Calculating the gravitational field due to a horizontal uniform thin disk

AI Thread Summary
The discussion focuses on calculating the gravitational field produced by a horizontal uniform thin disk at a distance above its center. The gravitational field's magnitude is derived using the formula 2πGρd(1-h/(R²+h²)^(1/2)). A related problem involves determining how much time a pendulum clock gains per year when placed above a 1cm thick lead layer. The user is attempting to integrate the gravitational potential equation but is uncertain about the integration limits and how to derive the necessary terms. Clarification on the integration process and limits is sought to complete the solution.
AaronKnight
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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.
 
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In the relevant equations sections ϕ=-Gdm/R should be dϕ=-Gdm/R
 

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