Results 1 to 2 of 2

Math Help - Optimal mass distribution for maximal gravitational field

  1. #1
    Junior Member
    Joined
    Jul 2010
    Posts
    52

    Optimal mass distribution for maximal gravitational field

    Let's say we have some finite mass M of constant density that can be shaped at will. How should we shape it and where do we place it for the gravitational field in some fixed point P to be maximal?

    My solution:
    Let's say P is the origin of the spherical coordinate system with the force pointing in the pozitive direction of the z-axis.

    We must maximize the integral \int_{\mathcal{V}}{\frac{\cos\theta}{r^2}dm}, where dm is an infinitesimal element of mass with the integral summing all over the shape of the volume V. If we express dm in spherical coordinates and disregard any constants, we have \int_{\mathcal{V}}{\cos\theta \sin\theta dr d\theta d\phi} for the force at the origin.

    The finite mass constraint gives \int_{\mathcal{V}}{r^2 \sin\theta dr d\theta d\phi}=const.

    How should I proceed?
    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jul 2010
    Posts
    52
    Nevermind, I solved it: r = sqrt(cos(theta))
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: July 12th 2011, 02:08 PM
  2. Poissons equation, gravitational field.
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: December 12th 2010, 12:54 PM
  3. Statistical Analysis of a Mass Distribution
    Posted in the Statistics Forum
    Replies: 8
    Last Post: March 28th 2010, 09:08 AM
  4. Maximal Ideal Problem within a Field
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 14th 2008, 09:33 AM
  5. Maximal Extension Field
    Posted in the Advanced Algebra Forum
    Replies: 12
    Last Post: August 17th 2006, 09:49 PM

/mathhelpforum @mathhelpforum