2PE8P — /(2)(48J) —061 = (95 N/m)(O.25 m)

Practice Problems14.1 Periodic Motion

pages 375—380page 378

1. How much force is necessaiy to stretch a

spring 0.25 m when the spring constant

is 95 N/rn?

F=kx

= (95 N/m)(O.25 m)

= 24 N

2. A spring has a spring constant of 56 N/m.

How far will it stretch when a block

weighing 18 N is hung from its end?

F=kx1BN 2

k 56NIm

3. What is the spring constant of a spring that

stretches 12 cm when an object weighing

24 N is hung from it?

F=kx

k=x

— 24N0.12 m

= 2.0X102N/rn

4. A spring with a spring constant of 144 N/rn

is compressed by a distance of 16.5 cm.

How much elastic potential energy is stored

in the spring?

PE = ikx2

= +(144 NIm)(0.165 rn)2 = 1.96 J

5. A spring has a spring constant of 256 N/rn.

How far must it be stretched to give it an

elastic potential energy of 48 J?

/2PE8P — /(2)(48J)—061

V k V256NIm. m

page 3796. What is the period on Earth of a pendulum

with a length of LO

T= 2irq = 290s2 = 2.0 s

7. How long must a pendulum be on the

Moon, where g = 1.6 rn/s2, to have a

period of 2.0 s?

T= 21TJ

1= (1.6 rnIs2)(20S)2= 0.16 m

8. On a planet with an unknown value of g,

the period of a 0.75-rn-long pendulum

is 1.8 s. What is g for this planet?

T= 21rq

g == (0.75 m)(12s)2= 9.1 mIs2

Section Review14.1 Periodic Motion

pages 375—380page 380

9. Ilooke’s Law Two springs look alike but

have different spring constants. How could

you determine which one has the greater

spring constant?

Hang the same object from both

springs. The one that stretches less

has the greater spring constant.

10. Hooke’s Law Objects of various weights

are hung from a rubber band that is sus

pended from a hook. The weights of the

objects are plotted on a graph against the

CHAPTER

1G lbronsdWayes

PE5 = 1kx2

Physics: Principles and Problems Solutions Manual 311

Chapter 14 continued

stretch of the rubber band. How can you 14. Critical Thinking How is uniform circular

tell from the graph whether or not the motion similar to simple harmonic

rubber band obeys Hooke’s law? motion? How are they different?

If the graph is a straight line, the rubber Both are periodic motions. In uniform

band obeys Hooke’s law. If the graph is circular motion, the accelerating force

curved, it does not, is not proportional to the displacement.

Also, simple harmonic motion is one

11. Pendulum How must the length of a pen- dimensional and uniform circular

dulum be changed to double its period? motion is two-dimensional.

How must the length be changed to halve

the period? Practice ProblemsPE — !kX2 so

S 2‘

14.2 Wave Properties

—

pages 381—386PE2 — x22 page 386

— (0.40 rn)2 15. A sound wave produced by a dock chime is

— (0.20 rn)2 heard 515 m away 1.50 s later.

= 4.0a. What is the speed of sound of the

The energy of the first spring isdock’s chime in air?

4.0 times greater than the energy of v = -

the second spring.t

_515m

12. Energy of a Spring What is the difference1.50 S

between the energy stored in a spring that is = 343 mIs

stretched 0.40 m and the energy stored in b. The sound wave has a frequency of

the same spring when it is stretched 0.20 m? 436 Hz. What is the period of the wave?

T=2,so-=,,J T=

To double the period:1

436 HzT2 I’2_r, ‘2_A =2.29X103s

The length must be quadrupled.c. What is the wave’s wavelength?

To halve the period:A =

ii_ W_! 4 — 1 — 343 mIs

T1 2’ I 4 — 436Hz

The length is reduced to one-fourth its = 0.787 m

original length.16. A hiker shouts toward a vertical duff 465 m

13. Resonance If a car’s wheel is out of bal- away. The echo is heard 2.75 s later.

ance, the car will shake strongly at a specific a. What is the speed of sound of the

speed, but not when it is moving faster or hiker’s voice in air?

slower than that speed. Explain. ‘2’465 ‘

At that speed, the tire’s rotation V 2.75 s= 338 mIS

frequency matches the resonant

frequency of the car.

