Exercises – Chapter 9
1. The acceleration due to gravity at the moon’s surface is only about one-sixth that at the earth’s surface. If you took a
pendulum clock to the moon, would it run fast, slow, or on time?
E.1
It would run slow.
2. A clothing rack hangs from the ceiling of a store and swings back and forth. Why doesn’t the period of this motion
depend on how many dresses the rack is holding? (Neglect the rack’s own mass.)
E.2
The rack is a pendulum and its period depends only on its length and on the strength of gravity.
E.2
While increasing the mass of the moving object lengthens the period of most harmonic oscillators, in this
case adding mass to the moving object also increases that object's weight and thus stiffens the restoring
force acting on the object. No matter how heavy the dresses, the rack will swing back and forth in the same
amount of time.
3. If a child stands up on the seat of a playground swing, how will the swing’s period be affected?
E.3
The period will decrease as the pendulum gets shorter.
4. If you pull a small tree to one side and suddenly let go, it will swing back and forth several times. The period of this
motion won’t depend on how far you bend the tree. How does the restoring force that returns the tree to its upright
position depend on how far you bend the tree?
E.4
The restoring force acting on the tree is proportional to how far it is bent away from its normal upright
orientation.
5. A flagpole is a harmonic oscillator, flexing back and forth with a steady period. If you want to increase the
amplitude of the pole’s motion by pushing it near its base, when should you push—as it flexes toward you or away
from you?
E.5
Push on it as it flexes away from you (so you do work on it).
6. Depending on how the base of a rocking chair was cut, the period of its motion may or may not depend on how hard
you’re rocking it. What can you say about the restoring forces acting in these two cases?
E.6
If the restoring force acting on the chair is proportional to how far it is displaced from its equilibrium
position, then the chair will be harmonic oscillator and its period of motion will not depend on the
amplitude of that motion. But if the restoring force is not proportional to its displacement, the chair will be
an anharmonic oscillator and its period will change with amplitude.
7. An electronic ruler measures the distance to a wall by bouncing sound off it. How can a ruler use a sound emitter, a
sound receiver, and a timer to measure how far away the wall is?
E.7
Multiplying sound’s roundtrip time by its speed gives twice the distance to the wall.
8. Which of the following clocks would keep accurate time if you took them to the moon: a pendulum clock, a balance
clock, and a quartz watch? Why?
E.8
The balance clock and quartz watch would continue to work, but the pendulum clock would run slow. The
first two timepieces don't depend on gravity for their restoring forces and are thus independent of the
strength of gravity. However, the pendulum clock depends on gravity for the stiffness of its restoring force.
In the moon's weak gravity, the restoring force will be weak and the pendulum will swing more slowly
than on earth.
9. To modify the pitch of a guitar string you could change its mass, tension, or length. To raise its pitch, how should
you change each of these three characteristics?
E.9
Less mass, more tension, or less length.
10. Why do changes in temperature affect a violin string’s pitch?
E.10 Changes in temperature cause materials to change length. If the string and violin change length by different
amounts, the tension in the string will change and so will the string's pitch.
11. The strings that play the lowest notes on a piano are made of thick steel wire wrapped with a spiral of heavy copper
wire. The copper wire doesn’t contribute to the tension in the string, so what is its purpose?
E.11 The copper wrap adds mass to the string to lower its pitch.
12. Why are the highest pitched strings on most instruments, including guitars, violins, and pianos, the most likely
strings to break?
E.12 The highest pitched strings are usually the thinnest (to reduce mass) and the tautest (to increase stiffness).
Thin, tight strings break easily.
13. Some wind chimes consist of sets of metal rods that emit tones when they’re struck by wind-driven clappers. These
rods vibrate like the baseball bat in Fig. 3.2.7. Why do the longer rods emit lower pitched tones than the shorter rods?
E.13 The longer rods have more mass vibrating and are less stiff.
14. Why would replacing the air in an organ pipe with helium raise its pitch?
E.14 Helium is less dense than air and responds more rapidly to forces. Helium will accelerate back and forth in
the pipe faster than air does.
15. A flute and a piccolo are both effectively pipes that are open at both ends, with holes in their sides to allow them to
produce more tones. The piccolo is very nearly a half-size version of the flute. How does this fact explain why the
piccolo’s tones are one octave above those of a flute?
E.15 A piccolo’s air columns are half the length of those in a flute, so they vibrate at twice the pitch or one
octave higher.
16. The most important difference between a trumpet and a tuba is in the lengths of their pipes. The tuba’s pipe is
much longer than that of the trumpet. How does this difference affect the relative pitches of the two instruments?
E.16 The tuba's longer air column vibrates more slowly than the trumpet's shorter air column.
17. In the Mediterranean Sea, high tide is only 30 cm above low tide. Why?
E.17 Ocean water can’t enter or leave the Mediterranean Sea quickly enough to allow its high tide to be very
different from its low tide. The tide simply rearranges the water levels within the sea itself.
18. Why are the tides relatively weak near the north and south poles?
E.18 The bulges in the earth's oceans are located near the equator--the nearest and farthest points to the moon.
Since the bulges aren't present at the north and south poles, the tides at those poles are weak.
