Gravitational Potential Calculation

[21boy]

Homework Statement

A meteorite impacts a flat-tabular Earth (with a density of 6gm/cm^3). During transit, the meteorite is observed to have a density of 8 gm/cm^3 and to be a perfect sphere of radius 1km. The meteorite penetrates the flat Earth to an unknown depth to center mass, z_0; z_0 > 1km. The meteorite is undeformed during emplacement and the Earth that is displaced magically vanishes and the hole above the meteorite is filled again so that the surface is flat and the density uniform everywhere except for the meteorite. Use relevant equations relating Gauss' Law to this geometry to derive the gravitational potential of the meteorite as a function of position along the Earth's (slab's) surface. Let x=0 be over the center of the mass of the meteorite.

*Positive z is pointing towards the Earth's surface
*z=0 at the surface of the slab
*The thickness of the Earth is t

Homework Equations

Δρ=ρ1-ρ => 8-6=2

g_z = 2.79E-10 (Δρ r_sphere h)/(3 (x^2+h^2)^(3/2) To find gravity in g_z direction

V= ∫g_z d? to find potential of gravity.

Question is I'm not sure of my bounds for the integral and what I should be integrating in respect to. Also, for calculating gravity, should I use x=0?

 

[Simon Bridge]

Question is I'm not sure of my bounds for the integral and what I should be integrating in respect to. Also, for calculating gravity, should I use x=0?

The answer depends on how you are using them - what is your reasoning?

 

[21boy]

Well I'm trying to calculate gravitational potential over the x-surface above a buried object. I would think that I would derive my integral in terms of dx from x=0(area directly above meteorite) to some positive value and do the same for a symmetric negative value. I'd define this area as my equipotential surface that's being affected by the meteorite.

Does that answer the problem statement?

 

[Simon Bridge]

OK - so the x-y plane is the surface, the com of the meteorite is at $(x,y,z)=(0,0,-z_0)$ ... The potential of the flat-Earth is $gz:z\geq 0$ but you want the potential of the meteorite.

You are asked to use Gausses Law - or "the relevant equations relating Gauss' Law to this geometry" - find the potential.

So what is the geometry in question?
What shape will the equipotential surfaces make in the plane?