There’s your pi. In fact, you can measure the mass, period, and spring 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 the mass as a function of time, where A is the amplitude of the motion and ω is the angular frequency.
This solution includes the trigonometric function cosine. If your trig is hazy, just remember that all trig functions tell us about the ratio of sides for right triangles. For instance, the cosine of 30 degrees says that if you have a right triangle with one angle of 30 degrees, the length of the side adjacent to this angle divided by the length of the hypotenuse would be some value. (In this case, it would be 0.866).
(You might think it’s weird that we need a mathematical function that 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 just so happens to be a solution to our equation. In short, we use it because it works. Anyway, stick with me.)
Now imagine that your right triangle has an angle that is constantly increasing. (That’s the ωt term.) Since the angle changes, you essentially have a triangle that rotates around in a circle. If you look at just one side of this right triangle and how it changes with time—there’s your trigonometric function. Here’s what that looks like:
Since this oscillation is related to a circle, it seems obvious you would have a pi in there.
In fact, you can find pi in any other kind of oscillation that can be modeled with a trig function that contains sines or cosines. For example, think of a pendulum, which is a mass swinging from a string, or the vibrations of a diatomic molecule (a molecule with two atoms, like nitrogen), or even the change in 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). This is essentially just the Planck constant (h) divided by 2π.
The Planck constant gives the relationship between energy and frequency for super tiny objects, like atoms—and you can measure this constant yourself with some LEDs. In fact, pi shows up so often in models dealing with tiny quantum things that physicists combined pi and h to make h-bar.