19. If you’re carrying a full cup of coffee and take your steps at just the wrong frequency, the coffee will begin to slosh
wildly in the cup. What is causing this energetic motion?
E.19 Resonant energy transfer occurs when your steps are synchronized with the rhythmic motion of the coffee
sloshing in the cup.
20. When you throw a stone into a pool of still water, small ring-shaped ripples begin to spread outward at a modest
pace. Why do these ripples travel so much more slowly than waves on the ocean?
E.20 The speed of a surface wave on water increases with increasing wavelength. The short-wavelength ripples
you create with a stone travel very slowly.
21. When you throw a rock into calm water, ripples head outward as several concentric circles. As the circumferences
of these circles grow larger, their heights grow smaller. Explain this effect in terms of conservation of energy.
E.21 The wave’s energy is proportional to its circumference and its height. As its circumference grows, its
height must diminish.
22. If you pull downward on the middle of a trampoline and let go, the surface will fluctuate up and down several
times. Why is this motion an example of a standing wave?
E.22 The vibrating trampoline has a region of maximum up and down motion (the antinode) and a ring of zero
motion (the node).
23. Why is a violin string vibrating up and down in its fundamental vibrational mode an example of a standing wave
rather than a traveling wave?
E.23 Crests and troughs don’t travel along the string. Instead, the center of the string becomes alternately a crest
and a trough.
24. When you pluck the end of a kite string, a ripple will head up the string toward the kite. Why is this motion an
example of a traveling wave rather than a standing wave?
E.24 The plucked kite string has a ripple that moves along its length. The region of maximum motion moves
along the string and there is no pattern that simply vibrates back and forth in place.
25. As waves pass over a shallow sand bar, the largest ones break. What causes these waves to break and why only the
largest ones?
E.25 There isn’t enough water in front of the largest waves to complete their crests as they pass over the sand
bar.
26. Even when waves don’t break as they pass over a sand bar (Exercise 25), the sand bar is noticeable because you
can see the wave crests move closer together. What is happening to cause this bunching?
E.26 Water surface waves travel more slowly when they begin to encounter the shallow bottom of the water.
This slowing causes the crest to bunch together.
27. Sound can travel from one paper cup to another through a long, taut string that connects their bottoms. Is the wave
passing through the string longitudinal or transverse?
E.27 Longitudinal.
28. If you stamp on a wooden floor, you can make objects on a nearby table jump slightly. Are the waves traveling
through the floor longitudinal or transverse?
E.28 Transverse.
29. The crash of brass cymbals is rich with overtones that are not harmonics of the fundamental. Why aren’t the
overtones harmonics?
E.29 A surface can’t vibrate as half- or third-surfaces, so its overtones are complicated and don’t occur at
harmonic frequencies.
30. A Chinese gong produces a loud ringing sound which has nonharmonic overtones. Why aren’t the overtones
harmonic?
E.30 The gong is a surface and since it can’t vibrate as simple fractions of itself, its overtones don’t occur at
harmonic frequencies.
31. A harp is not a loud instrument, but it would be even softer if its strings were not attached to a wide wooden base.
What purpose does the wooden surface serve?
E.31 The surface projects sound waves far better than strings alone.
32. A string bass is an enormous instrument to carry around. Why can’t you just support the four strings with a sturdy
metal bar and leave out the whole wooden structure?
E.32 Most of the bass’s sound is projected by its vibrating body. Without that wooden body, it would have
almost no volume.
33. You clap your hands and send out a sound wave. Is there anything you can do one second later to send out a sound
wave that will overtake the first one you sent out?
E.33 No.
34. You drop a coin in a calm pool and send out a water surface wave. Is there anything you can do one second later to
send out a water surface wave that will overtake the first one you sent out?
E.34 No.
35. If you stand in front of a stone building and clap your hands, you hear an echo. What is happening to the sound
wave to cause this echo?
E.35 The sound waves reflect from the stone surface.
36. If you stand behind a massive stone wall, you have trouble hearing a person on the other side clapping her hands.
What is happening to the sound wave to make it difficult for you to hear?
E.36 Most of the sound wave reflects from the stone surface and fails to reach your ears.
37. A harbor breakwater is a stone wall in the water that prevents waves from entering the harbor. Where does a
wave’s energy go after it hits the breakwater?
E.37 Some of the wave reflects but the rest of its energy becomes thermal energy.
38. Waves tend to bend toward points of land projecting into the ocean and erode those points. What causes the wave
to bend toward the points?
E.38 The waves refract in the shallow water near the points and approach those points more and more directly as
a result.
39. Surfers are well aware that waves bend as they pass over coral reefs. What causes this bending?
E.39 Refraction occurs as a wave slows down over the shallow coral.
40. Musical tones can linger for many seconds in a stone cathedral. Why?
E.40 The stone surfaces reflect the sound waves well and do not absorb much of the sound energy.
41. If two violinists play slightly different notes at the same time, the combined sound has a pulsing character. What
causes that pulsation?
E.41 Interference between the two sound waves causes a beating effect.
42. When an airplane’s two propellers are turning at almost the same rate, their combined sound can have a pulsing
character to it. Explain that pulsing.
E.42 Interference between the sound waves from the two propellers causes a beating effect.