Questions about springs on SAT II Physics are usually simple matters of a mass on a spring oscillating back and forth. However, spring motion is the most interesting of the four topics we will cover here because of its generality. The harmonic motion that springs exhibit applies equally to objects moving in a circular path and to the various wave phenomena that we’ll study later in this book. So before we dig in to the nitty-gritty of your typical SAT II Physics spring questions, let’s look at some general features of harmonic motion.
Oscillation and Harmonic Motion
Consider the following physical phenomena:
- When you drop a rock into a still pond, the rock makes a big splash, which causes ripples to spread out to the edges of the pond.
- When you pluck a guitar string, the string vibrates back and forth.
- When you rock a small boat, it wobbles to and fro in the water before coming to rest again.
- When you stretch out a spring and release it, the spring goes back and forth between being compressed and being stretched out.
There are just a few examples of the widespread phenomenon of oscillation. Oscillation is the natural world’s way of returning a system to its equilibrium position, the stable position of the system where the net force acting on it is zero. If you throw a system off-balance, it doesn’t simply return to the way it was; it oscillates back and forth about the equilibrium position.
A system oscillates as a way of giving off energy. A system that is thrown off-kilter has more energy than a system in its equilibrium position. To take the simple example of a spring, a stretched-out spring will start to move as soon as you let go of it: that motion is evidence of kinetic energy that the spring lacks in its equilibrium position. Because of the law of conservation of energy, a stretched-out spring cannot simply return to its equilibrium position; it must release some energy in order to do so. Usually, this energy is released as thermal energy caused by friction, but there are plenty of interesting exceptions. For instance, a plucked guitar string releases sound energy: the music we hear is the result of the string returning to its equilibrium position.
The movement of an oscillating body is called harmonic motion. If you were to graph the position, velocity, or acceleration of an oscillating body against time, the result would be a sinusoidal wave; that is, some variation of a y = a sin bx or a y = a cos bx graph. This generalized form of harmonic motion applies not only to springs and guitar strings, but to anything that moves in a cycle. Imagine placing a pebble on the edge of a turntable, and watching the turntable rotate while looking at it from the side. You will see the pebble moving back and forth in one dimension. The pebble will appear to oscillate just like a spring: it will appear to move fastest at the middle of its trajectory and slow to a halt and reverse direction as it reaches the edge of its trajectory.
This example serves two purposes. First, it shows you that the oscillation of springs is just one of a wide range of phenomena exhibiting harmonic motion. Anything that moves in a cyclic pattern exhibits harmonic motion. This includes the light and sound waves without which we would have a lot of trouble moving about in the world. Second, we bring it up because SAT II Physics has been known to test students on the nature of the horizontal or vertical component of the motion of an object in circular motion. As you can see, circular motion viewed in one dimension is harmonic motion.
