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

1. Jan 9, 2019

### TroyElliott

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!