Optimal mass distribution for maximal gravitational field

  1. 1. The problem statement, all variables and given/known data

    Suppose that one is given a lump of clay of total mass M and constant density. Let P denote a particular point in space. In what way should one shape and position the clay so that the gravitational field in point P is maximum possible? It is assumed that the clay stays in one piece during the shaping.

    2. Relevant equations

    Newton's law of gravity
    Variational principle (?)

    3. The attempt at a solution

    I don't know if this problem has an obvious and trivial solution, but I'm thinking along the lines of variational principle. Let P be the origin of coordinates with z axis pointing in the direction of the gravitational field. Then we have the following equations

    [tex]F=G \rho\int_{\mathcal{V}}dV\frac{\cos \theta}{r^2}[/tex]

    [tex]M=\rho V[/tex]

    Can I maximize the force by varying the volume of integration using the second equation as a constraint?
  2. jcsd
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