$($Only accurate and good explanation answer required $)$

a$)$ A perfect gas, enclosed by an insulated $($upright$)$ cylinder and piston, is at equilibrium condition

$p_1,v_1,T_1.$ A weight is placed on the piston. After a number of oscillations, the motion subsides,

and the gas reaches a new equilibrium at condition $p_2,v_2,T_2.$ Find the temperature ratio $\frac{T_2}{T_1}$ in

terms of the pressure ratio $=\frac{p_2}{p_1}.$ Show that the change of entropy is given by

$s_2-s_1=Rln(\frac{1+(-1)}{}{)}^{\frac{}{-1}}\frac{1}{}$

Show that, if the initial disturbance is small, so that $=1+lon,$ where $lon1,$ then

$\frac{s_2-s_1}{R}=\frac{lo{n}^2}{2}$

$($b$)$ Show that the frequency of small oscillation about the equilibrium position, using isentropic

relations, is

$=\frac{1}{2}\sqrt[\phantom{2}]{\frac{g}{l}}\sqrt[\phantom{2}]{(1+\frac{p_{1}A}{W})}$

Where $l$ is the height of the cylinder.