312 Solutions Manual Physics: Principles and Problems


b. The wavelength of the sound is 0.750 m.What is its frequency?

v=Af,sof== =451Hz

C. What is the period of the wave?

T=1=1 =222X1O3s

f 451Hz

17. If you want to increase the wavelength of

waves in a rope, should you shake it at a

higher or lower frequency?

at a lower frequency, because wave

length varies inversely with frequency

18. What is the speed of a periodic wave distur

bance that has a frequency of 3.50 Hz and a

wavelength of 0.700 m?

v = Af = (0.700 m)(3.50 Hz) = 2.45 mIs

19. The speed of a transverse wave in a string is

15.0 rn/s. If a source produces a disturbance

that has a frequency of 6.00 Hz, what is its

wavelength?

)v=Af,soA===2.50m

20. Five pulses are generated every 0.100 s in a

tank of water. What is the speed of propaga

tion of the wave if the wavelength of the

surface wave is 1.20 cm?

0.lOOs —

5 pulses — 0.0200 s/pulse, so

T= 0.0200 s

A = vT, so

VT

= 1.20 cm0.0200 s

= 60.0 cmls = 0.600 mIs

21. A periodic longitudinal wave that has a

frequency of 20.0 Hz travels along a coil

spring. If the distance between successive

compressions is 0.600 rn, what is the speed

of the wave?

v = Af = (0.600 m)(20.0 Hz) = 12.0 mIs

Section Review14.2 Wave Properties

pages 381—386page 38622. Speed in Different Media If you pull on

one end of a coiled-spring toy, does thepulse reach the other end instantaneously?What happens if you pull on a rope? What

happens if you hit the end of a metal rod?

Compare and contrast the pulses traveling

through these three materials.

It takes time for the pulse to reach the

other end in each case. It travels faster

on the rope than on the spring, and

fastest in the metal rod.

23. Wave Characteristics You are creating

transverse waves in a rope by shaking your

hand from side to side. Without changing

the distance that your hand moves, you

begin to shake it faster and faster. What

happens to the amplitude, wavelength,

frequency period, and velocity of the wave?

The amplitude and velocity remain

unchanged, but the frequency increases

while the period and the wavelength

decrease.

24. Waves Moving Energy Suppose that

you and your lab partner are asked to

demonstrate that a transverse wave trans

ports energy without transferring matter.

How could you do it?

Tie a piece of yarn somewhere near the

middle of a rope. With your partner

holding one end of the rope, shake the

other end up and down to create a

transverse wave. Note that while the

wave moves down the rope, the yarn

moves up and down but stays in the

same place on the rope.

25. Longitudinal Waves Describe longitudinal

waves. What types of media transmit

longitudinal waves?

In longitudinal waves, the particles of

the medium vibrate in a direction

parallel to the motion of the wave.

0L)

I!

000

0

S0

0.a



Nearly all media—solids, liquids, andgases—transmit longitudinal waves.

26. Critical Thinking If a raindrop falls into apool, it creates waves with small amplitudes.If a swimmer jumps into a pool, waves withlarge amplitudes are produced. Why doesn’tthe heavy rain in a thunderstorm producelarge waves?

The energy of the swimmer Is transferred to the wave in a small spaceover a short time, whereas the energyof the raindrops is spread out inarea and time.

Section Review14.3 Wave Behavior

pages 387—391page 39127. Waves at Boundaries Which of the

following wave characteristics remainunchanged when a wave crosses a boundaryinto a different medium: frequency amplitude, wavelength, velocity, and/or direction?

Frequency remains unchanged. Ingeneral, amplitude, wavelength, andvelocity will change when a wave entersa new medium. Direction may or maynot change, depending on the originaldirection of the wave.

28. Refraction of Waves Notice inFigure 14-17a how the wave changesdirection as it passes from one medium toanother. Can two-dimensional waves crossa boundaiy between two media withoutchanging direction? Explain.

Yes, if they strike the boundary whiletraveling normal to its surface, or if theyhave the same speed in both media.

29. Standing Waves In a standing wave on astring fixed at both ends, how is the numberof nodes related to the number of antinodes?The number of nodes is always onegreater than the number of antinodes.

30. CrItical Thinldng As another way tounderstand wave reflection, cover the right-hand side of each drawing in Figure 14-13awith a piece of paper. The edge of the papershould be at point N, the node. Now, concentrate on the resultant wave, shown indarker blue. Note that it acts like a wavereflected from a boundary. Is the boundary arigid wall, or is it open-ended? Repeat thisexercise for Figure 14.-13b.

Figure 14-14a behaves like a rigid wallbecause the reflected wave is inverted;14-14b behaves like an open endbecause the boundary is an antinodeand the reflected wave Is not inverted.

Chapter AssessmentConcept Mapping

Complete the concept map using the following terms and symbols: amplitude,frequency, v, A, T.

Mastering Conceptspage 39632. What is periodic motion? Give three exam

ples of periodic motion. (14.1)

Periodic motion is motion that repeatsin a regular cycle. Examples includeoscillation of a spring, swing of a simplependulum, and uniform circular motion.

