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Harmonic Convergence

When you think about the motion of an object attached a spring, you probably don't think about circles. But what if you had two spring-like motions, one pushing the object up and down and the other pushing the object left and right?

Click Up/Down. The turtle moves up and down, as if attached to a vertical spring. Click Up/Down again to stop to the turtle. Now try Left/Right.

Now turn on both springs. What do you see? It depends exactly when you clicked the buttons. You probably see a type of "squished circle" known as an ellipse. But if you start the two springs at just the right time, you will see a circle (which is actually a very special type of ellipse).

The motion of an object attached to a spring is called simple harmonic motion. When you combine two simple harmonic motions at right angles to one another, you get an ellipse. The exact shape of the ellipse depends on the "phase" between the two springs. If one spring is fully extended just when the other spring is at the center of its motion (moving at its fastest speed), then you get a circular motion.

Here's another way to think about it. Imagine an object moving around a circle clockwise at constant speed, and you continually measure how much the object has moved in the vertical direction. When the object is near the top of the circle (the 12 of the clock), it is moving mostly left-to-right, without much change in the vertical direction. When the object is near the 3 of the clock, it is changing quickly in the vertical direction (moving down quickly). It turns out that the vertical motion of the object is exactly the same as the motion of an object on a spring (simple harmonic motion). The same is true of the horizontal motion of the object.

So any circular motion can be broken apart into two spring-like motions. In fact, one of the most amazing discoveries in mathematics is that every possible motion can be broken apart into a combination of spring-like motions -- as long as you choose the right types of springs.

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Last modified: November 17, 2003
by Rupert Russell