Potential of spherical and non-spherical mass distributions?

[Schmetterling]

Homework Statement

Suppose a planet whose surface is spherical and the gravitational potential exterior to it is exactly -GM/r, like that of a point mass. Is it possible to know if the inner mass distribution is actually shperically symmetric? Can a non-spherical mass distribution produce such an external potential? If yes, give an example.

Homework Equations

Newton's shell theorems, Gauss' law

The Attempt at a Solution

"The potential out of any spherical distribution of mass is like if all the mass was in a point", but this is true for shells of uniform density. I remember that a particle inside of the sphere doesn't feel any forces regardless the mass distribution, but outside?
If we use a "gaussian surface" to enclose such a non-spherical mass distribution, Gauss' law gives the total mass, so internal distribution wouldn't be important, but, for example, inhomogeneities in Earth's density can affect nearby planetary bodies. Then? I'm confused.

 

[Simon Bridge]

The potential out of any distribution of mass is like if all the mass was in a point... at the center of mass of the distribution.

What happens to the center of mass if the distribution is not spherical?
Is the mass of spherical shells not spherically distributed?
Will a particle inside a spherical distribution of mass experience forces from non-spherical distortions?

 

[Schmetterling]

The potential out of any distribution of mass is like if all the mass was in a point... at the center of mass of the distribution.

What happens to the center of mass if the distribution is not spherical?

Its location will be displaced fron the geometric center towards where the density is greater... Oh, wait, I am confusing a spherical distribution with a homogeneous one!
If the distribution is uniform, either spherical or not, the center of mass will coincide with the geometric center, right?

Is the mass of spherical shells not spherically distributed?

It is spherically distributed, it can vary whit radius, but shells have spherical symmetry.

Will a particle inside a spherical distribution of mass experience forces from non-spherical distortions?

Inside? Uhmmm... No...

 

[Simon Bridge]

The bits you had trouble with are key to your problem.

You've figured that non-uniform distributions can be determined from outside by comparing the geometric center with the gravitational center. This raised the possibility that a uniform distribution will fit the criteria ... another characteristic of a spherical distribution of mass is that the gravitational equipotential surfaces are spheres (i.e. the grav field radiates equally in all directions). If you walk around the surface, gravity will have the same force everywhere you go. Is this true for non-spherical distributions?

Of course - if you are outside the object and you cannot tell by looking at it that it is non-spherical then it must be outwardly spherical (or blocked from view).For the particle inside a sphere of mass - see if you can work out if it feels a gravitational pull from, say, another sphere right next to it.