Cases – Chapter 9
1. Because sound carries energy, musical instruments need energy to operate. The musician, who does mechanical work on the
instrument, normally provides this energy. How does a musician transfer energy to
a. an accordion?
b. a piano?
c. a xylophone?
d. a violin?
2. When two people swing a jump rope between them, it forms a moving arc in the air. The rope’s behavior is similar to that of a
violin string, except that the jump rope swings around in a circle rather than vibrating directly back and forth.
a. What happens to the speed at which the jump rope turns if you increase the length of the rope but keep the tension on that
rope the same?
b. What happens to the speed at which the jump rope turns if you use a thicker, more massive rope but keep the rope’s length
and tension the same?
c. What happens to the rope’s motion if the people holding its ends pull harder on the rope and increase its tension?
d. How quickly must the people swing the rope to get it to form an S-shape, with a stationary node in the middle?
3. Imagine a stiff ruler that’s lying at the edge of a table so that one end of it is on the table and the other end is free in the air. If
you hold the ruler’s supported end against the table with one hand and pluck its free end with the other hand, the ruler will vibrate
up and down.
a. Why does the ruler’s frequency of oscillation decrease when you extend more of it off the table?
b. Why doesn’t the ruler’s frequency of oscillation depend on how hard you pluck it?
c. What forms does the ruler’s energy take as it vibrates up and down?
d. What eventually happens to the plucked ruler’s energy?
e. If the ruler vibrates slowly enough, you can increase the energy of its vibration by pushing down on it rhythmically, once per
cycle, just after it completes its upward motion and begins to head downward. Why must you push it downward as it heads
downward, rather than pushing it downward as it heads upward?
4. In bungee jumping, a strong elastic cord or “bungee” is used to break the fall of a person who has leapt from a tall platform.
This cord runs from the platform to the person’s ankles and is short enough that the person doesn’t hit the ground. After a few
terrifying seconds of freefall, the cord pulls taut and the jumper bounces back upward. On this first bounce, the cord goes slack
and the jumper once again experiences freefall. But in subsequent bounces, the cord remains taut and the jumper bounces up and
down smoothly.
a. Why must the bungee cord stretch relatively easily to avoid injuring the jumper?
b. As a jumper bounces up and down on the cord, the time it takes to complete each bounce doesn’t depend on how high that
bounce is. Explain.
c. Why does a heavier person bounce up and down more slowly than a lighter person?
d. Who experiences the more extreme feelings of heaviness while coming to a stop at the end of the cord: a heavier person or a
lighter person?
e. Why must the cords used in bungee jumping be able to withstand tensions that exceed the weights of the heaviest jumpers?
f. If the jumper used two bungee cords instead of one—perhaps one attached to each ankle—how would that affect the time it
takes to complete each bounce up and down?
5. The body of a car is suspended above its wheels by four strong springs. These springs allow the wheels to respond quickly to
bumps in the road without having the entire body respond, too. However, the wheels are also connected to the body by shock
absorbers. Unlike the springs, which store energy when work is done on them, the shock absorbers use frictionlike effects to
convert work into thermal energy.
a. You remove the shock absorbers from your parked car and then jump into the driver’s seat. The body bounces up and down
for a long time. Explain this behavior.
b. If the car’s first bounce takes 1 s, how long does its second bounce take? its tenth bounce?
c. You then drive the car off a high curb and onto the road. Describe the body’s motion for the next ten seconds.
d. You reinstall the shock absorbers. Now every time the car body moves toward or away from the wheels, it does work on the
shock absorbers and this work becomes thermal energy. You jump back into the driver’s seat of the parked car. How does it
respond?
e. You again drive the car off the high curb and onto the road. The shock absorbers make a difference. Describe the body’s
motion for the next ten seconds.
