Calculating the gravitational field due to a horizontal uniform thin disk

[AaronKnight]

Homework Statement

Show that the gravitational field due to a horizontal uniform thin disc (thickness D, radius R and density r) at a distance h vertically above the centre of the disc has magnitude

2πGρd(1-h/(R²+h²)^1/2)

A pendulum clock in the centre of a large room is observed to keep correct time. How many
seconds per year will the clock gain if the floor is covered by a 1cm thick layer of lead of density
11350kgm−3?
[Newton’s gravitational constant is G = 6.67×10−11Nm2 kg−2.]

Homework Equations

Gravitational potential, \phi=-Gdm/R
Where dm=2πRDρ.dR
Gravitational field, g= -\nabla\phi

The Attempt at a Solution

I have got to \phi=-2πDρGdR and I know I need to integrate with respect to R, then use the g= -\nabla\phi but I am unsure what my integration limits should be? I think I need to integrate between 0 and R but then I can't see how I would get the h/(R²+h²)^1/2 term?
Any hints would be very useful.

 

[AaronKnight]

In the relevant equations sections ϕ=-Gdm/R should be dϕ=-Gdm/R