A gas spring consists of a piston of area \(A\) moving in a cylinder of gas. As the piston moves, the gas expands and contracts, changing the pressure exerted on the piston. The process occurs adiabatically (without heat transfer), so

\[ p=C ho^{\gamma} \]

where \(p\) is the gas pressure, \(ho\) is the gas density, \(\gamma\) is the constant ratio of specific heats, and \(C\) is a constant dependent on the initial state. Consider a spring when the initial pressure is \(p_{0}\) and the initial temperature is \(T_{0}\). At this pressure, the height of the gas column in the cylinder is \(h\). let \(F=ho_{0} A+\delta F\) be the pressure force acting on the piston when it has displaced a distance \(x\) into the gas from its initial height.

(a) Determine the relation between \(\delta F\) and \(x\).

(b) Linearize the relationship of part

(a) to approximate the air spring by a linear spring. What is the equivalent stiffness of the spring?

(c) What is the required piston area for an air spring \((\gamma=1.4)\) to have a stiffness of \(300 \mathrm{~N} \cdot \mathrm{m}\) for a pressure of \(150 \mathrm{kPa}\) (absolute) with \(h=30 \mathrm{~cm}\).