# Extremizing the energy?

1. Nov 2, 2014

### grandpa2390

1. The problem statement, all variables and given/known data
I can not find anything online or in my textbooks to help me with extremizing. please help...

I need to find a ratio of R/H that will minimize the energy for a fixed volume
I am given the ground state energy of a particle that is inside a right circular cylinder. and has a height H and radius R.

2. Relevant equations
ground state energy http://www.wolframalpha.com/input/?i=E = C_1/R^2 + C_2/H^2
C_1 is http://www.wolframalpha.com/input/?i=C_1 = h^2 / (2m) * (2.4048)^2
C_2 is http://www.wolframalpha.com/input/?i=C_2 = h^2 / (2m) * pi^2
3. The attempt at a solution

I don't understand the problem or how to begin. I have never done anything with energy of particles in cylinders. I don't know how ground state energy is affected by the height and radius of the container. I am sure everything I need is right in front of me, but I don't understand. I am not looking for someone to just give me the answer, but to walk me through it please. Please help!

I feel like I should start by rewriting h and r in terms of x and y
h = ds = sqrt (1 + x'^2)dy
and r...

Last edited: Nov 2, 2014
2. Nov 2, 2014

### grandpa2390

I think I have something. I found a similar problem that find the ratio R to H that maximizes volume for a right circular cylinder... maybe I can do that and just swap the formula E for the volume formula?

but would I plug in C1 and C2 at the beginning, or can I just wait till the end? what would i do with that?

3. Nov 2, 2014

### grandpa2390

I attempted to do a switcheroo and I got 1.2R = H

4. Nov 2, 2014

### grandpa2390

whoops I forgot that volume is fixed.
brb

ok I got 1.85R = H

Last edited: Nov 2, 2014