$(60$ pts$)$ The force required to maintain a polymeric fiber at a length $L,$ when its unstretched length is $L_0,$ has been observed to be related to its temperature as follows:

$F=T(L-L_0),$

where $$ is a positive constant. The heat capacity of the fiber measured at the constant length $L$ is given by:

$C_L=+T,$

where $$ and $$ are parameters that depend only on the fiber length.

$($a$)$ Develop an equation that relates the change in entropy of the fiber with respect to temperature and length changes, i$.$e$.,$ evaluate the derivatives $(del\frac{S}{d}$elT${)}_L$ and $(del\frac{S}{d}$elL${)}_T.$

$($b$)$ Evaluate the derivatives $(del\frac{U}{d}$elT${)}_L$ and $(del\frac{U}{d}$elL${)}_T.$

$($c$)$ Develop an equation that relates the entropy of the fiber at a temperature $T_0$ and an extension $L_0$ to its entropy at any other temperature $T$ and extension $L.$

$($d$)$ If the fiber at $T=T_i$ and $L=L_i$ is stretched slowly and adiabatically until it attains a length $L_f,$ what is the fiber temperature $T_f?($Assume

$($e$)$ In polymer science and engineering, it is common to attribute the force necessary to stretch a fiber to energetic and entropic effects. The energetic force $($i$.$e$.,$ that part of the force which, upon an isothermal extension of the fiber, increases its internal energy$)$ is $F_U=(del\frac{U}{d}$elL${)}_T,$ and the entropic force is $F_S=-T(del\frac{S}{d}$elL${)}_T.$ Evaluate $F_U$ and $F_S$ for the fiber.