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Pressure inside the sphere

  1. Apr 22, 2009 #1
    1. The problem statement, all variables and given/known data

    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=ρo*(1-r/R), where R is radius of the sphere and is known: R=5*1010m, and r is a variable.
    • Gas has mol mass m=2.7 g/mol.
    • Mass of the whole sphere is M=3.3*1050 kg.
    • Gas is ideal.

    2. Relevant equations

    • pV=nRT
    • p=F/A


    3. 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/r2 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*r2o*(1-r/R)*dr
    Mass M=4/3*PI*r3o*(1-r/R)

    so dF=G*16/3*PI2*r3o2*(1-r/R)2*dr

    dp=dF/A
    dp=dF/4*PI*r2
    dp=4/3*PI*G*ρo2*(1-r/R)2*r*dr

    so
    p(r)=4/3*PI*G*ρo2*[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?
     
  2. jcsd
  3. Apr 23, 2009 #2
    too hard or did I do something wrong?
     
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