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Homework Help: Mass of air in cylinder column

  1. Jan 18, 2010 #1
    1. The problem statement, all variables and given/known data
    The density of air in the lower atmosphere decreases exponentially with height: ρ = ρ0e−z/H
    where ρ0 = 1.3 kg/m3 and H = 10 km. What is the mass of air in a cylindrical column of cross-sectional area 1 m2 and height z, as a function of z? How much mass is contained in such a column 1.0 km high?



    2. Relevant equations

    Volume of a Cylinder= pi*r*h


    3. The attempt at a solution

    I think I need to integrate from 0 to z to get the formula for mass. However, I'm not sure what the equation is that I need to integrate. I was thinking it might be something like intg[0,z] pi*r2e-z/Hρ0dz. Still if it that was, I see an issue with integrating e-z/H. I haven't gone to the second part because I need to know the first.
     
  2. jcsd
  3. Jan 18, 2010 #2
    Looks good to me.

    [tex]m=\int^{z}_{0}dm=\int^{z}_{0}\pi r^{2}e^{-z/H}\rho_{0}dz[/tex]

    of course a mathematician will tell you that integrating for 0 to z is bad notation but its unlikely to give you the wrong answer.
     
  4. Jan 18, 2010 #3
    How would you integrate e-z/h?
     
  5. Jan 18, 2010 #4
    By substitution for the argument of the exponential.
     
  6. Jan 18, 2010 #5
    I'm not quite sure what you mean. Do you integration by substitution? Could you elaborate or give an example?
     
  7. Jan 19, 2010 #6
    Yes integration by substitution. Its a pretty elementary integration, look in a calculus book. Sorry I don't have time right now to explain.
     
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