What is the Mass of Air in a Cylindrical Column as a Function of Height?

[~Sam~]

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.

 

[Phyisab****]

Looks good to me.

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

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.

 

[~Sam~]

How would you integrate e^-z/h?

 

[Phyisab****]

By substitution for the argument of the exponential.

 

[~Sam~]

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?

 

[Phyisab****]

Yes integration by substitution. Its a pretty elementary integration, look in a calculus book. Sorry I don't have time right now to explain.