Pendulum moving with constant veloctiy

[kraigandrews]

Homework Statement

Consider a pendulum that consists of a mass M hanging from a massless string of length r. The string is being pulled upward at constant velocity through a tiny hole in the ceiling, so the length of the pendulum is given by r = r0 - alpha*t, where alpha is a constant. Let theta be the angle of the string with respect to vertical. Assume that the motion is in a vertical plane, but do not make small angle approximations.
(a) Find the Lagrange equation of motion.

(b) Find the Hamilton equations of motion.

(c) Find the equation of motion in the approximation that the angle theta is small. (You do not have to solve the equation of motion -- its solution will be discussed in lecture.)

Homework Equations

L = T-U
r = $r_{o}$ - $\alpha$t

The Attempt at a Solution

So pretty much I just need help with getting started i.e. making sure that I have the correct equations for T and U.

I believe T would just be the Kinetic Energy of the pendulum and the rate at which the rope is moving to give:

T =1/2M($\dot{r^{2}}$ + $r^{2}$$\dot{\vartheta^{2}}$) + 1/2M$\dot{r^{2}}$
where the second r dot term is for the changing radius (however, this doesn't seem correct)

U = mgrsin$\vartheta$

then obviously L is just the difference of T and U and then carry out the necessary derivative to obtain the Lagrange equations and somewhat similar for the Hamiltonian equations. My main problem is making sure that the initial equations are correct, which they don't seem to be.

 

[BigJDubs]

Then for the approximation that the angle is small, I believe the equation of motion would just be \ddot{\vartheta} + \frac{g}{r_{o}}\vartheta = 0