# Lagrangian problem invovling velocity

1. Dec 11, 2011

### rabble88

1. The problem statement, all variables and given/known data

A particle of mass m is placed on the inside of a smooth paraboloid of revolution whose equation is
cz = x2 + y2 , where c is a constant, at a point P which is at a height H above the horizontal x-y plane.
Assuming that the particle starts from rest (a) find the speed with which it reaches the vertex O, (b) express
the time τ taken to travel from that height to the vertex in the form of an integral. Do not solve it. It leads to
an elliptic integral which cannot be solved analytically.

P.S. I am new to the forum and i posted the same post in the Classical Physics section titled Lagrangian. It contains a picture of the paraboloid

2. Relevant equations

L = T-V x = rcosθ
y = rsinθ

3. The attempt at a solution

I cannot figure out what should be my first step. I have a basic idea on how to do this problem.

I know that i have to use L = T - V. Should i change this to polar coordinates and then take the integral of T and V from h to 0?

2. Dec 11, 2011

### diazona

What is always the first step in any Lagrangian mechanics problem? Hint: what quantity do you have to write an expression for? (if it's not already given)

3. Dec 11, 2011

### JHamm

I posted a response in that thread but it seems to have been removed so I'll re post it here.
For part (a) I would just use conservation of energy, no need for any equations of motion.
For part (b) I would find the period of oscillation of the system and divide by 4.

Remember that all of the motion will take place in a plane.
:)

4. Dec 13, 2011

### rabble88

im assuming that delta and psi are changing since it is coming down the paraboloid and the paraboiloid is spining due to revolution. Should i use polar coordinates or cartesian form?

5. Dec 15, 2011

### diazona

It's spinning? You didn't say that before... is there any friction? If not, it doesn't matter that the paraboloid is spinning, and if there is, it becomes a more complicated problem.