33. What is the difference between frequencyand period? How are they related? (14.1)

Frequency is the number of cycles orrepetitions per second, and period is thetime required for one cycle. Frequencyis the inverse of the period.

(

page 39631.

I

•1314 Solutions Manual Physics: Principles and Problems

Simple harmonic motion is periodic

motion that results when the restoring

force on an object is directly propor

tional to its displacement. A block

bouncing on the end of a spring is one

example.

35. If a spring obeys Hooke’s law, how does it

behave? (14.1)

The spring stretches a distance that is

directly proportional to the force

applied to it.

36. How can the spring constant of a spring be

determined from a graph of force versus

displacement? (14.1)

The spring constant is the slope of the

graph of F versus x.

37. How can the potential energy in a spring be

determined from the graph of force versus

displacement? (14.1)

The potential energy is the area under

the curve of the graph of F versus x.

38. Does the period of a pendulum depend on

the mass of the bob? The length of the

string? Upon what else does the period

depend? (14.1)

no; yes; the acceleration of gravity, g

39. What conditions are necessary for

resonance to occur? (14.1)

Resonance will occur when a force is

applied to an oscillating system at the

same frequency as the natural frequency

of the system.

40. How many general methods of energy

transfer are there? Give two examples

of each. (14.2)

Two. Energy is transferred by particle

transfer and by waves. There are many

examples that can be given of each: a

baseball and a bullet for particle transfer;

sound waves and light waves.

41. What is the primary difference between amechanical wave and an electromagneticwave? (14.2)

The primary difference is that mechanical waves require a medium to travelthrough and electromagnetic waves do

not need a medium.

42. What are the differences among transverse,

longitudinal, and surface waves? (14.2)

A transverse wave causes the particles

of the medium to vibrate in a direction

that is perpendicular to the direction inwhich the wave is moving. A longitudi

nal wave causes the particles of the

medium to vibrate in a direction parallel

with the direction of the wave. Surface

waves have characteristics of both.

43. Waves are sent along a spring of fixed

length. (14.2)

a. Can the speed of the waves in the springbe changed? Explain.

Speed of the waves depends only

on the medium and cannot be

changed.

b. Can the frequency of a wave in the

spring be changed? Explain.

Frequency can be changed by

changing the frequency at which the

waves are generated.

44. What is the wavelength of a wave? (14.2)

Wavelength is the distance between two

adjacent points on a wave that are in

phase.

45. Suppose you send a pulse along a rope.

How does the position of a point on the

rope before the pulse arrives compare to the

point’s position after the pulse has passed?

(14.2)

Once the pulse has passed, the point is

exactly as it was prior to the advent of

the pulse.


34. What is simple harmonic motion? Give an

example of simple harmonic motion.

(14.1))

)

)

S8z

000

•0

00

.0

8


A pulse is a single disturbance in amedium, whereas a periodic wave consists of several adjacent disturbances.

53. When a wave crosses a boundary betweena thin and a thick rope, as shown inFigure 14-18, its wavelength and speedchange but its frequency does not. Explainwhy the frequency is constant. (14.3)

47. Describe the difference between wavefrequency and wave ve1odty (14.2)

Frequency is the number of vibrationsper second of a part of the medium.Velocity describes the motion of thewave through the medium.

48. Suppose you produce a transverse wave byshaking one end of a spring from side toside. How does the frequency of your handcompare with the frequency of the wave?(14.2)

They are the same.

49. When are points on a wave in phase witheach other? When are they out of phase?Give an example of each. (14.2)

Points are in phase when they havethe same displacement and the samevelocity. Otherwise, the points are outof phase. Two crests are in phase witheach other. A crest and a trough are outof phase with each other.

50. What is the amplitude of a wave and whatdoes it represent? (14.2)

Amplitude is the maximum displacement of a wave from the rest or equilibrium position. The amplitude of thewave represents the amount of energytransferred.

51. Describe the relationship between theamplitude of a wave and the energy itcarries. (14.2)

The energy carried by a wave is proportional to the square of its amplitude.

52. When a wave reaches the boundaiy of anew medium, what happens to it? (14.3)Part of the wave can be reflected andpart of the wave can be transmitted intothe new medium.

• Figure 14-18

The frequency depends only on the rateat which the thin rope is shaken and thethin rope causes the vibrations in thethick rope.

54. How does a spring pulse reflected from arigid wall differ from the incident pulse?(14.3)

The reflected pulse will be inverted.

