What is the optimal ratio of R/H to minimize the energy for a fixed volume?

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Homework Help Overview

The discussion revolves around finding the optimal ratio of radius (R) to height (H) that minimizes the energy of a particle confined within a right circular cylinder of fixed volume. The ground state energy of the particle is expressed in terms of R and H.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster expresses confusion about how to approach the problem of minimizing energy in relation to the dimensions of the cylinder. They consider rewriting the dimensions in terms of other variables.
  • Another participant suggests looking at a similar problem regarding maximizing volume for a cylinder, questioning whether the constants in the energy formula should be incorporated early or later in the process.
  • Subsequent posts indicate attempts to derive a relationship between R and H, with varying results.

Discussion Status

The discussion is ongoing, with participants exploring different approaches and interpretations of the problem. There is no explicit consensus yet, but attempts to relate the problem to known formulas are being made.

Contextual Notes

Participants note the constraint of fixed volume, which impacts their reasoning about the relationship between R and H. There is uncertainty regarding the application of constants in the energy equation.

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Homework Statement


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.

Homework 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

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:
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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?
 
I attempted to do a switcheroo and I got 1.2R = H
 
whoops I forgot that volume is fixed.
brb

ok I got 1.85R = H
 
Last edited:

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