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Unformatted text preview: (an a) 318 Chapter 15 Vibrations and Waves amplitude: The maximum distance from the equilibrium pow
sition that occurs in periodic motion. antinode: One of the positions in a standing wave or interfer
ence pattern where there is maximal movement; that is, the ain
plitude is a maximum. crest: The peak ofa wave disturbance. cycle: One complete repetition of a periodic motion. It may
start anyplace in the motion. diffraction: The spreading of waves passing through an open
ing or around a barrier. displacement: In wave {or oscillatory) motion, the distance of
the disturbance (or object} from its equilibrium position. equilibrium position: A position where the net force is zero. frequency: The number of times a periodic motion repeats in
a unit of time. It is equal to the inverse of the period fundamental frequency: The lowest resonant frequency for
an oscillating system. harmonic: A frequency that is a wholenumber multiple of the
fundamental frequency. in phase: Two or more waves with the same wavelength and
frequency that have their crests lined up. interference: The superposition of waves. longitudinal wave: A wave in which the vibrations of the
medium are parallel to the direction the wave is mming.
node: One of the positions in a standing wave or interference pattern where there is no movement: that is, the amplitude is
zero. l. Ifthe net force on a mass oscillating at the end ofa vertical
spring is zero at the equilibrium point, why doesn’t the
mass stop there? 2. If the restoring force on a pendulum is zero when it is ver—
tical, why doesn‘t it quit swinging at this point? @A mass is oscillating up and down on a vertical spring.
When the mass is above the equilibrium point and moving down, what direction is the net force on the mass? When
the mass is above the equilibrium point and moving up,
what direction is the net force on the mass? 4. A mass is oscillating up and down on a vertical spring.
When the mass is below the equilibrium point and moving
down, what direction is its acceleration? Is the mass speed
ing up or slowing down? mass is oscillating up and down on a vertical spring. if the
mass is increased, will the period of oscillation increase, def KEY TERMS CONCEPTUAL QUESTIONS oscillation: A vibration about an equilibrium position or
shape. period: The shortest length of time it takes a periodic motion
to repeat. It is equal to the inverse of the frequency. periodic wave: A wave in which all the pulses have the same
size and shape. The wave pattern repeats itself over a distance of
one wavelength and over a time of one period. resonance: A large increase in the amplitude of a vibratiop,
when :1 force is applied at a natural frequency of the medium or
object. spring constant: The amount of force required to stretch a
spring by one unit of length. Measured in newtons per meter, standing wave: The interference pattern produced by two
waves of equal amplitude and frequency trawling in opposite
directions. The pattern is characterized by alternating nodal
and antinodal regions. superposition: The combinng of two or more waves at a loca“
tion in space. transverse wave: A wave in which the vibrations of the
medium are perpendicular to the direction the wave is moving. trough: A valley of a wave disturbance. vibration: An oscillation about an equilibrium position or shape. wave: The movement of energy from one place to another
without any accompanying matter. wavelength: The shortest repetition length for a periodic
ware. For example, it is the distance from (rest to crest or
[rough to trough. crease, or stay the same? Will the frequency increase, de
crease, or stay the same? 6. You have a grandfather clock (with a pendulum) that'keeps
perfect time on Earth. if you were to transport this clock to
the Moon, would its period ot‘Oscillation increase, decrease,
or stay the same? would its frequency increase, decrease, 01‘
stay the same? Explain. 7. You hang a likilogram block from a spring and ﬁnd that
the spring stretches 15 centimeters. What mass of block
would you need to stretch the spring by 45 centimeters? 8. Which spring would you expect to have the greater spring
constant, the one in the suspension of your Chevy or the
one in the mechanism of your watch? Why? 9. Assume that you pull the mass on the spring 1 centimeter
from the equilibrium position, let go, and measure the [36‘
I‘iod of the oscillation. Would you expect the period to be m.mr» are; .4 ,u ._ "not” i I0. I2. [3. *I4. l5. l8. i larger, the same, or smaller if you pulled the mass 2 cen—
timeters from the equilibrium position? Why? The amplitudes of real pendula decrease because of fric—
tional forces. How does the period of a real pendulum
change as it dies down? tat is the period of the hand on a clock that measures
the seconds? What is its frequency? What is the period of the hand on a clock that measures
the minutes? What is its frequency? ' Suppose your grandfather clock runs too fast. If the mass
on the pendulum can be moved up or down, which way
would you move it to adjust the clock? Explain your
reasoning. How does the natural frequency of a swing change when
you move from sitting down to standing up? You find that the exhaust system on your 1979 Chrysler
Cordoba tends to rattle loudly when the tachometer,
which measures the engine's frequency, reads ‘20th rpm.
