Simple Harmonic Motion

December 17th, 2016

$\frac{d^2\theta}{dt^2}=-\frac{g}{l}\theta$
$\frac{d^2\theta}{dt^2}=-\frac{g}{l}\left(\theta-\frac{\theta^3}{3!}\right)$
$\frac{d^2\theta}{dt^2}=-\frac{g}{l}\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}\right)$

Physicists love using the "small angle approximation". Often when dealing with things that are being rotated by some angle $$\theta$$, describing the motion accurately will often involve taking the sine of that angle $$\sin(\theta)$$. However this often makes the math much more complicated, so physicists will use the fact that for small values of $$\theta$$ (like really small angles) $$\sin(\theta)\approx\theta$$. This comes from what is known as the Taylor series for the sine function: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ so if $$x$$ or $$\theta$$ is tiny, all of the stuff after the first term is pretty tiny, so you can write $\sin(x) = x \pm \text{some stuff.}$

But how much difference does that stuff make? This simulation suggests that it can actually make quite a big difference.