Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Two rings and soap water

  1. Nov 18, 2007 #1

    LMZ

    User Avatar

    Two rings with radius R and r are let down in the soap water. Between them pellicle appeared (like in image).
    Image: here all are symmetric ;)

    problem: need to find y=f(x)

    what i think:volume of figure is minim. maybe express volume via needed function, then get minimum of volume, and find this function.
     

    Attached Files:

  2. jcsd
  3. Nov 18, 2007 #2

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    Soap films try to minimize surface area, not volume.
     
  4. Nov 18, 2007 #3
    Yep, area ~ surface tension.
    If I'm not mistaken, for r=R your solution should look like a hyperbolic cosine.
     
  5. Nov 19, 2007 #4

    LMZ

    User Avatar

    please be more concrete
     
  6. Nov 19, 2007 #5
    Sure, but about which part?

    Nature "seeks" to minimize the energy of a system. In your case, your system's only energy (that changes as the surface changes shape) is the surface tension. I'm not an expert on "surface"-y things, but in this problem the surface tension energy is simply proportional to the area, A:

    [tex]E = C \int_{\mathrm{sample}} dA[/tex]

    Since your problem has rotational symmetry about (say) the x-axis, we can simplify this as follows: let y=y(x) be the function describing a cross section of your sample (by a plane passing through the axis of rotational symmetry, the x-axis). Then, given x, the total area contributed to the energy term is simply:

    [tex]\Delta E(x) = 2 \pi y(x) ds[/tex]

    where ds is the "length element" of the curve y(x). It is given by:

    [tex]ds = \sqrt{dx^2 + dy^2} = \left( 1+ \left( \frac{dy}{dx} \right)^2\right)^{1/2} dx[/tex]

    and so:

    [tex]E \propto \int_{x_1}^{x_2} y(x) \left( 1+ (y')^2\right)^{1/2} dx [/tex]

    Now all that is left to do is apply Euler-Lagrange's equations using the appropriate boundary conditions (y(start) = r, y(end)=R) and you're done!

    --------
    Assaf
    Physically Incorrect
     
  7. Nov 21, 2007 #6

    LMZ

    User Avatar

    thanks a lot!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Two rings and soap water
Loading...