it is relatively quiet at frequencies above or below this. Use
the concept of resonance to explain this. Why do soldiers “break step” before crossing a suspension
bridge? You hold one end of a spring in your hand and hang
a block from the other end. After lifting the block up
slightly and releasing it, you find that it oscillates up and
down at a frequency of 2 hertz. At which of the following
frequencies could youjiggle your hand tip and down and
produce resonance: 5 hertz, 4 hertz, l.5 hertz, l hertz, or
0.5 hertz? You stand to the side of the low point ofa child’s swing and
always push the child in the same direction. Which of the
following multiples of the fundamental frequency will not
produce resonance: %, 12, l, or 2? When you yell at your friend, are the air molecules that
strike his ear the same ones that were in your lungs?
Explain. What is being transported along a clothesline when a wave
moves from one end to the other? 21. 22. 23.
24. 25. 26. 27. 28. '\
Conceptual Questions 3 i 9 ‘ ._ Sonar devices use underwater sound to explore the ocean
floor. Would you expect sonar to he a longitudinal or a
transverse wave? Explain. You fasten one end ofa long spring to the base ofa wall and
stretch it out along the floor, holding the other end in your
hand. Describe how you would generate a transverse pulse
on the spring. Describe how you would generate a longitu
dinal pulse on the spring. Is it possible for a shout to overtake a whisper? Explain. You stretch a long spring between a doorknob and your
hand. You generate a small transverse pulse on the spring
traveling toward the doorknob. How could you generate a
second pulse that would overtake the first pulse? Which of the following properties affect the speed of waves
along a rope: amplitude of the pulse, shape of the pulse,
tension in the rope, and/or the mass per unit length of the
rope? Why? You stretch a long spring between a doorknob and your
hand. You jerk your hand up and down to send a pulse
down the spring. Ifyou want to generate a slowertraveling
pulse, which of the following woulcl you do? Move your
hand up and down the same distance as before but do it
more slowly; move your hand up and down a smaller dis—
tan ce at the same speed as before; or move slightly closer to
the doorknob to decrease the tension in the spring. You fasten one end of a long spring to the base of a wall and
stretch it out along the floor, holding the other end in your
hand. You send a pulse of amplitude 5 centimeters down
the right side of the spring, and a moment later you send
a second identical pulse on the same side. The ﬁrst pulse
reflects from the fixed boundary and returns along the
spring toward you. When the reflected pulse meets the secw
0nd pulse, will the resulting amplitude be less than, equal
to, or greater than 5 centimeters? Explain your reasoning. imagine that the string in Figure 1544 is tied to the pole
with a loose loop such that the end is free to move up and 320 Chapter 15 Vibrations and Waves down. A pulse of amplitude 10 centimeters is sent down the
top of the string, and a moment later a second identical
pulse is sent, also on the top. The first pulse reflects from
the free boundary and returns along the string. When the
rellected pulse meets the second pulse, will the resulting
amplitude be less than, equal to, or greater thart lU centir
meters? Explain your reasoning. 29. The pulse in the ﬁgure is traveling on a stringr to the right
toward a fixed end. Draw the shape of the pulse after it
reﬂects from the boundary. 30. A pulse in the shape of a crest is sent from left to right along
a stretched rope. A trough travels in the opposite direction
so that the pulses meet in the middle of the rope. Would
you expect to observe a crest or a trough arrive at the righb ; I hand end of the rope? Explain. . .Il‘shapcs a and b in the figure correspond to idealized wave
l
I pulses on a rope, what shape is produced when they conr
pletely overlap? (a) ’__‘ 32. Repeat Question 31 for shapes :1 and c. 33. Which of the following properties are meaningful for peri
odic waves but not for single pulses: frequency, wavelength,
speed, amplitude? 34. In the following list of properties of periodic waves, which
one is independent Of the others: frequency, wavelength,
speed, amplitude? 35. Two waves have the same speed but one has twice the free
quency of the other. Which one has the longer wavelength?
