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Homework Help: Thermodynamics and simple harmonic motion

  1. Dec 5, 2009 #1
    1. The problem statement, all variables and given/known data
    A horizontal piston-cylinder system containing n mole of ideal gas is surrounded by air at temperature [itex]T_{0}[/itex] and pressure [itex]P_{0}[/itex]. If the piston is displaced slightly from equilibrium, show that it executes simple harmonic motion with angular frequency [itex]\omega=\frac{AP_{0}}{\sqrt{MnRT_{0}}}[/itex], where A and M are the piston area and mass, respectively. Assume the gas temperature remains constant.

    2. Relevant equations
    x=amount by which piston moves
    L=length of cylinder
    F = force on piston
    3. The attempt at a solution
    Popping all this into the mix gives
    If I get rid of the [itex]P_{0}Ax}[/itex] term then we arrive at the correct answer but not sure why one should throw away this term

  2. jcsd
  3. Dec 6, 2009 #2


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    Homework Helper

    The denominator is equal to [tex]
    nRT_0 (1-x/L)

    It is assumed that the piston is only slightly displaced from equilibrium, so that x/L<<1. That means that its second or higher power can be neglected with respect to it.
    We often apply this method in Physics when working with small quantities: use Taylor expansion and keep linear terms.
    Your expression of force can be written in terms of (x/L),


    Expanding the the fraction with respect to x/L :



    Omitting all terms but linear, we get:



  4. Dec 6, 2009 #3
    thanks for that
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