Thermodynamics: Consider N particles of an ideal gas with adiabatic coefcient 7 in a tall cylinder of cross-sectional area A. The surrounding environment is at temperature To. There is an air-tight cap of mass M in the cylinder as shown which can slide up and down frictionlessly. Assume that gravity acts on the cap but that the length of the cylinder is small enough that the pressure in the cylinder is uniform. We approximate the region above the cap as vacuum. The cap is supported from below by the pressure in the gas and the equilibrium height is 5 above the bottom of the cylinder. If we were to displace the cap vertically up or down and release it, it will rapidly oscillate about the equilibrium position. Q: Determine the equilibrium height 6 of the pistOn. As the cap rapidly oscillates, explain whether the gas in the cylinder change adiabatically or isothermally. What is the resulting frequency of small oscillations? [Hint: For the final bit a good rst step would be to determine the gas pressure when the cap has been displaced by a distance a: above its equilibrium height]