Mass of air in cylinder column

  • Thread starter ~Sam~
  • Start date
  • #1
80
0

Homework Statement


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?



Homework Equations



Volume of a Cylinder= pi*r*h


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.
 

Answers and Replies

  • #2
585
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.
 
  • #3
80
0
How would you integrate e-z/h?
 
  • #4
585
2
By substitution for the argument of the exponential.
 
  • #5
80
0
By substitution for the argument of the exponential.

I'm not quite sure what you mean. Do you integration by substitution? Could you elaborate or give an example?
 
  • #6
585
2
Yes integration by substitution. Its a pretty elementary integration, look in a calculus book. Sorry I don't have time right now to explain.
 

Related Threads on Mass of air in cylinder column

Replies
4
Views
2K
Replies
7
Views
9K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
5
Views
7K
Replies
17
Views
6K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
3
Views
718
  • Last Post
Replies
8
Views
10K
Replies
2
Views
4K
Top