6. An earthquake generates waves that travel through the earth in all directions. While the earth’s internal structure complicates
the passages of these waves to distant sites, they follow simple paths to nearby places. However, there are at least two types of
waves, and they travel at different speeds.
a. The waves that travel fastest in the earth’s crust are those that resemble sound. They’re called “P” or “primary” waves
because they arrive first. As these waves travel forward, the ground moves alternately forward and backward, along the
direction of the wave velocity. For example, if a “P” wave is heading north, the ground oscillates north and south as the wave
passes. A seismometer can detect this motion by hanging a pendulum above a sheet of paper that’s resting on the ground. When
the wave passes, the pendulum marks the paper along the direction of the wave velocity. Why does the pendulum shift back
and forth relative to the paper?
b. The ground along the path of a “P” wave doesn’t all shift north or south at once. These shifts are actually the “P” wave’s
crests and troughs, and they move northward with the wave. When the ground underfoot shifts first northward and then
southward, it’s because a crest passed by, followed by a trough. At the moment the crest is under you, the trough is still a long
way behind it. In fact, it’s exactly half a wavelength behind it. How could you use several seismometers to determine the
wavelength of a passing “P” wave?
c. How could you use one seismometer to determine the frequency of the “P” wave?
d. How could you use several seismometers to measure the wave velocity of the “P” wave?
e. Waves that involve sideways or vertical motion of the ground travel more slowly and arrive after the “P” waves. They’re
called “S” or “secondary” waves. For example, in one type of “S” wave that is heading north, the ground oscillates east and
west as the wave passes. How will a hanging pendulum seismometer respond when this “S” wave passes?
f. In another northbound “S” wave, the ground moves up and down. How could you use a weight and a spring to detect this
wave as it passes by?
g. The wave from an earthquake spreads out in all directions like a ripple on a pond. Use conservation of energy to explain
why the wave’s amplitude of motion must decrease as it gets farther from the earthquake’s center and why the energy passing
under a city near the earthquake must be much greater than the energy passing under a similar-sized city far away from the
same earthquake. (This weakening with distance occurs even when no energy is lost as thermal energy. It protects us from all
but relatively nearby earthquakes.)
7. When the paper cone in your stereo speaker moves forward and backward rhythmically, it produces a sound wave that travels
through the air at about 331 m/s. The crests of this wave consist of slightly compressed air and the troughs consist of slightly
rarefied air.
a. Suppose that you have an instrument that can monitor the pressure at one point in space. You place this instrument in front
of the speaker and begin measuring. Draw a rough graph of the pressure that the instrument measures versus the time that has
elapsed since the measurement started.
b. Suppose that instead of keeping the instrument in one place, you began to move the instrument so that it continues to detect
a region of slightly compressed air (one of the crests in the wave). How fast must you move the instrument to keep up with the
crest?
c. You spray a mist of water into the air so that you can see if the air itself moves as the sound travels through it. The mist
remains still, so you know that the air isn’t moving significantly, even though the sound wave is. Explain this surprising result.
d. How could you determine the wavelength of the sound wave emerging from the speaker?
e. What two types of energy does a sound wave contain?
f. The oscillation of air inside an organ pipe is actually a standing sound wave. How is it different from the traveling sound
wave that’s passing through your room?
8. If astronauts landing on Mars were to find liquid water on the surface, the waves that they would see would behave somewhat
differently. That’s because the acceleration due to gravity on Mars is only 3.71 m/s2.
*a. A wave always needs two forms for its energy. In a surface wave on water, one of those forms is gravitational potential
energy. How would the gravitational potential energy of a wave on Mars compare with that of an identical looking wave on
Earth?
b. The “stiffness” of the water’s surface depends on gravity. When gravity is weak, it takes less work to deform the water’s
surface. Use your answer to part a to show that this is the case.
c. The surface of water on Mars is more easily deformed and thus less “stiff” than the surface of water on Earth. On which
planet would surface waves on water travel faster, Mars or Earth?
d. Which wave carries more energy, a surface wave on Mars or an identical looking wave on Earth?
e. On a spaceship that’s far away from stars, planets, and other celestial objects, there is almost no gravity. If the astronauts on
that spaceship put water in a small basin, would they be able to produce normal surface waves in that water? Why or why not?
