(Piston oscillator) Let a piston of mass m be place at the midpoint of a closed cylinder o...

(Piston oscillator) Let a piston of mass m be place at the midpoint of a closed cylinder of cross-sectional area A and length 2L, as sketched. Assume that the pressure p on either

side of the piston satisfies Boyle’s law (namely, that the pres­sure times the volume is constant), and let po be the pressure on both sides when x = 0.

Expand the x/(L2 − x2) term in a Taylor series about x = 0, up to the third-order term. Keeping only the leading term, derive the linearized version

of (10.1), which is restricted to the case of small oscillations - that is, where the amplitude of oscillation is small compared to L.