55. Describe interference. Is interference a property of only some types of waves or all typesof waves? (14.3)

The superposition of two or morewaves is interference. The superposition of two waves with equal but opposite amplitudes results in destructiveinterference. The superposition of twowaves with amplitudes in the samedirection results in constructive interference; all waves; it Is a prime test forwave nature.

56. What happens to a spring at the nodes of astanding wave? (14.3)

Nothing, the spring does not move.

57. Violins A metal plate is held fixed in thecenter and sprinkled with sugar. With aviolin bow, the plate is stroked along oneedge and made to vibrate. The sugar beginsto collect in certain areas and move awayfrom others. Describe these regions in termsof standing waves. (14.3)

Bare areas are antinodal regions wherethere is maximum vibration. Sugar-covered areas are nodal regions wherethere is no vibration.


46. What is the difference between a wave pulseand a periodic wave? (14.2)

I316 Solutions Manual Physics: Principles and Problems


)

)

58. If a string is vibrating in four parts, there arepoints where it can be touched withoutdisturbing its motion. Explain. How manyof these points exist? (14.3)

A standing wave exists and the stringcan be touched at any of its five nodalpoints.

59. Wave fronts pass at an angle from onemedium into a second medium, where theytravel with a different speed. Describe twochanges in the wave fronts. What does notchange? (14.3)

The wavelength and direction of thewave fronts change. The frequencydoes not change.

Applying Conceptspage 39760. A bail bounces up and down on the end of

a spring. Describe the energy changes thattake place during one complete cyde. Doesthe total mechanical energy change?At the bottom of the motion, the elasticpotential energy is at a maximum, whilegravitational potential energy is at aminimum and the kinetic energy Is zero.At the equilibrium position, the KE is ata maximum and the elastic potentialenergy is zero. At the top of the bounce,the KE is zero, the gravitational potential energy is at a maximum, and theelastic potential energy is at a maximum. The total mechanical energy isconserved.

61. Can a pendulum clock be used in the orbiting International Space Station? Explain.

No, the space station is in free-fail, andtherefore, the apparent value of g iszero. The pendulum will not swing.

62. Suppose you hold a 1-m metal bar in yourhand and hit its end with a hammer, first,in a direction parallel to its length, andsecond, in a direction at right angles to itslength. Describe the waves produced in thetwo cases.

In the first case, longitudinal waves; inthe second case, transverse waves.

63. Suppose you repeatedly dip your finger intoa sink full of water to make circular waves.What happens to the wavelength as youmove your finger faster?

The frequency of the waves willincrease; the speed will remain thesame; the wavelength will decrease.

64. What happens to the period of a wave asthe frequency increases?

As the frequency increases, the perioddecreases.

65. What happens to the wavelength of a waveas the frequency increases?

As the frequency increases, the wavelength decreases.

66. Suppose you make a single pulse on astretched spring. How much energy isrequired to make a pulse with twice theamplitude?

approximately two squared, or fourtimes the energy

67. You can make water slosh back and forth ina shallow pan only if you shake the panwith the correct frequency Explain.

The period of the vibration must equalthe time for the wave to go back andforth across the pan to create constructive Interference.



68. In each of the four waves in Figure 14-19,

the pulse on the left is the original pulse

moving toward the right. The center pulse is

a reflected pulse; the pulse on the right is a

transmitted pulse. Describe the rigidity of

the boundaries at A, B, C, and D.

J2\

\:7

I Figure 14-19

Boundary A is more rigid; boundary Bis less rigid; boundary C is less rigid;boundary D is more rigid.

Mastering Problems14.1 Periodic Motion

pages 397—398

Level 169. A spring stretches by 0.12 m when some

apples weighing 3.2 N are suspended fromit, as shown in Figure 14-20. What is thespring constant of the spring?

70. Car Shocks Each of the coil springs of acar has a spring constant of 25,000 N/rn.How much is each spring compressed if it

supports one-fourth of the car’s 12,000-N

weight?

F= kx,

so x =

71. How much potential energy is stored in aspring with a spring constant of 27 N/rn if

it is stretched by 16 cm?

= (1)(27 NIm)(0.16 rn)2 = 0.35 J

Rocket Launcher A toy rocket-launcher

contains a spring with a spring constant of35 N/rn. How far must the spring becompressed to store 1.5 J of energy?

PE = kX2,

=0.29m

Level 373. Force-versus-length data for a spring are

plotted on the graph in Figure 14-21.

EZ7

A/

— (j-)12ooo N)

— 25,000 NIm

=0.12m

PE5 = kX2

Level 272.

/ 2PE8sox=q k

/ (2)(1.5 J)35NIm

zci.’VI0

U.

