Center of mass of infinite cylinder of air

[phosgene]

Homework Statement

The density of air at height z above the Earth’s surface is proportional to e^(−az) , where a is a constant > 0. Find the centre of mass of an infinite cylinder of air above a small flat area on the Earth’s surface. Hint : Consider line density and the identities:

$\frac{d}{dz}e^{-az}=-ae^{-az}$

$\frac{d}{dz}((az+1)e^{-az})=-a^{2}ze^{-az}$

Homework Equations

Center of mass = $\frac{1}{M}\sum{m_{i}x_{i}}=\frac{1}{M}\int{xdm}$

The Attempt at a Solution

I have no idea how to get started because I don't know how to use the e^(-az) expression. Could I just write that the density of air at height z = be^(-az) where b is some constant of proportionality? Then I think I would try to find M and dm/dx, plug it into the center of mass equation and integrate from 0 to infinity?

 

[ehild]

Homework Statement

The density of air at height z above the Earth’s surface is proportional to e^(−az) , where a is a constant > 0. Find the centre of mass of an infinite cylinder of air above a small flat area on the Earth’s surface. Hint : Consider line density and the identities:

$\frac{d}{dz}e^{-az}=-ae^{-az}$

$\frac{d}{dz}((az+1)e^{-az})=-a^{2}ze^{-az}$

Homework Equations

Center of mass = $\frac{1}{M}\sum{m_{i}x_{i}}=\frac{1}{M}\int{xdm}$

The Attempt at a Solution

I have no idea how to get started because I don't know how to use the e^(-az) expression. Could I just write that the density of air at height z = be^(-az) where b is some constant of proportionality? Then I think I would try to find M and dm/dx, plug it into the center of mass equation and integrate from 0 to infinity?

Yes, taking into account that dm=ρ(z)dz, and you integrate with respect to z.

ehild

 

[phosgene]

Thanks :) I did the calculation and got 1/a, is that correct?

 

[ehild]

Thanks :) I did the calculation and got 1/a, is that correct?

It is correct. Well done!

ehild

 

[phosgene]

Thanks again!