$(30\%)$ Considering a piston$-$cylinder system as shown in the figure below, an amount of heat $Q$ is supplied

to the fluid inside the cylinder. The spring constant is $(\frac{N}{m}).$ Please estimate the work done by the fluid

inside the cylinder to the surroundings and the spring caused by this heating if $($a$)$ the spring is extremely

stiff, $($b$)$ the spring is reasonably compressible.

Analysis:

$($a$)$ For an extremely stiff spring, the piston can be assumed nearly stationary, i$.$e$.$

there is negligible change in volume of the fluid contained in the cylinder. Thus,

there is no work done by the fluid to the surroundings and the spring.

$($b$)$ Please refer to the following problem discussed in TA sessions:

EXAMPLE $5-4$ Expansion of a Gas Against a Spring

A piston$-$cylinder device contains $0\mathrm{.}05m^3$ of a gas initially at $200$ kPa $.$ At this

state, a linear spring that has a spring constant of $150k\frac{N}{m}$ is touching

the piston but exerting no force on it$.$ Now heat is transferred to the gas,

causing the piston to rise and to compress the spring until the volume inside

the cylinder doubles. If the cross$-$sectional area of the piston is $0\mathrm{.}25m^2,$

determine $($a$)$ the final pressure inside the cylinder, $($b$)$ the total work done by

the gas, and $($c$)$ the fraction of this work done against the spring to compress it$.$