12.0

8.0

4.0

0C,n

I

I3.2 N

I Figure 14-20

F=kx,

so k = = 3.2 N =27 N/rnx 0.12m

//

0.0 0.20 0.40 0.60

Length (m)

• Figure 14-21



_AF_ 12.ON—4.ONAx 0.6m—0.2m

=20 N/rn

b What is the energy stored in the springwhen it is stretched to a length of50.0 cm?

PE5 = area = ibh

= (+)o.500 m)(10.0 N) = 2.50 J

74. How long must a pendulum be to have aperiod of 2.3 s on the Moon, where

g = 1.6 m/s2?

T2ir,J,soI=T2g

— (2.3s)2(1.6m1s2)—

— 47.2 —0.21m

T

0.12Hz =8.3s

76. Ocean Waves An ocean wave has a lengthof 12.0 m. Awave passes a fixed locationevery 3.0 s. What is the speed of the wave?

v=Af=A(+)= (12.0 m)(3)

= 4.0 mIs

77. Water waves in a shallow dish are 6.0-cmlong. At one point, the water moves up anddown at a rate of 4.8 oscillations/s.

a. What is the speed of the water waves?

v = Af

= (0.060 m)(4.8 Hz) = 0.29 rn/s

b. What is the period of the water waves?

= 4.8Hz = 0.21 s

78. Water waves in a lake travel 3.4 m in 1.8 s.The period of osdllation is 1.1 s.

a. What is the speed of the water waves?

d 3.4m l9mIst 1.Bs

b. What is their wavelength?

A= J= vT

= (1.9 m/s)(1.1 s)

=2.1 m

Level 279. Sonar A sonar signal of frequency

1.00X106Hz has a wavelength of 1.50mmin water.

a. What is the speed of the signal in water?

v = Af

= (1.50X103rn)(1.00X106Hz)

= 1.50X103rn/sb. What is its period in water?

1— J — 1.00x106Hz

= 1.00x10—6S

a. What is the spring constant of thespring?

k= slope)

14.2 Wave Pmperties

page 398

Level 175. Building Motion The Sears Tower in

Chicago, shown in Figure 14-22, swaysback and forth in the wind with a frequencyof about 0.12 Hz. What is its period ofvibration?

I Figure 14-22



C. What is its period in air?

i.ooxio—6s

The period and frequency remain

unchanged.

80. A sound wave of wavelength 0.60 m and a

velocity of 330 rn/s is produced for 0.50 s.

a. What is the frequency of the wave?

v = Af

._ -

= 330 mIs‘A 0.60m

= 550 Hz

b. How many complete waves are emittedin this time interval?

ft = (550 Hz)(0.50 s)

= 280 complete waves

c. After 0.50 s, how far is the front of thewave from the source of the sound?

d=vt

= (330 mls)(0.50 s)

= 1.6X102m

81. The speed of sound in water is 1498 rn/s. Asonar signal is sent straight down from aship at a point just below the water surface,and 1.80 s later, the reflected signal isdetected. How deep is the water?

The time for the wave to travel downand back up is 1.80 s. The time one wayis half 1.80 s or 0.900 s.

d= vt

= (1498 m/s)(0.900 s)

= 1350 m

Pepe and Aifredo are resting on an offshoreraft after a swim. They estimate that 3.0 mseparates a trough and an adjacent crest ofeach surface wave on the lake. They count12 crests that pass by the raft in 20.0 s.Calculate how fast the waves are moving.

A = (2)(3.0 m) = 6.0 m

.12 waves

20.Os —0.60 Hz

V = Af

= (6.0 m)(0.60 Hz)

= 3.6 mIs

83. Earthquakes The velocity of the transverse waves produced by an earthquake is

8.9 km/s, and that of the longitudinalwaves is 5.1 km/s. A seismograph recordsthe arrival of the transverse waves 68 s

before the arrival of the longitudinalwaves. How far away is the earthquake?

d = vt. We do not know t, only thedifference in time, it. The transversedistance, c11. = VTt, is the same as the

longitudinal distance, dL = vL(t + At).

Use VTt = vL(t + At), and solve for t:

VAt

VT — VL

(5.1 kmls)(68 s)8.9km/s—5.1 km/s

91s

Sketch the result for each of the three casesshown in Figure 14-23, when the centersof the two approaching wave pulses lie onthe dashed line so that the pulses exactlyoverlap.

I

$11

I

.1

I

Then putting t back into

dT VTt = (8.9 kmls)(91 s)

8.1X102 km

14.3 Wave Behavior

pages 398—399

Level 184.

Level 382.

1 I2

L i3

• Figure 14-23



2. The amplitudes cancel each other.

3. If the amplitude of the first pulse Isone-half of the second, the resultantpulse is one-half the amplitude of thesecond.

