What is the Maximum Height of a Pendulum with Given Initial Conditions?

[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.

Thank you for your help.

 

[Spinnor]

The second problem on the following page may be useful, it starts near equation 736. I think equation 743 when set to zero and you input your initial velocity your answer will pop out? See,

http://farside.ph.utexas.edu/teaching/336k/Newton/node90.html

From

https://www.google.com/webhp?hl=en#...b15419fa31a4bb&bpcl=38625945&biw=1093&bih=491

 

[99wattr89]

Thank you for that link, I haven't had a chance to go through it in detail (it's been a hectic week), but it does look promising, I just need to figure it all out!