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

[grandpa2390]

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...

 

[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?

 

[grandpa2390]

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

 

[grandpa2390]

whoops I forgot that volume is fixed.
brb

ok I got 1.85R = H

 

[HalfLostAndSearching]

The optimal ratio of R/H to minimize the energy for a fixed volume can be found by using the principles of variational calculus and the Euler-Lagrange equation. This technique is commonly used in physics to find the extremum of a functional, which in this case is the ground state energy.

To begin, we need to understand the physical meaning of the ground state energy. The ground state energy is the lowest possible energy that a particle can have in a given system. In this case, the system is a right circular cylinder with a fixed volume. The height and radius of the cylinder will determine the possible energy states of the particle.

Using the given equations for the ground state energy, we can rewrite the functional as:

E = h^2/[2mR^2(2.4048)^2] + h^2/[2mH^2(pi)^2]

Now, we can define the functional F as:

F = E - λ(V - πR^2H)

where λ is a Lagrange multiplier and V is the fixed volume of the cylinder.

Next, we need to find the Euler-Lagrange equation for F, which is given by:

∂F/∂R - d/dx(∂F/∂R') + ∂F/∂H - d/dy(∂F/∂H') = 0

where R' and H' are the derivatives of R and H with respect to x and y, respectively.

Solving this equation will give us the optimal values of R and H that will minimize the energy for a fixed volume. Once we have these values, we can then find the ratio R/H that will give us the minimum energy.

Note that this is a general approach and the exact steps may vary depending on the specific problem and equations given. It is important to carefully define the functional and use the appropriate Euler-Lagrange equation. I hope this helps in understanding how to approach this problem.