A perfect gas enclosed by an insulated upright cylinder with a weightless piston on top is at equilibrium conditions p1 V1 and T1 A weight is placed on the piston After a number of oscillations the motion subsides and the gas reaches a new equilibrium at conditions p2 V2 and 72 Find the temperature ratio T2 71 in terms of the pressure ratio p2 p1 a Show that the change in entropy is given by S S Rln 2 S S1 R b Show that if the initial disturbance is small such that 1 where 1 then f 1 y 1 a Y 27 x 1 c Using isentropic relations show that the frequency of small oscillations about the equilibrium position is 178 PA 27 L W d Justify the isentropic assumption in part c using your result from part b How does the result in part c compare with the frequency of a pendulum where is the height of the cylinder and Wis the weight of the mass on the piston