Question #43462

A composite rod is formed by joining two rods of equal length $L = 1 \text{ m}$ end-to-end. Rod 1 has Young's modulus $Y_1 = Y$ and coefficient of linear expansion $\alpha_1 = 2\alpha$. Rod 2 has $Y_2 = 2Y$ and $\alpha_2 = \alpha$. The composite rod has a uniform cross-sectional area $A = 10^{-4} \text{ m}^2$ and is rigidly clamped between two immovable walls at $T_0 = 300 \text{ K}$. Given $Y = 10^{11} \text{ N/m}^2$ and $\alpha = 10^{-5} \text{ K}^{-1}$. A thermistor is attached to the rod, whose resistance varies as $R(T) = C T^{-3/2} \exp(E_g / 2kT)$, where $E_g$ is the band gap energy and $k$ is the Boltzmann constant. At $T_0 = 300 \text{ K}$, its resistance is $R_0$. A safety valve is controlled by a logic circuit with an XOR gate: $Z = X \oplus Y$. The valve opens if $Z = 1$. Input $X$ becomes HIGH (1) when the compressive thermal force in the rod $F \ge 60 \text{ kN}$. Input $Y$ becomes HIGH (1) when $R(T) \le R_0 2^{-3/2} e^{-2}$. As the system is slowly heated from $300 \text{ K}$, the safety valve NEVER opens. What is the value of $E_g / k$?