9. A trumpet consists mainly of a long brass tube with a mouthpiece at one end and a flared opening or “bell” at the other end.
This tube is coiled to make the trumpet more compact. The trumpet also has three valves that splice additional segments of tubing
into the main tube, thus extending its overall length. When a musician seals his lips together and presses them against the
mouthpiece of the trumpet, he creates a resonant system. The air can oscillate in and out of the open bell while the air pressure in
the mouthpiece oscillates up and down.
a. To sustain the oscillation, the musician blows small bursts of high-pressure air into the mouthpiece. Why are these bursts
timed to coincide with the moments when the pressure inside the mouthpiece is already at its maximum?
b. As the musician presses the valves and lengthens the trumpet tube, the oscillation slows and the trumpet’s pitch goes down.
Explain.
c. Even without touching the valves, a good trumpet player can produce a variety of higher pitched notes using higher-order
vibrational modes. What is the air inside the trumpet doing while it’s oscillating in one of these higher-order modes?
d. The best way for a musical instrument to project sound is to move a large amount of air back and forth at low speeds rather
than a small amount of air back and forth at high speeds. That’s because room air can’t respond quickly enough to the highspeed air near an instrument to extract its energy. Instead, that energy stays with the instrument. If the trumpet had no bell, the
air at its open end would move back and forth very rapidly. Describe how the bell helps the trumpet project its sound.
10. Diving off a springboard is much more complicated than diving off a platform because the springboard can bounce up and
down. When you stand on the springboard without bouncing, it sags downward to a new equilibrium position. From that
position you can begin experimenting.
a. If you begin to bounce up and down gently about the equilibrium position, what factors will determine the frequency of the
bounce?
b. If you bounce more vigorously than in a, while still keeping your feet in contact with the board, how will the frequency of
the bounce change?
c. If you begin to bounce so vigorously that your feet leave the surface of the board, the board’s frequency of vibration changes
abruptly. Why?
d. You’re bouncing gently again. If you want to bounce more vigorously, you must add energy to the board. Your friend is
floating under the board and can push on the board every time it bounces. When should your friend push on the board to add
energy to it?
e. If you want to make the board bounce less vigorously, when should your friend push on the board?
10. A music box uses a row of small metal strips to play music. These strips are part of a metal plate that resembles a comb with
many teeth. Pins projecting from the surface of a rotating metal cylinder pluck the free ends of these strips. These pins are
carefully placed so that, as the cylinder rotates, the music box plays a tune.
a. The metal strips are all of different lengths so that they play different notes. Why do the longer strips vibrate more slowly
than the shorter ones?
b. The sound of each note starts abruptly and decays gradually. Why doesn’t the sound build gradually?
c. How does the cylinder transfer energy to the strips?
d. Draw several sketches of a strip as it vibrates up and down. How is it similar to a violin string? How is it different?
e. When the strip is vibrating in its fundamental mode, where is its anti-node and where is its node?
f. A violin string can vibrate as two half-strings. Why can’t a strip vibrate as though it were two half-strips?
g. Why does f imply that the strip can’t produce the second harmonic of its fundamental pitch?
11. A toddler enjoys bouncing up and down in a seat that’s suspended by a strong spring. The toddler pushes its feet on the floor
and leaps up and down.
a. A toddler’s bouncing motion in the seat is rhythmic. What determines the frequency of this bouncing?
b. As the toddler grows, what happens to the frequency of the bouncing or does it remain unchanged?
c. The toddler can change the amplitude of the bouncing motion by pushing on the floor. When should the toddler push on the
floor to make the seat bounce higher?
d. When should the toddler push on the floor to make the seat bounce less high?