Gas Compression in piston and resulting oscillations

Homework Statement

A cylinder is filled with .1 moles of an ideal gas at STP, and a piston of mass 1.4Kg seals the gas in the cylinder with a frictionless seal, as shown in the figure below. The trapped column of gas has an initial height 2.4. The piston and cylinder are surrounded by air, also at STP. The piston is released from rest and starts to fall. The motion of the piston ceases after a series of oscillations that ends with the piston and the trapped air in thermal equilibrium with
the surrounding air (which is at STP). (a) Find the final height h of the gas column.
(b) Suppose the piston is pushed down below the equilibrium position by a small amount and then released. Assuming that the temperature of the gas remains constant, find the frequency of the vibration.

PV=nRT
P=F/A

The Attempt at a Solution

a) The initial height, H, can be described by PVi=nRT=PAH
The final height, h, can be described by the Ʃf=0, since its in equilibrium, = Pinternal*A-mg-P(A) where P is standard pressure.

Pin=mg/A +P = (nRT)/Vf using nRT from above the internal pressure can be written as
Pin=PAH/Ah = PH/h so

h= PH/[P+mg/PA] and using PAH=nRT for the original conditions we can solve for A=nRT/PH
so h= PH/[P+mg/P{nRT/PH}] using 300K for T and 1.01e5Pa for P I get h = 2.12m

h = [(2.4)]/[1+((1.4*9.8)/{(1.01e5)([(.1*8.3*300)/[(1.01e5)(2.4)]}]

How does that look?

b) This part confuses me. I am trying to use hookes law for a linear oscillator but I can not figure out the the k value.
F=kx
F=PA so k=PA/x
so ω=√[(PA/x)/m], I am using x=2.1 from part a but I am sure that is incorrect.
I get f=ω/(2pi) = [sqrt[((.00103)(1.01e5)/2.1)/2.4]]/(2pi) = .723Hz

What am I doing wrong for the second part. How should I solve for omega?

Thanks

Does anybody have any idea how to approach the second problem?

I understand that I need a restoring force in order to find a k value for the system. Basically the sum of the forces will equal ma, not zero, and P'inA - PinA=ma but I am not sure how to go about actually solving for the the a value.