Here’s your pi. In fact, you possibly can independently measure the mass, period, and spring constant and use them to calculate pi for fun.
However, we also can use a mathematical function to represent this oscillation. Here is the best equation for the position of the mass as a function of time, where A is the amplitude of the motion and ω is the angular frequency.
Illustration: Rhett Allain
This solution incorporates the cosine trigonometric function. If your trig is hazy, just do not forget that all trigonometric functions tell us the ratio of the perimeters for right triangles. For example, the cosine of 30 degrees says that if you could have a right triangle with one 30-degree angle, the length of the side adjoining to that angle divided by the length of the hypotenuse will likely be some value. (In this case, it will be 0.866).
(You might imagine it’s weird that we want a math function that is also used for triangles to grasp the motion of a spring – which is, in any case, a circular object. But, in any case, this function just so happens to be the answer to our equation. Briefly saying, we use it because it really works. Stick with me anyway.)
Now imagine your right triangle has an angle that keeps increasing. (This is the term ωt.) Since the angle changes, we mainly have a triangle that rotates in a circle. If you simply take a look at one side of this right triangle and the way it changes over time, here’s your trigonometric function. Here’s what it looks like:
Video: Rhett Allain
Since this oscillation is expounded to a circle, it seems obvious that you just would have pi there.
In fact, pi may be present in another style of oscillation that may be modeled with a trigonometric function containing sines or cosines. Think, for instance, of a pendulum, which is a mass swinging on a string, or the vibration of a diatomic molecule (a molecule with two atoms, like nitrogen), or perhaps a change in the electrical current in something like a circuit inside a radio that causes it to oscillate.
The Uncertainty Principle
For physics geeks, probably the most well-liked fundamental principle is the h-bar (ħ). Basically, it’s just Planck’s constant (h) divided by 2π.
Planck’s constant gives the connection between energy and frequency for very small objects like atoms – and you possibly can measure this constant yourself with some LEDs. In fact, pi comes up so often in models coping with small quantum things that physicists have combined pi and h to make h-bar.