Explain. 36. If the frequency of a periodic wave is cut in half while the
speed remains the same, what happens to the wavelength? 37. lfthe speed ofa periodic wave doubles while the period re
mains the Same, what happens to the wavelength? 38. What happens to the wavelength ofa periodic wave if both
the speed of the wave and the frequency are cttt in half? 39. 40. 4. 42. 43. 44. 45. Cuustesv cl Philadelphia International Airport Travelers are spaced 10 feet apart on a moving sidewalk in
an airport. They are all walking at exactly 3 mph relative to
the sidewalk. \Nhen the moving sidewalk ends, they con
tinue to walk at 3 mph. An obseiver standing still next to
the movingr sidewalk notes that the travelers are passing at
a frequency of l hertz. A second obsewer stands just be
yond the end of the moving sidewalk and notes the fre
quency at which the travelers pass. Would this frequency be
greater than, equal to, or less than i hertz? is the spacing
between the travelers after leaving the moving sidewalk
greater than, equal to, or less than 10 feet? Explain. A waterproof electric buzzer has a membrane that vibrates
at a constant frequency of 440 hertz. The buzzer is placed
in a bucket of water. Knowing that the speed of sound is
much greater in water than in air, will the frequency of the
sound heard in th * air be greater than, equal to, or less
than 440 hertz? \Nill the wavelength of the sound in air be
greater than, equal to, or less than what it was in the water?
Explain (Hint: Review Question 39 and think of the travel
ers as the wave crests.) Draw a diagram to represent the standingwave pattern for
the third harmonic of a rope fixed at both ends. How many antinodes are there when a rope ﬁxed at both
ends vibrates in its third harmonic? Draw a diagram to represent the standingwave pattern for
the fourth harmonic ofa rope ﬁxed at both ends. How many nodes are there when a rope ﬁxed at both ends
vibrates in its fourth harmonic? How much higher is the frequency of the fifth harmonic on
a rope than the fundamental frequency? How much higher is the frequency of the sixth harmonic
on a rope than that of the second? Standing waves can be established on a rope that is fixed on
one end but free to slide up and down a pole on the other.
The fixed end remains a node, while the free end must
be an antiuode. Draw diagrams to represent the standing
wave patterns for the two lowest frequencies. How does the fundamental wavelength of standing waves
on a string with one end ﬁxed and the other free compare to the fundamental wavelength if the same sn‘ing‘is held
with both ends ﬁxed? How does the wavelength of the fourth harmonic on a rope
with both ends fixed compare with the length of the rope? How does the wavelength of the fourth harmonic on a
rope with both ends ﬁxed compare with that of the second
harmonic? A longitudinal standing wave can be established in a long
aluminum rod by stroking it with rosin on your fingers. If
the rod is held tightly at its midpoint, what is the wave
length of the fundamental standing wave? Assume that
there are antinodes at each end of the rod and a node
where the rod is held. What is the wavelength of the fundamental standing wave
for the rod in Question 51 ifit is held midway between the
center and one end? Will the resulting pitch be higher or
lower than when the rod was held at its midpoint? Explain. Two point sources produce waves of the same wavelength
and are in phase. At a point midway between the sources,
Would you expect to ﬁnd a node or an antinode? Explain. Two point sources produce waves of the same wavelength
and are completely out of phase (that is, one produces
a crest at the same time as the other produces a trough}. *55. *56. 57. Cnurtesy at Central Scientiﬁc Company 58. Exercises 3 2 I At a point midway between the sources, would you expect
to ﬁnd a node or an antinode? Why? What happens to the spacing of the antinodal lines in an
interference pattern when the two sources are moved far~
ther apart? Explain. As you increase the frequency of the sources, what hap
pens to the spacing of the nodal lilies in an interference
pattern produced by two sources? Explain. An interference pattern is produced in a ripple tank. As
the two sources are brought closer together, does the sep
aration of the locations of maximum amplitude along the
far edge of the tank decrease, increase, or remain the
same? Why? As the frequency of the two sources forming an inter
ference pattern in a ripple tank increases, does the sepa~
ration of the locations of minimum amplitude along the
far edge of the tank increase, decrease, or remain the
same? Why? EXERCISES If a mass on a spring takes 6 s to complete two cycles, what
is its period?
