Gravitation Problem: Force of Sphere on Sheet

[awri]

Homework Statement

If a uniform sphere of mass M and radius R, is height h above an infinite sheet of uniform density $$\rho$$$$_{s}$$, what is the gravitational force of the sphere, on the sheet.

Homework Equations

F=$$\frac{GMm}{R^{2}}$$; $$\Phi$$=$$\frac{GM}{R}$$; $$\nabla$$$$\bullet$$g=4$$\pi$$G$$\rho$$; U=m$$\Phi$$; F=-$$\nabla$$U

The Attempt at a Solution

My professor advised me to find the force of the sheet on the sphere since that force would be equal to the force of the sphere on the sheet. So I drew a gaussian cylinder around my "sheet" and attempted to calculate g by saying that $$\nabla$$$$\bullet$$g = $$\frac{dg}{dz}$$ since all relavent field lines were in the z direction. All I think I need to know is whether or not that was the correct assumption. I don't think it is because it yeilds an answer that looks like 4$$\pi$$G$$\rho$$z. And the z is an issue. the answer in the back of the book is 2$$\pi$$GM$$\rho$$. Any help would be much appreciated.

 

[Redbelly98]

There is another way to use the gaussian cylinder. There's an equation that relates the surface integral of g·dA over the cylinder, to the mass within the cylinder. I think that will be useful here.