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## Homework Statement

I have a sphere filled with unknown gas. I have to estimate the pressure inside of it.

- The tricky part is that density of the sphere changes as we go deeper inside of it with the given formula:

ρ=ρ_{o}*(1-r/R), where R is radius of the sphere and is known: R=5*10^{10}m, and r is a variable.

- Gas has mol mass m=2.7 g/mol.

- Mass of the whole sphere is M=3.3*10
^{50}kg.

- Gas is ideal.

## Homework Equations

- pV=nRT

- p=F/A

## The Attempt at a Solution

The force of gravity of an infinitesimal layer of thickness dr inside the sphere at some radius r caused by the inner sphere is:

dF=GmM/r

^{2}where m is the mass of the infinitesimal layer and M is the mass of inner part of the sphere.

Mass m=V*ρ=4*PI*r

^{2}*ρ

_{o}*(1-r/R)*dr

Mass M=4/3*PI*r

^{3}*ρ

_{o}*(1-r/R)

so dF=G*16/3*PI

^{2}*r

^{3}*ρ

_{o}

^{2}*(1-r/R)

^{2}*dr

dp=dF/A

dp=dF/4*PI*r

^{2}

dp=4/3*PI*G*ρ

_{o}

^{2}*(1-r/R)

^{2}*r*dr

so

p(r)=4/3*PI*G*ρ

_{o}

^{2}*[tex]\int[/tex](1-r/R)

^{2}*r*dr from r to R

After making the calculations I will put p(r=0) and get the pressure inside the sphere. Is my method good?