If a mass on a spring has a frequency of 4 Hz, what is its period? ®A Foucault pendulum with a length of 9 m has a period of
6 5. What is its frequency? 4 A mass on a spring bobs up and down over a distance of 30 cm from the top to the bottom ofits path twice each sec
ond. What are its period and amplitude? 5 A spring hanging from the ceiling has an unstretched
length of 80 cm. A mass is then suspended at rest from the
spring, causing its length to increase to 89 cm. The mass is
pulled down an additional 3 cm and released. What is the
amplitude of the resulting oscillation? 6. '\. IO. / A mass oscillates up and down on a vertical spring with an
amplitude of 4 cm and a period of 2 5. What total distance
does the mass travel in 10 seconds? ‘Nhat is the period of a 0.4—kg mass suspended from a
spring with a spring constant of 40 N/rn? A boy with a mass of 50 kg is hanging from a spring with a
spring constant of 200 N/m. With what frequency does the
boy bounce up and down? oi By what factor would you have to increase the spring con stant to double the frequency for a mass on a spring? By what factor would you have to increase the mass to triple
the period for a mass on a spring? A pendulum has a length ofﬁ m. What is its period? l/R 322 I2. *I3. *l4. I5. “3. I9. Chapter 15 Vibrations and Waves 8 distance, cm A girl with a mass of 40 kg is swingingfrom a rope with a
length of 2.5 m. What is the frequency of her swinging? The highly idealized wave pulses shown in the ﬁgure at a
Lime equal to zero have the same amplitudes and travel at
1 cm/s. Draw the shape of the rope at 2, 4, 5, and 8 5. Work Exercise 13 but change the rectangular pulse from
a crest to a trough. A train, consisting of identical lOm boxcars, passes you
such that 2:3 boxcars pass you each minute. Find the speed
of the train. . You observe that 25 crests of a water wave pass you each minute. If the wavelength is 10 In, what is the speed of
the wave? A periodic wave on a string has a wavelength of 25 cm and
a frequency of 3 Hz. What is the speed of the wave? If the breakers at a beach are separated by 5 m and hit
shore with a frequency of 0.3 Hz, at what speed are they
traveling? What is the distance between adjacent crests of ocean
waves that have a frequency 0f().‘2 Hz if the waves have a
speed of 3 Iii/s? 20. 2. /2.
.A rope is tied between two posts separated by 3 in. What *24. *25. *26. 10 12 16 Sound waves in iron have a speed of about 5] 00 ni/s. if the
waves have a frequency of‘iOO Hz, what is their wavelength? For sound waves, which travel at 343 m/s in air at room
temperature, what frequency corresponds to awavelength
of l m? What is the period of waves on a rope if their wavelength
is 0.8 In and their speed is 2 m/s? possible wavelengths will produce standing waves on the
rope? A 3mlong rope is tied to a very thin string so that one
end is essentially free. “That possible wavelengths will pro—
duce standing waves on this rope? ‘Nhat is the fundamental frequency on 21 6—111 rope that is
tied at both ends if the speed of the waves is 18 m/s? Tweety Bird hops up and down at a frequency of0.5 Hz on
a power line at the midpoint between the poles, which are
separated by 20 In. Assuming Tweety is exciting the fun
damental standing wave, ﬁnd the speed of transverse
waves on the power line. (Hint: What is the wavelength for
this standing wave?) ”"iiilnfoTrac‘E College Edition For additional readings, explore InfoTrac College Edition, your online
library. Go to http1//V\ww.infotrac—coliege.com/wadsworth and use the
passcode that came on the card with your book. Try these search terms:
atomic clock, Christiaan Huygens,john Harrison and chronometerJohn
Harrison and clock, pendulum clock, seismic waves, standing waves, ‘ Tacoma Narrows Bridge. ...
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 Fall '08
 GRIFFIN
 Physics

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