# Optimal mass distribution for maximal gravitational field

by Heirot
Tags: distribution, field, gravitational, mass, maximal, optimal
 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 $$F=G \rho\int_{\mathcal{V}}dV\frac{\cos \theta}{r^2}$$ $$M=\rho V$$ Can I maximize the force by varying the volume of integration using the second equation as a constraint?

 Related Discussions Classical Physics 11 Special & General Relativity 5 High Energy, Nuclear, Particle Physics 4 Classical Physics 4