Expressions - Geometry

CONTENTS

OTHER MATERIAL

Simple Harmonic Motion

We saw in the previous section that we can create circular motion by using sine and cosine to define x- and y-coordinates. Here, we'll do the reverse: show that a sine (or cosine) wave results from extracting one component of circular motion.

In this animation, imagine that the small dot on the circle's perimeter controls a plotting pen to its left. As the circle rotates, this pen draws a graph of the dot's vertical motion, ignoring any horizontal motion.

You'll notice that the 'plotting pen' dot moves in a rhythmic pattern that decelerates at the extremes and is fastest towards the middle, similar to what you might expect from the 'Easy-ease' Keyframe Assistant. This kind of motion is called 'simple harmonic motion'. One good real-world example of simple harmonic motion would be the path traced on a floor by a swinging pendulum.

This motion might be easier to recognize by focusing on either of the two smaller yellow dots in this next animation. The dot on the right mirrors the vertical component of the larger dot's circular motion, and is controlled by a sine function; the dot at the bottom mirrors the horizontal component and is controlled by a cosine function:

Simple harmonic motion shows up wherever an object is subject to forces that increase linearly as the object is offset from its rest position—such as at the end of a spring. In this animation, layer B's x-coordinate is controlled by a sine function and the spring layer uses a 'span' script to match its scale to the space between A and B. The end result is a fairly convincing spring animation:

The results would be far more realistic, however, if the spring's oscillation slowed over time and eventually stopped—as a real-world spring would. To accomplish this, we'll need to understand the frequency and amplitude of the spring's motion, as discussed in the next section.