Getting the Period of a Spring-Operated Physical Pendulum
A juicy physics problem that requires delving into the differential equation for pendulum motion. A rod of mass M on a pivot a distance r from the end is driven by a spring of constant k pulling the end back and forth to make an (admittedly stupid) physical pendulum. We have to show that its period T is given by $latex T^2 = 4\pi \frac{M}{3kr^2} (L^2 + 3r^2 -3rl) &s=3$.
This video walks through doing $latex \int \frac{x^3}{1+x^4}\,dx&s=3$ using u-substitution.
Once you’ve established that curl F = 0, you know that F is the gradient of some potential ⱷ. You can use the Fundamental Theorem to evaluate ∫F.dr — if only you could find the potential function. In this video I go over a straighforward method of finding the potential function using partial integration. (Note: the sound quality gets wonky near the end but you can still hear what I’m explaining perfectly well. Apologies and I’ll figure out eventually why that happens.)
A clear explanation of the epsilon-delta definition of a limit
Lots of calculus students learn to solve differential equations without understanding what they really mean. Here’s a simple example of a differential equation so you can see how it works.