\sect{Oscillations of a piston in a cylinder filled with gas}

Consider a vertically aligned cylinder whose basis has an area~$\mathsf{A}$. 
A piston that has mass~$M$ is pushed from above. 
The piston is held by a spring that has an elastic constant~$K$. 
If the cylinder is empty the piston is down at zero height (${x=0}$).
The cylinder is filled with~$N$ gas particles. 
Each particle has mass $\mass$ and the temperature is~$T$.
Consequently the the piston goes up a distance~$x$, 
such that the gas occupies a volume~$\mathsf{A}x$. \\
Consider the following 3~cases:

\begin{enumerate}
\item[(a)] The temperature is high, such that Boltzmann approximation can be applied. 
\item[(b)] The particles are condensed Bosons, $T$ is lower than the condensation temperature.  
\item[(c)] The particles are spinless Fermions, and the temperature is zero. 
\end{enumerate}

Answer the following questions, relating to each case separately.

\begin{enumerate}
\item What is the equilibrium position $x_{eq}$ of the piston?
\item What is the frequency $\omega$ of small oscillations?
\item Plot schematic drawing of $\omega$ versus $T$.
\end{enumerate}

Express answers using $\mathsf{A}$, $M$, $K$, $N$, $T$.
The schematic drawing is required to be be clearly displayed.

