Here is your pi. In fact, you can measure the spring’s mass, period, and constant independently and use this to calculate pi just for fun.
However, we can also use a mathematical function to represent this oscillation. Here is the simplest equation that gives the position of mass as a function of time, where A is the amplitude of motion and ω is the pulsation.
This solution includes the cosine trigonometric function. If your trigonometric is fuzzy, just remember that all trigonometric functions tell us about the ratio of the sides of right triangles. For example, the cosine of 30 degrees says that if you have a right triangle with an angle of 30 degrees, the length of the side adjacent to that angle divided by the length of the hypotenuse would be a certain value. (In this case it would be 0.866).
(You might think it’s weird that we need a mathematical function which is also used for triangles to understand the motion of a spring – which is a circular object, after all. But in the end, this function happens to be a solution to our equation. In short, we use it because it works. Anyway, bear with me.)
Now imagine that your right triangle has an angle that is constantly increasing. (That’s the term ωt.) Since the angle changes, you essentially have a triangle that revolves around a circle. If you look at just one side of this right triangle and how it changes over time, there’s your trigonometric function. Here is what it looks like:
Since this oscillation is related to a circle, it seems obvious that you would have a pi in there.
In fact, you can find pi in any other type of oscillation that can be modeled with a trigonometric function containing sines or cosines. For example, think of a pendulumwhich is mass swinging on a string, or the vibrations of a diatomic molecule (a molecule with two atoms, like nitrogen), or even the change of electric current in something like a circuit inside a radio that makes an oscillation.
For physics geeks, perhaps the most popular fundamental is called h-bar (ħ). It is basically Planck’s constant (h) divided by 2π.
Planck’s constant gives the relationship between energy and frequency for super tiny objects, like atoms—and you can measure this constant yourself with LEDs. In fact, pi appears so often in models dealing with tiny quantum things that physicists have combined pi and h to create h-bar.