$(23\%)$

A frictionless piston has a mass of $200l{b}_m.$ It starts at the top of a $20$ ft high cylinder from which point it is lowered. The air in the cylinder is at $80F$ and remains at that temperature as the piston is lowered. None of the air leaks past the piston during its descent. The cylinder is $0\mathrm{.}5f{t}^2$ in cross$-$sectional area, and the air inside the cylinder is initially at $1$ atm $(14\mathrm{.}7$ psia $)$; the outside air is at $1$ atm $.$ Assume earth gravity of $32\mathrm{.}0f\frac{t}{s^2}.$ Assume variable specific heat for the air.

a$)(12\%)$ What will be the equilibrium height of the piston in the cylinder when the rope goes slack, i$.$e$.,$ no longer supporting the weight of the piston $($ft$)?$

b$)$

c$)$

$(4\%)$ What is the initial potential energy of the piston relative to the ground $(ft1b_f)?$

$(3\%)$ Sketch the $P-V$ diagram $(1b\frac{f}{f}{t}^2$ vs$.f{t}^3)$ for the air in the shaft. Label the initial and final states with numerical values on the axes.

d$)(4\%)$ If the cylinder were instead insulated and the compression of the air isentropic$,$ what would be the temperature of the air after the rope is longer supporting the weight of the piston $(($:$R\}?$