# Oscillations and Simple Harmonic Motion

### Introduction and Summary

We have already studied the most common types of motion: linear and rotational motion. We have developed the concepts of work, energy, and momentum for these types of motion. To complete our study of classical mechanics we must finally examine the complicated case of oscillations. Unlike the other types of motion we have studied, oscillations generally do not have constant acceleration, are many times chaotic, and require far more advanced mathematics to handle. As such, we give the most complete treatment to the subject as possible, concentrating on the kinds of oscillations that are easiest to examine.

We begin be defining oscillations, and the variables associated with this motion. Next we take a closer look at a special kind of oscillation, simple harmonic motion. It is this kind of oscillation that will form the bulk of our study of oscillations. We derive the motion of simple harmonic systems, and relate this motion to the concept of oscillation that we have already defined. This derivation is quite complex, and to complete it we must use some complex calculus. The derivation itself is not as important as the end product, but if one can understand the mathematics, it can greatly increase understanding of the topic.

Deriving the equations for simple harmonic motion will allow us to take an in depth look at various kinds of harmonic motion, as seen in the next section.

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