Physics 101: How Things Work

These are the answers to the even-numbered exercises and problems, and all of the cases assigned in Chapter 7. The answers to the others appear in the back of the textbook.

Exercises

2. The rack is a pendulum and its period depends only on the length of the clothes hangin from it and on the strength of gravity.

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.

4. Since the period is independent of the amplitude, the tree is executing simple harmonic motion. This means the restoring force acting on the tree is proportional to how far it is bent away from its normal upright orientation.

10. The pendulum's restoring force is directly related to the fact that it accelerates so as to minimize its gravitational potential energy. Without gravity, there is no gravitational potential energy, no restoring force, and no oscillation about a stable equilibrium.

12. The pendulum's restoring force is directly related to the fact that it accelerates so as to minimize its gravitational potential energy. Without gravity, there is no gravitational potential energy, no restoring force, and no oscillation about a stable equilibrium.

14. The pendulum's restoring force is directly related to the fact that it accelerates so as to minimize its gravitational potential energy. Without gravity, there is no gravitational potential energy, no restoring force, and no oscillation about a stable equilibrium.

16. The pendulum's restoring force is directly related to the fact that it accelerates so as to minimize its gravitational potential energy. Without gravity, there is no gravitational potential energy, no restoring force, and no oscillation about a stable equilibrium.

In terms of equations, the frequency is equal to the wave velocity divided by the wavelength. The wavelength is proportional to L, the length of the rubber band. The wave velocity is the square root of the tension divided by the mass per unit length. The tension is proportional to L, and the mass per unit length is inversely proportional to L. This means the velocity is proportional to L. Therefore, the factors of L cancel when you calculate the period.

This is different from stretching a violin string. In that case, the tension depends on how far the string is stretched from equilibrium, not on the original length. The string doesn't stretch very much, so L can be treated as fixed while the tension increases. Stretching a violin string leaves the wavelength and mass per unit length approximately the same, since it doesn't stretch very much, but the tension changes a lot. Thus, the wave velocity increases, and so does the frequency.

20. The highest pitched strings are usually the thinnest (to reduce mass) and the tautest (to increase stiffness). Thin, tight strings break easily.

24. While creating the different tones, the air column in the bugle is vibrating in different modes. The more the total air column is divided into smaller segments, the higher the pitch.

30. If all the instruments played at the same instant but each produced a different pitch, the different-pitched sounds would reach their audience at different times. What started as a single cord would be spread out into an arpeggio.

---

Cases

2a. If the length is longer but everything else fixed, the jump rope will slow down.
2b. If the rope is made more massive, with everything else fixed, the jump rope will slow down.
2c. Increasing the tension makes the jump rope speed up.
2d. It must move twice as fast to get the S shape with a node in the middle, since now its wavelength is half as long.

4a. A stretchy bungee cord will spread out the momentum change of the jumper over a longer time, which means a smaller force must act over this time to make the person bounce. In other words, the same impulse can be delivered with a smaller force over a longer time when the cord is stretchier.
4b. When the person is bouncing so that the cord stays taut, it obeys Hooke's Law, so there is a linear restoring force. That means the person is executing simple harmonic motion, so the amplitude is independent of the period.
4c. A heavier person has more inertia, so their acceleration is less, and their period is longer.
4d. The heavier person feels a stronger restoring force. The restoring force is proportional to the amplitude of the bounce, which is greater for a heavier person, since they have more gravitational potential energy when jumping. (It turns out, however, that the upward acceleration of the heavy person is less than for the light person, because the increase in inertia is greater than the increase in restoring force. So the lighter person feels more "g"s.)
4e. The tension is the sum of the person's weight and the upward restoring force at the bottom of the bounce. This is greater than the weight alone. The tension must be greater than the weight, since the person is accelerating upward at the lowest point.
4f. Two bungee cords would double the amount of force it takes to stretch the cords a given length, effectively doubling the spring constant in Hooke's Law. This will speed up the bouncing (by a factor of the square root of 2).

5a. This is an example of a driven harmonic oscillator. It bounces at the resonant frequency of the automobile on the springs, which depends on the stiffness of the springs and the mass of the automobile. The bouncing persists, because this is an example of simple harmonic motion.
5b. Each bounce will take it just as high, assuming energy is conserved.
5c. The body will bounce up and down at a constant speed and amplitude.
5d. The bouncing is damped out by the shock observers. The car will bounce fewer times, and with an amplitude that decreases with time.
5e. The car bounces up and down (slightly slower than before) with a decreasing amplitude, as the oscillations are damped out by the shock absorbers.

6a. The pendulum has inertia, and remains at rest as the paper moves under it.
6b. You can get the period of the wave by measuring the time between crests at a single seismometer. You can get the velocity of the wave by measuring the time it takes the same crest to get from one seismometer to another. The wavelength is the velocity times the period.
6c. The time between crests is the period. The frequency is the reciprocal of the period.
6d. The wave velocity is the time it takes a crest to get from one of the seismographs to the other.
6e. It swings east and west, perpendicular to the direction of travel of the wave.
6f. The up and down motion will impart momentum to the mass on the spring, causing it to bounce up and down at its resonant frequency.
6g. The energy in a wave is proportional to the square of its amplitude. (It is enough to remember that the energy increases with the amplitude.) As the wave travels outward, its energy gets spread out over an area which increases as the square of the distance from the origin. Therefore, the amplitude must decrease as the wave spreads out.

8a. The gravitational potential energy of a wave on Mars would be smaller than for a similar wave on Earth by a factor of 3.71/9.8 = 0.379.
8b. The amount of work to deform the surface is equal to the amount of potential energy in the wave that is built up in the process. The amount of work to build the same wave on Mars is 38% of the work needed to build this wave on Earth.
8c. Water waves will travel faster on Earth, due to the greater stiffness of the surface in the stronger gravity.
8d. The identical wave on Earth carries more energy.
8e. No. Producing normal waves requires a restoring force to draw the water back down after it is pushed up. This force is gravity.

---

Department of Physics

University of Tennessee