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
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?