Calculate acceleration due to gravity of hemisphere

[Kara386]

Homework Statement

This isn't actually a coursework problem, but I can't solve it and I'd definitely be interested in the answer!

The acceleration due to gravity at the surface of a uniform, spherical asteroid is $g_0$. Half of the asteroid is destroyed in a collision, leaving only a hemisphere with the same density and radius as the original sphere. Determine the value, in terms of $g_0$, for the gravitational acceleration at the centre of the plane face of the hemisphere.

Homework Equations

The Attempt at a Solution

I'm a bit stumped because I've only ever dealt with situations where $g = \frac{GM}{r^2}$. So is that equation a kind of point mass thing? Because the hint says you can't just use that equation. The spherical symmetry that I'm guessing is usually assumed doesn't work, but how does the equation get adapted then? It probably involves calculus and maybe integrating $dr$ $d\theta$ $d\phi$ or something, but I can't quite work out exactly what you'd integrate!

Thanks for any help/hints! :)

 

[kuruman]

Find the gravitational field on the axis of a disk, then add disks of varying radii over the hemisphere.