A simple gravity pendulum is the simplified model for a pendulum: a point mass suspended from a massless cord suspended from a pivot in a vacuum subject to gravity.
In my class, I studied very simplified spring mass systems where the solution to the second order linear homogeneous equation $y''+C_1y'+C_2y=0$ was the position function of the point mass attached to the spring (where $C_1$ and $C_2$ are the damping constant and the spring constant respectively).
I was told that a simple gravity pendulum could be modeled with a similar second order differential equation, and I would appreciate if someone could derive this equation in a didactic and systematic manner, so I could fully appreciate and understand this model (which, I know, is not very realistic, but interesting to me nonetheless). Thank you.
Best Answer
Unfortunately I don't have enough reputation to post images, so you'll have to look at this picture instead.
Consider a pendulum of length $L$ and mass $m$. We denote the angle the pendulum makes with the vertical $\theta$, so that the pendulum is at rest when $\theta = 0$.
The dynamics of the pendulum are governed by Newton's second law, $\tau = I \alpha$. Here $\tau$ is the torque around the pivot, $I$ is the rotational inertia of the pendulum and $\alpha$ is its angular acceleration.
Make the following substitutions:
These substitutions give: $$-mg\sin(\theta)L = mL^{2}\theta''.$$
We cancel $mL$ on both sides and rearrange: $$\theta'' + \frac{g}{L}\sin(\theta) = 0.$$
This equation has solutions in terms of elliptic functions, which are complicated, but we can avoid all that unpleasantness by making the small-angle approximation: $\sin(\theta) \sim \theta.$ So provided that the pendulum only makes small oscillations its motion is described by $$\theta'' + \frac{g}{L}\theta = 0.$$