Having established the basics of oscillations, we now turn to the special case of simple harmonic motion. We will describe the conditions of a simple harmonic oscillator, derive its resultant motion, and finally derive the energy of such a system.

### The Simple Harmonic Oscillator

Of all the different types of oscillating systems, the simplest, mathematically speaking, is that of harmonic oscillations. The motion of such systems can be described using sine and cosine functions, as we shall derive later. For now, however, we simply define simple harmonic motion, and describe the force involved in such oscillation.

To develop the idea of a harmonic oscillator we will use the most common example of harmonic oscillation: a mass on a spring. For a given spring with constant k, the spring always puts a force on the mass to return it to the equilibrium position. Recall also that the magnitude of this force is always given by:

 F(x) = - kx

where the equilibrium point is denoted by x = 0. In other words, the more the spring is stretched or compressed, the harder the spring pushes to return the block to its equilibrium position. This equation is only valid if there are no other forces acting on the block. If there is friction between the block and the ground, or air resistance, the motion is not simple harmonic, and the force on the block cannot be described by the above equation.

Though the spring is the most common example of simple harmonic motion, a pendulum can be approximated by simple harmonic motion, and the torsional oscillator obeys simple harmonic motion. Both of these examples will be examined in depth in Applications of Simple Harmonic Motion.

### Simple Harmonic Motion

>From our concept of a simple harmonic oscillator we can derive rules for the motion of such a system. We start with our basic force formula, F = - kx. Using Newton's Second Law, we can substitute for force in terms of acceleration:

ma = - kx

Here we have a direct relation between position and acceleration. For you calculus types, the above equation is a differential equation, and can be solved quite easily. Note: The following derivation is not important for a non- calculus based course, but allows us to fully describe the motion of a simple harmonic oscillator.

### Deriving the Equation for Simple Harmonic Motion

Rearranging our equation in terms of derivatives, we see that:

m   = - kx

or + x = 0

Let us interpret this equation. The second derivative of a function of x plus the function itself (times a constant) is equal to zero. Thus the second derivative of our function must have the same form as the function itself. What readily comes to mind is the sine and cosine function. Let us come up with a trial solution to our differential equation, and see if it works.