Question #42568

A monatomic ideal gas is confined in a cylindrical chamber by a movable vane. The vane is connected to a torsional spring of stiffness $\kappa$ (relaxed at $\theta=0$). The gas volume $V$ is related to the vane angle $\theta$ by $V = c\theta$, where $c$ is a constant. The vane also experiences a constant frictional torque $\tau_f = \frac{1}{2}\kappa\theta_0$ that always opposes its motion. The system undergoes a quasi-static thermodynamic cycle starting from an initial angle $\theta_0$:

1. The gas is heated at constant volume until the vane is on the verge of moving outward.
2. The gas is further heated, expanding the volume to $2V_0$ (where $V_0 = c\theta_0$).
3. The gas is cooled at constant volume until the vane is on the verge of moving inward.
4. The gas is further cooled, compressing the volume back to $V_0$.

What is the thermodynamic efficiency of this cycle?