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Center of mass of infinite cylinder of air

  1. Jul 29, 2012 #1
    1. The problem statement, all variables and given/known data

    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:

    [itex]\frac{d}{dz}e^{-az}=-ae^{-az}[/itex]

    [itex]\frac{d}{dz}((az+1)e^{-az})=-a^{2}ze^{-az}[/itex]

    2. Relevant equations

    Center of mass = [itex]\frac{1}{M}\sum{m_{i}x_{i}}=\frac{1}{M}\int{xdm}[/itex]

    3. 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?
     
  2. jcsd
  3. Jul 29, 2012 #2

    ehild

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

    ehild
     
  4. Jul 29, 2012 #3
    Thanks :) I did the calculation and got 1/a, is that correct?
     
  5. Jul 30, 2012 #4

    ehild

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    It is correct. Well done!

    ehild
     
  6. Jul 30, 2012 #5
    Thanks again!
     
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