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Optimal mass distribution for maximal gravitational field 
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#1
Nov1210, 12:36 PM

P: 151

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? 


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