# Three Solutions for the Simple Harmonic Oscillator

Let’s be honest — the world is made of springs. OK, maybe they aren’t literally springs but they are approximately springs. Yes, for small oscillations we can approximate stuff as a simple harmonic oscillator.

**The Physics Problem**

Suppose you have a mass (m) connected to a spring and the mass can ONLY move in the x-direction and the spring is the only force acting on the mass. Here’s what that looks like (I’ll show you how to make this animation later in this post).

If I set x = 0 meters (you can always move the origin to make this true) to be the equilibrium position, then the spring force will always pull the mass back towards the origin with a force that’s proportional to the displacement. It looks like this:

Since this is the only force (in our make-believe situation) it is equal to the mass multiplied by the acceleration. Remember that the acceleration is the second derivative of the position.

OK, wait. Since I’m a physicist, I have special permission to write the second derivative of x (with respect to time) in this “dot notation”. Check it out.