# Falling (toppling) rigid tower (uniform rod)

by zaphat
Tags: falling, rigid, toppling, tower, uniform
 Thanks P: 5,687 Your tower is a pendulum, whose initial angle from "pointing down" is 180 degree. Starting from the pendulum's differential equation, $$x'' + a \sin x = 0 \\ x''x' + a (\sin x)x' = 0 \\ \frac {x'^2 - {x'}_0^2} {2} - a(\cos x - \cos x_0) = 0$$ ## x_0 = \pi ## (the pendulum is upward from "pointing down"), so $$\\ x' = \sqrt {{x'}_0^2 + 2a(\cos x + 1)} \\ \int_{\pi}^x ({x'}_0^2 + 2a(\cos x + 1))^{-1/2} dx = t$$ The latter integral, as far as I can tell, does not exist in the closed form, but it can be tabulated between ## \pi ## (upward) and ## \pi/2 ## (toppled), which will give you the trajectory you want. You will need ## {x'}_0 ## which is the initial angular velocity, and you will need ## a ##, which is ## \frac {3g} {2L} ## for a uniform rod, ## L ## being the length.