I I85. If you slosh the water in a bathtub at the

correct frequency, the water rises first at oneend and then at the other. Suppose you canmake a standing wave in a 150-cm-long tubwith a frequency of 0.30 Hz. What is thevelocity of the water wave?A = 2(1.5 m) = 3.0 mv = Af

= (3.0 m)(0.30 Hz)= 0.90 mIs

Guitars The wave speed in a guitar string is265 rn/s. The length of the string is 63 an. Youpluck the center of the string by pulling it upand letting go. Pulses move in both directionsand are reflected off the ends of the string.a. How long does it take for the pulse to

move to the string end and return to thecenter?

d= (2X63 cm)

=63 cm

O.63m =24X103s— v 265m!s

b. When the pulses return, is the stringabove or below its resting location?Pulses are inverted when reflectedfrom a more dense medium, soreturning pulse is down (below).

c. If you plucked the string 15 an fromone end of the string, where would thetwo pulses meet?

15 cm from the other end, where thedistances traveled are the same.

87.. Sketch the result for each of the four casesshown in Figure 14-24, when the centers ofeach of the two wave pulses lie on the dashedline so that the pulses exactly overlap.

21

31_

_

:

Mixed Reviewpage 399—400Level 188. What is the period of a pendulum with a

length of 1.4 m?

T=2ir/

= 21q9.80 mIs2 = 2.4 S

89. The frequency of yellow light is 5.1X 1014 Hz.Find the wavelength ofyellow light. Thespeed of light is 3.OOX 108 rn/s.

Solutions Manual 321

1. The amplitude is doubled.

4-1 —

• Figure 14-24

Level 286.

1

2

3

4 -I.

Physics: Principles and Problems

J


ACf

— 3.00Xl 08 mIs— 5.1X1O14Hz

= 5.9x107m

90. Radio Wave AM-radio signals are broad

cast at frequencies between 550 kI-Lz(kilohertz) and 1600 kHz and travel3.OX 108 rn/s.

a. What is the range of wavelengths forthese signals?

v=Af

— V — 3.0X108mis— f1 — 5.5X105Hz

b. FM frequencies range between 88 MHz(megahertz) and 108 MHz and travelat the same speed. What is the range ofFM wavelengths?

3.0X108mIs8.8X1O7Hz

=3.4m

3.0X108rn/s1.O8X1O8Hz

=2.8m

Range is 2.8 m to 3.4 m.

91. You are floating just offshore at the beach.Even though the waves are steadilymoving in toward the beach, you don’tmove any doser to the beach.

a. What type of wave are you experiencingas you float in the water?

transverse waves

b. Explain why the energy in the wavedoes not move you doser to shore.

The displacement is perpendicularto the direction of the wave—inthis case, up and down.

c. In the course of 15 s you count tenwaves that pass you. What is the periodof the waves?

T 15s 15s10 waves

d. What is the frequency of the waves?

f= - == 0.67 Hz

a. You estimate that the wave crests are 3 mapart. What is the velocity of the waves?

v=Af= (3 m)(0.67 Hz) = 2 m/s

1. After returning to the beach, you learnthat the waves are moving at 1.8 m/s.What is the actual wavelength of thewaves?

A V 1.8m/s 27mf 0.67Hz

Bungee Jumper A high-altitude bungeejumper jumps from a hot-air balloon usinga 540-m-bungee cord. When the jump iscomplete and the jumper is just suspendedfrom the cord, it is stretched 1710 m. Whatis the spring constant of the bungee cord ifthe jumper has a mass of 68 kg?

k — F — mg — (68 kg)(9.80 mis2)

x x — 1710m—540m

= 0.57 N/rn

93. The time needed for a water wave to changefrom the equilibrium level to the crest is0.18 s.

a. What fraction of a wavelength is this?

wavelength

b. What is the period of the wave?

T= (4)(0.18 s) = 0.72 s

c. What is the frequency of the wave?

f= =1.4 Hz

94. When a 225-g mass is hung from a spring,the spring stretches 9.4 cm. The spring andmass then are pulled 8.0 cm from this newequilibrium position and released. Find thespring constant of the spring and the maximum speed of the mass.

=550m

— V — 3.0X108misA2—1—1.6X1O6Hz

=190m

Range is 190 m to 550 m.

Level 292.

