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Extremizing the energy?

  1. Nov 2, 2014 #1

    grandpa2390

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    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. jcsd
  3. Nov 2, 2014 #2

    grandpa2390

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    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?
     
  4. Nov 2, 2014 #3

    grandpa2390

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    I attempted to do a switcheroo and I got 1.2R = H
     
  5. Nov 2, 2014 #4

    grandpa2390

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    whoops I forgot that volume is fixed.
    brb

    ok I got 1.85R = H
     
    Last edited: Nov 2, 2014
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