}$ and traveling with a speed of $30.0 \mathrm{m} / \mathrm{s} ?$

Sandeep Desai

Numerade Educator

Problem 35

A two-dimensional water wave spreads in circular ripples. Show that the amplitude $A$ at a distance $r$ from the initial disturbance is proportional to $1 / \sqrt{r}$. (Suggestion: Consider the energy carried by one outward-moving ripple.)

Shoukat Ali

Other Schools

Problem 36

Transverse waves are being generated on a rope under constant tension. By what factor is the required power increased or decreased if (a) the length of the rope is doubled and the angular frequency remains constant, (b) the amplitude is doubled and the angular frequency is halved, (c) both the wavelength and the amplitude are doubled, and (d) both the length of the rope and the wavelength are halved?

Vidhi Bhatt

Numerade Educator

Problem 37

Sinusoidal waves $5.00 \mathrm{cm}$ in amplitude are to be transmitted along a string that has a linear mass density of $4.00 \times 10^{-2} \mathrm{kg} / \mathrm{m} .$ If the source can deliver a maximum power of $300 \mathrm{W}$ and the string is under a tension of $100 \mathrm{N}$ what is the highest frequency at which the source can operate?

Sandeep Desai

Numerade Educator

Problem 38

It is found that a $6.00-\mathrm{m}$ segment of a long string contains four complete waves and has a mass of $180 \mathrm{g} .$ The string is vibrating sinusoidally with a frequency of $50.0 \mathrm{Hz}$ and a peak-to-valley distance of $15.0 \mathrm{cm} .$ (The "peak-to-valley" distance is the vertical distance from the farthest positive position to the farthest negative position.) (a) Write the function that describes this wave traveling in the positive $x$ direction. (b) Determine the power being supplied to the string.

Sandeep Desai

Numerade Educator

Problem 39

A sinusoidal wave on a string is described by the equation $$y=(0.15 \mathrm{m}) \sin (0.80 x-50 t)$$ where $x$ and $y$ are in meters and $t$ is in seconds. If the mass per unit length of this string is $12.0 \mathrm{g} / \mathrm{m},$ determine (a) the speed of the wave, (b) the wavelength, (c) the frequency, and (d) the power transmitted to the wave.

Sandeep Desai

Numerade Educator

Problem 40

The wave function for a wave on a taut string is $$y(x, t)=(0.350 \mathrm{m}) \sin (10 \pi t-3 \pi x+\pi / 4)$$ where $x$ is in meters and $t$ in seconds. (a) What is the average rate at which energy is transmitted along the string if the linear mass density is $75.0 \mathrm{g} / \mathrm{m} ?$ (b) What is the energy contained in each cycle of the wave?

Nishant Kumar

Numerade Educator

Problem 41

A horizontal string can transmit a maximum power $\mathscr{P}_{0}$ (without breaking) if a wave with amplitude $A$ and angular frequency $\omega$ is traveling along it. In order to increase this maximum power, a student folds the string and uses this "double string" as a medium. Determine the maximum power that can be transmitted along the "double string," assuming that the tension is constant.

Neil Pandya

Numerade Educator

Problem 42

In a region far from the epicenter of an earthquake, a seismic wave can be modeled as transporting energy in a single direction without absorption, just as a string wave does. Suppose the seismic wave moves from granite into mudfill with similar density but with a much lower bulk modulus. Assume the speed of the wave gradually drops by a factor of $25.0,$ with negligible reflection of the wave. Will the amplitude of the ground shaking increase or decrease? By what factor? This phenomenon led to the collapse of part of the Nimitz Freeway in Oakland, California, during the Loma Prieta earthquake of 1989.

Sandeep Desai

Numerade Educator

Problem 43

(a) Evaluate $A$ in the scalar equality $(7+3) 4=A$. (b) Evaluate $A, B,$ and $C$ in the vector equality $7.00 \hat{\mathbf{i}}+$ $3.00 \hat{\mathbf{k}}=A \hat{\mathbf{i}}+B \hat{\mathbf{j}}+C \hat{\mathbf{k}} .$ Explain how you arrive at the answers to convince a student who thinks that you cannot solve a single equation for three different unknowns. (c) What If? The functional equality or identity $A+B \cos (C x+D t+E)=(7.00 \mathrm{mm}) \cos (3 x+4 t+2)$ is true for all values of the variables $x$ and $t,$ which are measured in meters and in seconds, respectively. Evaluate the constants $A, B, C, D,$ and $E .$ Explain how you arrive at the answers.