4 H


j


k=-=

)— (0.225kg)(9.80 mis2)

— 23 N/0.094m m

Maximum velocity occurs when the masspasses through the equilibrium point,where all the energy is kinetic energy.Using the conservation of energy:

PE = KEmass

+kX2 =

= ic

— — /(23 Nlrn)(0.080 rn)2V—.. m — ‘4/ 0.225kg

= 0.81 m/s

95. Amusement Ride You notice that yourfavorite amusement-park ride seems bigger.The ride consists of a carriage that isattached to a structure so it swings like apendulum. You remember that the carriageused to swing from one position to anotherand back again eight times in exactly 1 mm.Now it only swings six times in 1 mm. Giveyour answers to the following questions totwo significant digits.

a. What was the original period of the ride?

1 1T=—= =7.5sf (8 swIngs

\ 60.Os

=10 101f f6swlngs X S

1 60.Os

c. What is the new frequency?

1.0X10s=0.10Hz

d. How much longer is the arm supportingthe carriage on the larger ride?Original:

T2I—g--

= (9.80 m/s2)(1 .OX1O s)2

4w

=25m

The arm on the new structure is11 m longer.

e. If the park owners wanted to double theperiod of the ride, what percentageincrease would need to be made to thelength of the pendulum?

Because of the square relationship,there would need to be a 4 timesincrease in the length of the pendulum, or a 300% Increase.

96. Clocks The speed at which a grandfatherdock runs is controlled by a swingingpendulum.

a. If you find that the dock loses timeeach day, what adjustment would youneed to make to the pendulum so it willkeep better time?

The clock must be made to runfaster. The period of the pendulumcan be shortened, thus Increasingthe speed of the clock, by shortening the length of the pendulum.

b. If the pendulum currently is 15.0 cm,by how much would you need tochange the length to make the periodlessen by 0.0400 s?

iT= 2iTJ — 21r/

iiT[1 [2w i/g ijg

2ir’/ ‘2 ‘1

New:

I—g-

)

.8

Ib. What is the new period of the ride?

(7.5 s)2

) = (9.80 mIs2) 2

=14m



‘2(2)

(

= ((_0.0400 s)V9.80 rn/s2+

= 0.135 m

The length would need to shorten by

I.—12=0.150m—0.135m=0.015m

97. Bridge Swinging In the summer over the New River in West Virginia, severalteens swing from bridges with ropes, then drop into the river after a few swingsback and forth.

a.. If Pam is using a 10.0-rn length of rope, how long will it take her to reach thepeak of her swing at the other end of the bridge?

swing to peak =

fT I 10.Omlr,—1r1 2—3.17s

‘‘g ‘9.8OmIs

b. If Mike has a mass that is 20 kg more than Pam, how would you expect theperiod of his swing to differ from Pam’s?

There should be no difference. T is not affected by mass.

c. At what point in the swing is KE at a maximum?

At the bottom of the swing, KE is at a maximum.

d. At what point in the swing is PE at a maximum?

At the top of the swing, PE is at a maximum.

a.. At what point in the swing is KE at a minimum?

At the top of the swing, KE is at a minimum.

f. At what point in the swing is PE at a minimum?

At the bottom of the swing, PE is at a minimum.

98. You have a mechanical fish scale that is made with a spring that compresses jwhen weight is added to a hook attached below the scale. Unfortunately, thecalibrations have completely worn off of the scale. However, you have oneknown mass of 500.0 g that displaces the spring 2.0 cm.

a. What is the spring constant for the spring?

F=mg=kx

k-_x

— (0.5000 kg)(9.80 mis2)— 0.020 m

= 2.4X102N/m

b. If a fish displaces the spring 4.5 cm, what is the mass of the fish?

F= mg= kx


J


= 1.1 kg

99. Car Springs When you add a 45-kg loadto the trunk of a new small cai; the two rearsprings compress an additional 1.0 an.a. What is the spring constant for each of

the springs?

F = mg = (45 kg)(9.80 mis2)= 440 Nforce per spring = 220 N

F=kx,sok—-

k=

= 22,000 NIm

b. How much additional potential energyis stored in each of the car springsafter loading the trunk?

PE= 1kx2

= (I)(22,000 NIm)(0.010 rn)2

= 1.1 J

The velocity of a wave on a string dependson how tightly the string is stretched, andon the mass per unit 1enth of the string. If

is the tension in the string, and p. is themass/unit length, then the velocity, v, canbe determined by the following equation.

Now v= so

FT V211.L

= (1.50X102m/s)2(2.83X103kg/rn)= 63.7 N

Thinking Criticallypage 400101. Analyze and Condude A 20-N force is

required to stretch a spring by 0.5 m.a. What is the spring constant?

F=kx,sok=-= =4ONIrn

b. How much energy is stored in thespring?

