15.1 Simple Harmonic Motion
What conditions must be met to produce SHM?
(a) If frequency is not constant for some oscillation, can the oscillation be SHM? (b) Can you think of any examples of harmonic motion where the frequency may depend on the amplitude?
Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.
Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a more pliable material.
As you pass a freight truck with a trailer on a highway, you notice that its trailer is bouncing up and down slowly. Is it more likely that the trailer is heavily loaded or nearly empty? Explain your answer.
Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer springs? Explain your answer.
15.2 Energy in Simple Harmonic Motion
Describe a system in which elastic potential energy is stored.
Explain in terms of energy how dissipative forces such as friction reduce the amplitude of a harmonic oscillator. Also explain how a driving mechanism can compensate. (A pendulum clock is such a system.)
The temperature of the atmosphere oscillates from a maximum near noontime and a minimum near sunrise. Would you consider the atmosphere to be in stable or unstable equilibrium?
15.3 Comparing Simple Harmonic Motion and Circular Motion
Can this analogy of SHM to circular motion be carried out with an object oscillating on a spring vertically hung from the ceiling? Why or why not? If given the choice, would you prefer to use a sine function or a cosine function to model the motion?
If the maximum speed of the mass attached to a spring, oscillating on a frictionless table, was increased, what characteristics of the rotating disk would need to be changed?
Pendulum clocks are made to run at the correct rate by adjusting the pendulum’s length. Suppose you move from one city to another where the acceleration due to gravity is slightly greater, taking your pendulum clock with you, will you have to lengthen or shorten the pendulum to keep the correct time, other factors remaining constant? Explain your answer.
A pendulum clock works by measuring the period of a pendulum. In the springtime the clock runs with perfect time, but in the summer and winter the length of the pendulum changes. When most materials are heated, they expand. Does the clock run too fast or too slow in the summer? What about the winter?
With the use of a phase shift, the position of an object may be modeled as a cosine or sine function. If given the option, which function would you choose? Assuming that the phase shift is zero, what are the initial conditions of function; that is, the initial position, velocity, and acceleration, when using a sine function? How about when a cosine function is used?
15.5 Damped Oscillations
Give an example of a damped harmonic oscillator. (They are more common than undamped or simple harmonic oscillators.)
How would a car bounce after a bump under each of these conditions?
(c) critical damping
Most harmonic oscillators are damped and, if undriven, eventually come to a stop. Why?
15.6 Forced Oscillations
Why are soldiers in general ordered to “route step” (walk out of step) across a bridge?
Do you think there is any harmonic motion in the physical world that is not damped harmonic motion? Try to make a list of five examples of undamped harmonic motion and damped harmonic motion. Which list was easier to make?
Some engineers use sound to diagnose performance problems with car engines. Occasionally, a part of the engine is designed that resonates at the frequency of the engine. The unwanted oscillations can cause noise that irritates the driver or could lead to the part failing prematurely. In one case, a part was located that had a length L made of a material with a mass M. What can be done to correct this problem?