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## Homework Statement

find the acceleration due to gravity at the centre of a solid hemisphere.

## Homework Equations

##F=\frac{GMm}{r^2}##

## The Attempt at a Solution

i decided to go for cylindrical coordinayes (which is way beyond my syllabus). I did some research though.

let me take a point P(r,θ,z) inside the sphere amd an elemental volume dV at P. This P exerts a force dF. But there is also a point Q(r,θ+180°,z) which cancells out the horizontal component of dF. let the line OP make an angle φ with the Z axis. (assuming the hemisphere lies on the xy plane with centre at O)

##dg=\frac{Gdm}{r^2+z^2}cos\phi##

##dg=\frac{G\rho dV}{{(r^2+z^2)}^{3/2}}##

now dV=dz.dr.rdθ

**Is this expresion for dV true for all cases? how do you get that expression for dV?**

(i found that expression on some video).

(i found that expression on some video).

now i have to integrate the expression.

so $$g=\int_{\theta=0}^{2\pi}\int_{r=0}^R\int_{z=0}^?f(r,\theta,z)dz.dr.rd\theta$$

**im finding it difficult to find upper limit for z.**

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