Question

What is the position of a pendulum with respect to time?

Answer 1

It's clear that the acceleration in terms of theta is given by \[
g\sin(\theta)
\]But that doesn't really help unless we know \(\theta(t)\).

Answer 2

And if \(\theta(t)\) is a tricky function, then integrating will be tricky as well.

Answer 3

We can say: \[
\theta(t) = \arctan\left(\frac{y(t)}{x(t)}\right)
\]

Answer 4

I particularly want to find \(\mathbf s(t) =\langle x(t),y(t)\rangle\)

Answer 5

We can say \(\mathbf s(0) = \langle L, 0\rangle \)

Answer 6

We can say acceleration of gravity is \(g\), length is \(L\) and mass is \(m\).
Also: \(\theta(0) = 0\).

Answer 7

My feeling is that we try to find some solution where we only swing by \(-\frac{\pi}{2} /n\). We approximate for this triangle, then we let \(n\to \infty\)... or something.