# Simple Harmonic Oscillator: An intuition

You have a mass suspended on a spring. We want to know where the mass will be at any instant of time.

## The physical solution

Now before we get on to the math, let us first visualize the motion by attaching a spray paint bottle as the mass.

Oh, wait that seems like a function that we are familiar with – The sinusoid.

Without even having to write down a single equation, we have found out the solution to our problem. The motion that is traced  by the mass is a sinusoid.

## But what do I mean by a sinusoid ?

If you took the plotted paper and tried to create that function with the help of sum of polynomials i.e x, x2, x3 … Now you this what it would like :

By taking an infinite of these polynomial sums you get the function Since this series of polynomial occurs a lot, its given the name – sine.

## A note on the cosine

Since cosine is merely the sine function pushed to the side, the analogy works the same way.

Therefore we can conclude that the motion described a mass attached to a spring is a sine or a cosine function:

$y(t) = A \sin( \omega t + \phi) \ \text{or} \ A \cos(\omega t + \phi)$

Hopefully this gives you an intuition on why the solution to a simple harmonic oscillator is given by the sine or the cosine function. Following runs through how you would go about getting it from solving a differential equation.

$\text{solution} \ x = A \cos(\omega t + \phi) \ \text{or} A \sin(\omega t + \phi)$