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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
    PV=nRT
    F=Ma
    P=F/A
    x=amount by which piston moves
    L=length of cylinder
    F = force on piston
    3. The attempt at a solution
    [tex]V_{0}=AL[/tex]
    [tex]V=A(L-x)[/tex]
    [tex]F_{net}=A(P_{0}-P)[/tex]
    [tex]L=\frac{nRT_{0}}{P_{0}A}[/tex]
    [tex]P=\frac{ALP_{0}}{V}[/tex]
    Popping all this into the mix gives
    [tex]F_{net}=\frac{-P_{0}^2A^2x}{nRT_{0}-P_{0}Ax}[/tex]
    [tex]M\frac{d^2x}{dt^2}=\frac{-P_{0}^2A^2x}{nRT_{0}-P_{0}Ax}[/tex]
    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

    Thanks
     
  2. jcsd
  3. Dec 6, 2009 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member

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

    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),

    [tex]F=AP_0(1-\frac{1}{1-x/L})
    [/tex]

    Expanding the the fraction with respect to x/L :

    [tex]F=AP_0(1-(1+x/L+(x/L)^2+(x/L)^3+....))

    [/tex]

    Omitting all terms but linear, we get:

    [tex]F=-AP_0(x/L)

    [/tex]

    ehild
     
  4. Dec 6, 2009 #3
    ok
    thanks for that
     
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