Though harmonic motion is one of the most widespread and important of physical phenomena, your understanding of it will not be taxed to any great extent on SAT II Physics. In fact, beyond the motion of springs and pendulums, everything you will need to know will be covered in this book in the chapter on Waves. The above discussion is mostly meant to fit your understanding of the oscillation of springs into a wider context.
The Oscillation of a Spring
Now let’s focus on the harmonic motion exhibited by a spring. To start with, we’ll imagine a mass, m, placed on a frictionless surface, and attached to a wall by a spring. In its equilibrium position, where no forces act upon it, the mass is at rest. Let’s label this equilibrium position x = 0. Intuitively, you know that if you compress or stretch out the spring it will begin to oscillate.
Suppose you push the mass toward the wall, compressing the spring, until the mass is in position
.
When you release the mass, the spring will exert a force, pushing the mass back until it reaches position
, which is called the
amplitude of the spring’s motion, or the maximum displacement of the oscillator. Note that
.
By that point, the spring will be stretched out, and will be exerting a force to pull the mass back in toward the wall. Because we are dealing with an idealized frictionless surface, the mass will not be slowed by the force of friction, and will oscillate back and forth repeatedly between
and
.
Hooke’s Law
This is all well and good, but we can’t get very far in sorting out the amplitude, the velocity, the energy, or anything else about the mass’s motion if we don’t understand the manner in which the spring exerts a force on the mass attached to it. The force, F, that the spring exerts on the mass is defined by Hooke’s Law:
where x is the spring’s displacement from its equilibrium position and k is a constant of proportionality called the spring constant. The spring constant is a measure of “springiness”: a greater value for k signifies a “tighter” spring, one that is more resistant to being stretched.
Hooke’s Law tells us that the further the spring is displaced from its equilibrium position (x) the greater the force the spring will exert in the direction of its equilibrium position (F). We call F a restoring force: it is always directed toward equilibrium. Because F and x are directly proportional, a graph of F vs. x is a line with slope –k.
Simple Harmonic Oscillation
A mass oscillating on a spring is one example of a simple harmonic oscillator. Specifically, a simple harmonic oscillator is any object that moves about a stable equilibrium point and experiences a restoring force proportional to the oscillator’s displacement.
