Here we learn that the compressible behavior of an ideal gas

can be put to use in the design of a spring. Consider a cylinder filled with an

ideal gas (cP, cv). In the center of this cylinder there

is a piston, which is pushed and displaced to the axial distance x by the

external force F = A(P1 _ P2), where A is the area

of the piston face, P1 is the pressure on the compressed side of the

piston, and P2 is the pressure on the expanded aside. Find the relationship

between F and x for two different designs:

(a) The process executed by the entire ideal gas system is

reversible and isothermal.

»

Here we learn that the compressible behavior of an ideal gas

can be put to use in the design of a spring. Consider a cylinder filled with an

ideal gas (cP, cv). In the center of this cylinder there

is a piston, which is pushed and displaced to the axial distance x by the

external force F = A(P1 _ P2), where A is the area

of the piston face, P1 is the pressure on the compressed side of the

piston, and P2 is the pressure on the expanded aside. Find the relationship

between F and x for two different designs:

(a) The process executed by the entire ideal gas system is

reversible and isothermal. Show that F = kT x, where the kT

factor increases monotonically with x. Determine the form of the kT(x)

function in the limit of small displacements, x _

V0_A, where V0 is the initial size of the gas

volume on either side of the piston.

(b) The process executed by the entire ideal-gas system is

reversible and adiabatic. Invoke from the start the limit of small

displacements, show that F = kSx, and that the factor kS

increases as x increases. Compare the two designs, (a) and (b): which air

spring is stiffer?

(c) In both designs, the work transferred to the system is

Fx. Where does this amount of energy reside at the end of the process in design

(a) and in design (b)?

(d) Imagine that after the process of deforming the spring

the piston returns to its original position quasi-statically. What is the work

transfer from the ideal-gas system to the external entity that opposes by force

the return motion of the piston?

»