Problem $1$:

A piston$/$cylinder system has air inside, initially at $4$ bar with a volume of $0\mathrm{.}4$ L $.$ The mass of the piston is $5$ kg and its diameter is $0\mathrm{.}1$ m $.$ The piston is loaded with a linear spring and the outside atmospheric pressure of $1$ bar. The spring exerts no force on the piston when it is at the bottom of the cylinder. The valve is opened slowly to let some air in$,$ causing the piston to rise $2$ cm and then it is closed. Find the new pressure in the cylinder assuming the system is always in mechaniral an$$ilihri$...$m