Neil Pandya

Numerade Educator

Problem 44

Show that the wave function $y=e^{b(x-v t)}$ is a solution of the linear wave equation (Eq. 16.27 ), where $b$ is a constant.

Sandeep Desai

Numerade Educator

Problem 45

Show that the wave function $y=\ln [b(x-v t)]$ is a solution to Equation 16.27 , where $b$ is a constant.

Shoukat Ali

Other Schools

Problem 46

(a) Show that the function $y(x, t)=x^{2}+v^{2} t^{2}$ is a solution to the wave equation. (b) Show that the function in part (a) can be written as $f(x+v t)+g(x-v t),$ and determine the functional forms for $f$ and $g$. (c) What If? Repeat parts (a) and (b) for the function $y(x, t)=\sin (x) \cos (v t)$.

Vidhi Bhatt

Numerade Educator

Problem 47

"The wave" is a particular type of pulse that can propagate through a large crowd gathered at a sports arena to watch a soccer or American football match (Figure P16.47). The elements of the medium are the spectators, with zero position corresponding to their being seated and maximum position corresponding to their standing and raising their arms. When a large fraction of the spectators participate in the wave motion, a somewhat stable pulse shape can develop. The wave speed depends on people's reaction time, which is typically on the order of 0.1 s. Estimate the order of magnitude, in minutes, of the time required for such a pulse to make one circuit around a large sports stadium. State the quantities you measure or estimate and their values.

Neil Pandya

Numerade Educator

Problem 48

A traveling wave propagates according to the expression $y=(4.0 \mathrm{cm}) \sin (2.0 x-3.0 t),$ where $x$ is in centimeters and $t$ is in seconds. Determine (a) the amplitude, (b) the wavelength, (c) the frequency, (d) the period, and (e) the direction of travel of the wave.

Vishal Gupta

Numerade Educator

Problem 49

The wave function for a traveling wave on a taut string is (in SI units)
$$y(x, t)=(0.350 \mathrm{m}) \sin (10 \pi t-3 \pi x+\pi / 4)$$
(a) What are the speed and direction of travel of the wave?
(b) What is the vertical position of an element of the string at $t=0, x=0.100 \mathrm{m} ?$ (c) What are the wavelength and frequency of the wave? (d) What is the maximum magnitude of the transverse speed of the string?

Neil Pandya

Numerade Educator

Problem 50

A transverse wave on a string is described by the equation $y(x, t)=(0.350 \mathrm{m}) \sin [(1.25 \mathrm{rad} / \mathrm{m}) x+(99.6 \mathrm{rad} / \mathrm{s}) t]$ Consider the element of the string at $x=0 .$ (a) What is the time interval between the first two instants when this element has a position of $y=0.175 \mathrm{m} ?$ (b) What distance does the wave travel during this time interval?

Neil Pandya

Numerade Educator

Problem 51

Motion picture film is projected at 24.0 frames per second. Each frame is a photograph $19.0 \mathrm{mm}$ high. At what constant speed does the film pass into the projector?

Neil Pandya

Numerade Educator

Problem 52

A block of mass $M,$ supported by a string, rests on an incline making an angle $\theta$ with the horizontal (Fig. P16.52). The length of the string is $L$, and its mass is $m<M .$ Derive an expression for the time interval required for a transverse wave to travel from one end of the string to the other. (FIGURE CAN'T COPY)

Sandeep Desai

Numerade Educator

Problem 53

A $2.00-\mathrm{kg}$ block hangs from a rubber cord, being supported so that the cord is not stretched. The unstretched length of the cord is $0.500 \mathrm{m},$ and its mass is 5.00 g. The "spring constant" for the cord is $100 \mathrm{N} / \mathrm{m} .$ The block is released and stops at the lowest point. (a) Determine the tension in the cord when the block is at this lowest point. (b) What is the length of the cord in this "stretched" position? (c) Find the speed of a transverse wave in the cord if the block is held in this lowest position.

