Find thermodynamic generalized force

[damarkk]

Homework Statement

Statistical Mechanics

Relevant Equations

Canonical Ensemble, Thermodynamic generalized forces

Assume you have a microscopic pendulum you can suppose is like quantum harmonic oscillator. If the length of pendulum has variation of $dl$, calculate the work on the pendulum and thermodynamic generalized force.
Find also the variation of mean number of extitations.


My Attempt

Firstly, I find the partition function: $Z= \sum_n e^{-\beta\epsilon_n}$, with $\epsilon_n = \hbar\omega(1/2 + n)$ and this is the result:

$Z= \frac{1}{2sinh(\beta\hbar\omega/2)}$.

In this result $\beta= \frac{1}{kT}$.

Then, I can write Helmholtz Free Energy: $F = U -TS = -kT\ln{Z}$

And of course $U = F-T\frac{\partial F}{\partial T}$


I know that $-\frac{\partial U}{\partial x} = F_x$ but if i don't know how thermodynamic variables depends on $l$ how I can compute the thermodynamic generalized force?

 

[anuttarasammyak]

$\omega$ depends on l.