One aspect of probability I’ve always found to be a little tricky is the part where you need to count things. In theory, this sounds easy enough. After all, it’s just looking at the complete list of things you’re studying, and enumerating them, right?

The pendulum is a classic physical object that is modeled in many introductory physics courses. It’s a nice system to study because it is so simple, yet still allows students to see how to study the motion of a system. Here, I want to do through the steps of deriving what is usually seen as an elementary result, the period of a pendulum, and show how it is actually more complicated than what most students see.

I’ll let you in on a bit of a secret. For most of my life, I hated doing experiments in science.

When I reflect on my education in science (and in physics in particular), the common theme I see is just how the amount of sophistication present in the computations and concepts I learned each year kept increasing. If there was one thing I could count on, it wasn’t learning something “new”. Instead, it was about viewing things I might have once taken for granted as a process that was much more deep than I realized.