Question #42552

A cylinder contains $n$ moles of an ideal monatomic gas. A piston of mass $m$ and cross-sectional area $A$ is attached to a spring of constant $k$ and a damper. The damper exerts a non-linear drag force $F_d = -\alpha v^3$ on the piston, where $v$ is its velocity and $\alpha$ is a positive constant. The gas undergoes a forced thermodynamic cycle such that the piston's state traces an ellipse in phase space given by $\frac{x^2}{x_0^2} + \frac{p^2}{p_0^2} = 1$, where $x$ is the displacement from equilibrium and $p$ is the momentum. At the state of maximum momentum, the piston's kinetic energy equals the internal energy that $n$ moles of this gas would have at a reference temperature $T_0$. Assuming the external pressure is zero, what is the net heat $\Delta Q$ absorbed by the gas per cycle?