A state of an elastic rod is described by the state variables entropy S and length L.

Question:

A state of an elastic rod is described by the state variables entropy S and length L. The differential of the internal energy U(S, L) of the rod is written as,Note that f (S, L) has the units of a force. The longitudinal stress τ on the rod is, where A is the cross-section of the rod. We neglect any change of A due to f. The physical properties of the rod material are given by the linear thermal expansion coefficient at constant stress,and the isothermal Young modulus,Make use of these two physical properties of the material to answer the following questions:a) Compute the partial derivative of the rod’s stress τ in the rod changes with respect to its temperature when its length is fixed. Consider that the cross-section A is independent of the temperature.b) Determine the heat transfer during an isothermal variation of the length of the rod ΔLif from an initial state i to a final state f in terms of α and E.c) Compute the partial derivative of the rod’s length L with respect to its temperature T.