Sandeep Desai

Numerade Educator

Problem 54

A block of mass $M$ hangs from a rubber cord. The block is supported so that the cord is not stretched. The unstretched length of the cord is $L_{0}$ and its mass is $m,$ much less than $M .$ The "spring constant" for the cord is $k .$ The block is released and stops at the lowest point. (a) Determine the tension in the string when the block is at this lowest point. (b) What is the length of the cord in this "stretched" position? (c) Find the speed of a transverse wave in the cord if the block is held in this lowest position.

Sandeep Desai

Numerade Educator

Problem 55

(a) Determine the speed of transverse waves on a string under a tension of $80.0 \mathrm{N}$ if the string has a length of $2.00 \mathrm{m}$ and a mass of $5.00 \mathrm{g} .$ (b) Calculate the power required to generate these waves if they have a wavelength of $16.0 \mathrm{cm}$ and an amplitude of $4.00 \mathrm{cm} .$

Neil Pandya

Numerade Educator

Problem 56

A sinusoidal wave in a rope is described by the wave function $$y=(0.20 \mathrm{m}) \sin (0.75 \pi x+18 \pi t)$$ where $x$ and $y$ are in meters and $t$ is in seconds. The rope has a linear mass density of $0.250 \mathrm{kg} / \mathrm{m} .$ If the tension in the rope is provided by an arrangement like the one illustrated in Figure $16.12,$ what is the value of the suspended mass?

Sandeep Desai

Numerade Educator

Problem 57

A block of mass $0.450 \mathrm{kg}$ is attached to one end of a cord of mass $0.00320 \mathrm{kg} ;$ the other end of the cord is attached to a fixed point. The block rotates with constant angular speed in a circle on a horizontal frictionless table. Through what angle does the block rotate in the time that a transverse wave takes to travel along the string from the center of the circle to the block?

Neil Pandya

Numerade Educator

Problem 58

A wire of density $\rho$ is tapered so that its cross-sectional area varies with $x$ according to $$A=\left(1.0 \times 10^{-3} x+0.010\right) \mathrm{cm}^{2}$$
(a) If the wire is subject to a tension $T$, derive a relationship for the speed of a wave as a function of position.
(b) What If? If the wire is aluminum and is subject to a tension of $24.0 \mathrm{N},$ determine the speed at the origin and at $x=10.0 \mathrm{m}$.

Sandeep Desai

Numerade Educator

Problem 59

A rope of total mass $m$ and length $L$ is suspended vertically. Show that a transverse pulse travels the length of the rope in a time interval $\Delta t=2 \sqrt{L} / g .$ (Suggestion: First find an expression for the wave speed at any point a distance $x$ from the lower end by considering the tension in the rope as resulting from the weight of the segment below that point.)

Sandeep Desai

Numerade Educator

Problem 60

If an object of mass $M$ is suspended from the bottom of the rope in Problem $59,$ (a) show that the time interval for a transverse pulse to travel the length of the rope is $$\Delta t=2 \sqrt{\frac{L}{m g}}(\sqrt{M+m}-\sqrt{M})$$ What If? (b) Show that this reduces to the result of Problem 59 when $M=0 .$ (c) Show that for $m<<M,$ the expression in part (a) reduces to $$\Delta t=\sqrt{\frac{m L}{M g}}$$

Supratim Pal

Numerade Educator

Problem 61

It is stated in Problem 59 that a pulse travels from the bottom to the top of a hanging rope of length $L$ in a time interval $\Delta t=2 \sqrt{L / g} .$ Use this result to answer the following questions. (It is not necessary to set up any new integrations.) (a) How long does it take for a pulse to travel halfway up the rope? Give your answer as a fraction of the quantity $2 \sqrt{L / g}$. (b) A pulse starts traveling up the rope. How far has it traveled after a time interval $\sqrt{L / g ?}$

Keshav Singh

Numerade Educator

Problem 62

Determine the speed and direction of propagation of each of the following sinusoidal waves, assuming that $x$ and $y$ are measured in meters and $t$ in seconds.
(a) $y=0.60 \cos (3.0 x-15 t+2)$
(b) $y=0.40 \cos (3.0 x+15 t-2)$
(c) $y=1.2 \sin (15 t+2.0 x)$
(d) $y=0.20 \sin [12 t-(x / 2)+\pi]$

Sandeep Desai

Numerade Educator

Problem 63

An aluminum wire is clamped at each end under zero tension at room temperature. The tension in the wire is increased by reducing the temperature, which results in a decrease in the wire's equilibrium length. What strain $(\Delta L / L)$ results in a transverse wave speed of $100 \mathrm{m} / \mathrm{s}$ ? Take the cross-sectional area of the wire to be $5.00 \times 10^{-6} \mathrm{m}^{2}$ the density to be $2.70 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3},$ and Young's modulus to be $7.00 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}$.

