Question #42126

A faulty thermometer reads $5^\circ\text{X}$ at the melting point of ice and $95^\circ\text{X}$ at the boiling point of water at standard atmospheric pressure. A uniform metal rod is perfectly constrained between two rigid, immovable walls. The rod is heated, and the faulty thermometer records its initial and final temperatures as $14^\circ\text{X}$ and $86^\circ\text{X}$, respectively. The rod has a constant coefficient of linear thermal expansion $\alpha = 2 \times 10^{-5} \text{ } ^\circ\text{C}^{-1}$. Its Young's modulus varies with the true Celsius temperature $T$ as $Y(T) = Y_0(1 - \beta T)$, where $Y_0 = 2 \times 10^{11} \text{ N/m}^2$ and $\beta = 5 \times 10^{-3} \text{ } ^\circ\text{C}^{-1}$. Assuming the rod does not buckle, what is the magnitude of the thermal stress developed in the rod?