Question #42500

A vertical cylinder of cross-sectional area $A = 0.1 \, \text{m}^2$ is completely submerged in a fluid bath. The top of the cylinder is fixed at a depth $H_0 = 2.0 \, \text{m}$ below the fluid surface. The bottom of the cylinder is sealed by a movable piston of mass $M$ and volume $V_p = 0.05 \, \text{m}^3$. An external mechanism moves the piston to perform a cycle.

The cycle consists of four steps:

1. The piston is slowly pushed down, increasing the cylinder's internal volume from $V_1 = 0.1 \, \text{m}^3$ to $V_2 = 0.3 \, \text{m}^3$, while the fluid is maintained at $T_H = 350 \, \text{K}$.
2. The piston is held in place while the fluid is cooled to $T_C = 300 \, \text{K}$.
3. The piston is slowly pulled up, decreasing the volume from $V_2$ to $V_1$, while the fluid is maintained at $T_C$.
4. The piston is held in place while the fluid is heated back to $T_H$.

The fluid density is $\rho(300 \, \text{K}) = 1000 \, \text{kg/m}^3$ and $\rho(350 \, \text{K}) = 960 \, \text{kg/m}^3$. Calculate the magnitude of the net work done BY THE FLUID on the piston during one complete cycle. (Assume $g = 10 \, \text{m/s}^2$ and $P_{\text{atm}} = 10^5 \, \text{Pa}$)