PE = kX2

= (1-)4o NIrn)(0.5 rn)2 = 5 J

c. Why isn’t the work done to stretch thespring equal to the force times thedistance, or 10 1?The force is not constant as thespring is stretched. The averageforce, 10 N, times the distancedoes give the correct work.

102. Make and Use Graphs Several weightswere suspended from a spring, and theresulting extensions of the spring weremeasured. Table 14-1 shows thecollected data.

Table 14-I

)

A piece of string 5.30-m long has a massof 15.0 g. What must the tension inthe string be to make the wavelength of a125-Hz wave 120.0 cm?

v = Af= (1.200 m)(125 Hz)

= 1.50X102rn/s

andL=

— 1.50X102kg— 5.30m

= 2.83X103kg/rn

Weights on a Spring

Force, F (N) Extension, x Cm)

2.5 0.12

5.0 0.26

7.5 0.35

10.0 0.50

12.5 0.60

15.0 0.71

m kxg

— (2.4x102NIm)(4.5X102m)— 9.80 mIs2

Level 3100.



a. Make a graph of the force applied to the spring versus the spring length. Plot

the force on the y.axis.

20.0

•_ 15.0z

10.00

5.0

0.0

Stretch (m)

b. Determine the spring constant from the graph.

The spring constant is the slope.

iiF — 15.0 N — 2.5 N= 21 N!mk = slope

— — 0.71 m — 0.12 m

c. Using the graph, find the elastic potential energy stored in the spring when

it is stretched to 0.50 m.

The potential energy is the area under the graph.

PE = area = Ibh

= (-)(o.5o m)(10.0 N)

= 2.5 J

103. Apply Concepts Gravel roads often develop regularly spaced ridges that

are perpendicular to the road, as shown in Figure 14.25. This effect, called

washboarding, occurs because most cars travel at about the same speed and

the springs that connect the wheels to the cars oscillate at about the same

frequency. If the ridges on a road are 1.5 m apart and cars travel on it at

about 5 m/s, what is the frequency of the springs’ oscillation?

v=Af

.- 5mIs

‘ A 1.5m


I

j

0.20 0.40 0.60 0.80

• Figure 14-25


Writing in Physicspage 400104. Research Christiaan Huygens’ work on

waves and the controversy between himand Newton over the nature of light.Compare and contrast their explanationsof such phenomena as reflection and refraction. Whose model would you choose asthe best explanation? Explain why.Huygens proposed the wave theoryof light and Newton proposed theparticle theory of light. The law ofreflection can be explained usingboth theories. Huygen’s principle andNewton’s particle theory are opposed,however, in their explanation of thelaw of refraction.

A 1400-kg drag racer automobile cancomplete a one-quarter mile (402 m)course in 9.8 s. The final speed of theautomobile is 250 mi/h (112 m/s).(Chapter 11)

a. What is the kinetic energy of theautomobile?

KE= mv2

= (+)1400 kg)(112 rn/s)2

= 8.8X106Jb. What is the minimum amount of work

that was done by its engine? Why can’tyou cakulate the total amount of workdone?

The minimum amount of workmust equal KE, or 8.8X 106 J. Theengine had to do more work thanwas dissipated in work doneagainst friction.

c. What was the average acceleration ofthe automobile?—a At

106. How much water would a steam enginehave to evaporate in 1 s to produce 1 kWof power? Assume that the engine is20 percent efficient. (Chapter 12)

1000 JIs

If the engine is only 20 percentefficient it must use five times moreheat to produce the 1000 JIs.

q = 5000 J!s =

Therefore, - = 5000 JIs

— 5000 JIs— 2.26X106Jl’.cg

= 2x103kg/s

Challenge Problempage 380A car of mass m rests at the top of a hill ofheight h before rolling without friction into acrash barrier located at the bottom of the hill.The crash barrier contains a spring with a springconstant, k, which is designed to bring the car torest with minimum damage.

1. Determine, in terms of m, h, k, and g, themaximum distance, x, that the spring willbe compressed when the car hits it.Conservation of energy implies that thegravitational potential energy of the carat the top of the hill will be equal to theelastic potential energy in the springwhen it has brought the car to rest. Theequations for these energies can be setequal and solved for x.

PE9 = PE5, so mgh =

/2mghk

= 112 mIs9.8 s

= 11 mis

Cumulative Reviewpage 400105.

)

)



2. If the car rolls down a hill that is twice as

high, how much farther will the spring be

compressed?

The height is doubled and x is propor

tional to the square root of the height,

so xwlll increase by \/i3. What will happen after the car has been

brought to rest?

In the case of an ideal spring, the

spring will propel the car back to the

top of the hill.


Documents

2PE8P — /(2)(48J) —061 = (95 N/m)(O.25 m)