Vishal Gupta

Numerade Educator

Problem 64

If a loop of chain is spun at high speed, it can roll along the ground like a circular hoop without slipping or collapsing. Consider a chain of uniform linear mass density $\mu$ whose center of mass travels to the right at a high speed $v_{0}$. (a) Determine the tension in the chain in terms of $\mu$ and $v_{0}$. (b) If the loop rolls over a bump, the resulting deformation of the chain causes two transverse pulses to propagate along the chain, one moving clockwise and one moving counterclockwise. What is the speed of the pulses traveling along the chain? (c) Through what angle does each pulse travel during the time it takes the loop to make one revolution?

Sandeep Desai

Numerade Educator

Problem 65

(a) Show that the speed of longitudinal waves along a spring of force constant $k$ is $v=\sqrt{k L / \mu},$ where $L$ is the unstretched length of the spring and $\mu$ is the mass per unit length. (b) A spring with a mass of $0.400 \mathrm{kg}$ has an unstretched length of $2.00 \mathrm{m}$ and a force constant of $100 \mathrm{N} / \mathrm{m} .$ Using the result you obtained in (a), determine the speed of longitudinal waves along this spring.

Sandeep Desai

Numerade Educator

Problem 66

A string of length $L$ consists of two sections. The left half has mass per unit length $\mu=\mu_{0} / 2,$ while the right has a mass per unit length $\mu^{\prime}=3 \mu=3 \mu_{0} / 2 .$ Tension in the string is $T_{0} .$ Notice from the data given that this string has the same total mass as a uniform string of length $L$ and mass per unit length $\mu_{0}$. (a) Find the speeds $v$ and $v^{\prime}$ at which transverse pulses travel in the two sections. Express the speeds in terms of $T_{0}$ and $\mu_{0},$ and also as multiples of the speed $v_{0}=\left(T_{0} / \mu_{0}\right)^{1 / 2} .$ (b) Find the time interval required for a pulse to travel from one end of the string to the other. Give your result as a multiple of $\Delta t_{0}=L / v_{0}$.

Sandeep Desai

Numerade Educator

Problem 67

A pulse traveling along a string of linear mass density $\mu$ is described by the wave function $$y=\left[A_{0} e^{-b x}\right] \sin (k x-\omega t)$$ where the factor in brackets before the sine function is said to be the amplitude. (a) What is the power $\mathscr{P}(x)$ carried by this wave at a point $x ?$ (b) What is the power carried by this wave at the origin? (c) Compute the ratio $\mathscr{P}(x) / \mathscr{P}(0)$.

Sandeep Desai

Numerade Educator

Problem 68

An earthquake on the ocean floor in the Gulf of Alaska produces a tsunami (sometimes incorrectly called a "tidal wave") that reaches Hilo, Hawaii, 4 450 km away, in a time interval of $9 \mathrm{h}$ 30 min. Tsunamis have enormous wavelengths $(100 \text { to } 200 \mathrm{km}),$ and the propagation speed for these waves is $v \approx \sqrt{g d},$ where $\bar{d}$ is the average depth of the water. From the information given, find the average wave speed and the average ocean depth between Alaska and Hawaii. (This method was used in 1856 to estimate the average depth of the Pacific Ocean long before soundings were made to give a direct determination.)

Sandeep Desai

Numerade Educator

Problem 69

A string on a musical instrument is held under tension $T$ and extends from the point $x=0$ to the point $x=L .$ The string is overwound with wire in such a way that its mass per unit length $\mu(x)$ increases uniformly from $\mu_{0}$ at $x=0$ to $\mu_{L}$ at $x=L .$ (a) Find an expression for $\mu(x)$ as a function of $x$ over the range $0 \leq x \leq L$. (b) Show that the time interval required for a transverse pulse to travel the length of the string is given by $$\Delta t=\frac{2 L\left(\mu_{L}+\mu_{0}+\sqrt{\mu_{L} \mu_{0}}\right)}{3 \sqrt{T}(\sqrt{\mu_{L}}+\sqrt{\mu_{0}})}$$

Sandeep Desai

Numerade Educator