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Spring and mass with varying force -- what is the change in temperature?

  1. Jan 9, 2019 #1
    1. The problem statement, all variables and given/known data
    A system consists of a mass m moving in one dimension and attached to a rigid wall by a spring having stiffness constant ##K##, as shown. The mass is subjected to a constant force ##F##, and is in equilibrium with the surroundings at a temperature ##T##. The partition function at constant ##T## and ##F## is given by

    $$Z = \frac{e^{\frac{\beta F^{2}}{2K}}}{\hbar \beta \sqrt{(K/m)}}.$$

    If the system is insulated from the surroundings, and the force is slowly and reversibly decreased from ##F## to zero, what will be the new temperature (in terms of the initial temperature ##T##)? What will be the final temperature if the force is abruptly reduced to zero?

    2. Relevant equations
    ##\Delta S = 0##
    ##\ln{(Z)} = \frac{\beta F^{2}}{2K}-\ln{(\beta)}-\ln{(\hbar \sqrt{(K/m)})}##
    ##U = -\frac{F^{2}}{2K}+k_{b}T##

    3. The attempt at a solution
    Using the equation ##S = k_{b}ln{(Z)}+\frac{U}{T},## where ##U = -\frac{\partial \ln{(Z)}}{\partial \beta},## I get that ##S = k_{b}(1-\ln{(\beta \hbar \sqrt{(K/m)})}).## This equation is independent of force. Since we are dealing with an adiabatic reversible situation we know ##\Delta S = 0##, thus the temperature doesn't change? Does this sound correct?

    As for the part about the final temperature if he force is abruptly reduced to zero, I am not sure where to begin. Any suggestions?

    Thank you very much!
     
  2. jcsd
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