Optimal mass distribution for maximal gravitational field

[Heirot]

Homework Statement

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.

Homework Equations

Newton's law of gravity
Variational principle (?)

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?

 

[mark726]

Dear fellow scientist,

Thank you for bringing up this interesting problem. Your approach using the variational principle is certainly a valid one. However, I believe there may be a simpler solution to this problem.

Since we are dealing with a lump of clay with constant density, we can think of it as a continuous distribution of mass. This means that the gravitational field at point P will be directly proportional to the amount of mass within a certain distance from P.

To maximize the gravitational field at P, we need to maximize the amount of mass within this distance. Therefore, the best way to shape and position the clay would be to mold it into a spherical shape with P at its center. This way, all parts of the clay will be equidistant from P and contribute equally to the gravitational field.

I hope this helps. Let me know if you have any further thoughts or questions.