\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 particle 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.