28: Oscillations
- Page ID
- 166285
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On September 19, 1985, a powerful earthquake struck off Mexico’s Pacific coast, hundreds of kilometers from Mexico City. By the time the seismic waves reached the capital they should have been feeble, and on solid ground they were. But much of the city is built on the soft, water-soaked clay of an ancient lake bed, and that clay has a natural rhythm of its own: shaken, it sways back and forth about once every two seconds. The arriving waves happened to contain plenty of shaking at just that period, so the lake bed amplified them, rocking the city for minutes. Then came a second, cruel coincidence. Buildings, too, have natural periods of sway, and those roughly six to fifteen stories tall swayed at about the same two-second rhythm as the ground beneath them. Hundreds of them collapsed, while many shorter and taller buildings on the same streets came through with far less damage. What destroyed them was not simply the strength of the shaking but its timing: the ground pushed each building at very nearly the building’s own natural rhythm, and a system driven at its natural rhythm responds with ever-growing amplitude. That runaway response has a name (resonance) and by the end of this chapter you will be able to say exactly what it is and why it is so dangerous.
A swaying building is an extreme example of something utterly ordinary. A mass bobbing on a spring, a child on a swing, the balance wheel of a wristwatch, a guitar string, the atoms in a crystal, the charge sloshing back and forth in the LC circuits of Chapter 26: all of these oscillate, repeating the same motion over and over about a stable equilibrium. Remarkably, an enormous number of these oscillators, despite their physical differences, obey the same equation and trace out the same motion in time. That shared motion is called simple harmonic motion, and it is the most important single model in all of physics. Master it here, for a mass on a spring, and you will meet it again in waves, in alternating-current circuits, in molecular vibrations, and in quantum mechanics.
Learning Objectives
- Recognize simple harmonic motion (SHM) as the motion produced by a linear restoring force, and write the position, velocity, and acceleration of an oscillator as functions of time.
- Relate the angular frequency of a mass–spring system to its mass and spring constant, and relate angular frequency, period, and frequency to one another.
- Compute the kinetic, potential, and total mechanical energy of an oscillator, and use energy conservation to find speeds and amplitudes.
- Show that a simple pendulum undergoes SHM for small angles, and find its period.
- Extend the analysis to torsion and physical pendulums.
- Describe how damping and an external driving force modify the motion, and explain the condition for resonance.
We will return to the Mexico City earthquake at the very end of the chapter, and in the final problems, once we have the tools to understand what went wrong.