The force required to maintain a rubber band or polymer fiber at a length L when its unstretched length is L0 has observed to be related to temperature as follows F=k(T)[LL0] where k(T) is positive and is of the form T, with a constant. The heat capacity of the rubber band or fiber measured at the constant length L0 is given by CL=+T where and are parameters that depend on the rubber band or fiber length. (a) What should happen to the temperature when the rubber band is streched adiabatically? Show mathematical results to justify your answer. (b) If you hang a weight on a rubber band and measure the length in the laboratory at room temperature and then you repeat the process, but in the refrigerator, the length of the rubber band will change. Does the length increase or decrease? Show mathematical results to justify your answer. (c) If CF is defined as CF=T(TS)F derive a relation for CFCL and show whether this difference is positive, negative, or zero. (d) The same amount of heat flows into two rubber bands, but one is held at constant tension and the other at constant length. Which has the largest increase in temperature? Justify your answer. (e) Show that the dependence of k(T) on temperature at constant length is related to the dependence of entropy on length at constant temperature. Offer a physical description for the signs of the derivatives. (f) In polymer science 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 that, on an isothermal extension of the fiber, increases its internal energy) is FU=(U/L)T, and the entropic force is FS=T(S/L)T. Evaluate FU and FS for the fiber or rubber band being considered here