Gravitational potential energy problem

[dmk90]

Homework Statement

A broken spaceship is located h=10 km above the center of a large circular thin sheet of unknown dust. The sheet has a radius of R=10⁶ km and a density of σ = 7*10¹¹ kg/m². The spaceship and the dust attract each other due to the gravitational force.

a) Find the initial acceleration of the spaceship.
b) Find the acceleration of the spaceship right before the spaceship hits the dust.
c) What will the astronauts in the spaceship experience right before the spaceship hits the dust?1

Homework Equations

The Attempt at a Solution

For a), consider each particle of the dust sheet, the x component of the gravitational force cancels each other, leaving the y component. Therefore, the force of each particle mass M is
f = GmM/x² = GmM/h² * 1/cos(θ) = GmM/h² * x/h = GmMx/h³

For entire dust sheet: df = (Gmx/h³)*dM

dM = σdA = σ2πr dr

a = F/m

Combine the 3, I get this:

Which looks very off. What did I do wrong?

For b), I think I can solve it with conservation of energy.

For c), I have no clue.

Can someone help me?

 

[SteamKing]

Are you sure the density of the dust cloud is really 7*10¹¹ kg/m²? That's like the density of ground up neutron stars.

 

[gneill]

Are you sure the density of the dust cloud is really 7*10¹¹ kg/m²? That's like the density of ground up neutron stars.

Maybe the dust particles are individual neutrons :)

@dmk90: Since the sheet is so extensive compared to the distance of the puny spaceship from its surface, it will effectively look like an infinite sheet of mass density σ. I don't know about you, but when I see infinite sheets of anything producing a field I think Gauss' Law. There's a Gauss' Law for gravity! Look it up.

By the way, by analogy to the field produced by an infinite sheet of electrical charge, what can you say about the acceleration that the ship experiences over the whole distance to the surface of the sheet? Remind you of anything?