# Maximum height of a pendulum

1. Nov 14, 2012

### 99wattr89

I'm trying to solve this problem via Lagrangian mechanics, but I'm having trouble correctly formulating the problem, I'm hoping one of you kindly people can show me where I'm going wrong.

A pendulum bob of mass m hangs in the equilibrium position from a light, inextensible string of length l. It is given a horizontal velocity of (3gl)1/2. Find the vertical displacement of the bob when the string becomes slack.

So the bob has one degree of freedom, theta, the angle at which it hangs, and to find the height when the string goes slack we want the maximum height the bob can reach, which can be found if the maximum value of theta is calculated.

So L = T - U = (1/2)m(lθ.)2 - mgl(1-cosθ)

Is that correct so far?

If it is, then the problem I run into is that the Euler-Lagrange equation for this motion doesn't seem to be a differential I can solve, since I have θ.. +(g/l)sinθ = 0, and that doesn't seem to be solvable with the current methods I know.