angular movement

Not knowing much about pendulums, I assumed that the answer was simple and easily found using Google. The truth is though that it is not so simple. The motion of an undamped pendulum is expressed by this differential equation:

    d<sup>2</sup>θ/dt<sup>2</sup> + g/L sin θ = 0 

This 2nd order non-linear equation cannot be solved for θ, so that means if you want to find θ at a given time, you must use numerical integration.

However, for small angles, you can substitute sin θ = θ and the result is

    θ = θ<sub>0</sub> cos ω<sub>0</sub>t, where ω<sub>0</sub> = sqrt( g/L )