Question #82124

A metallic rod of initial length $L_0$ (in $\text{m}$) and cross-sectional area $A$ (in $\text{m}^2$) is at an initial temperature $T_0$ (in $\text{K}$). Its coefficient of linear thermal expansion is $\alpha$ (in $\text{K}^{-1}$). The Young's modulus of the material varies with temperature $T$ as $Y(T) = \frac{Y_0}{1 + \beta (T - T_0)}$, where $Y_0$ (in $\text{N m}^{-2}$) and $\beta$ (in $\text{K}^{-1}$) are positive constants. A tensile force $F$ (in $\text{N}$) is gradually applied to the rod, increasing from $0$ to $F_0$. Simultaneously, the temperature of the rod is increased from $T_0$ to $T_0 + \Delta T$. The force is increased such that it is always proportional to the temperature change, i.e., $F(T) = \frac{F_0}{\Delta T} (T - T_0)$. Assuming the rod is free to expand and strains are small, what is the total work done (in $\text{J}$) by the applied force $F$ during this process?