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Specific Heat of a nonlinear, temperature dependent spring

  1. Sep 11, 2014 #1
    1. The problem statement, all variables and given/known data
    A nonlinear spring has a temperature dependent force law,

    [itex]F = -\frac{K}{T}(L-L_o)^3[/itex]

    At a temperature [itex]T = T_o[/itex] and length [itex]L = L_o[/itex] the specific heat at a constant length is [itex]C_L = C_o[/itex]. What is the specific heat at [itex]T = T_o[/itex] when the spring is stretched to length [itex]2L_o[/itex]?

    2. Relevant equations



    3. The attempt at a solution
    I am really not sure where to start. Am I supposed to use the general dU=dQ-dW equation and somehow work the spring into that using the force equation and then solve for a specific heat equation? Any advice would be great. Thanks.
     
  2. jcsd
  3. Sep 11, 2014 #2
    Since the force is the negative gradien of its potential energy, in a one dimensional case as this we can write,

    [itex]F=-\frac{dU}{dL}[/itex] or

    [itex]U=-\int FdL[/itex]

    Once obtained the potential energy, use the defenition of the heat capacity at constant length,

    [itex]C_L=(\frac{dU}{dT})_L[/itex]

    With the given information you should be able to express your result in terms of [itex]C_o[/itex].
     
  4. Sep 12, 2014 #3
    If you do that doesn't [itex]C_L=(\frac{dU}{dT})_L[/itex] come out to be [itex]C_L = -\frac{K}{4T^2}(L-L_o)^4[/itex]? How can you have a negative specific heat?
     
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