Question #42434

A uniform rod of cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear expansion $\alpha$ is initially at a uniform temperature of $10^\circ\text{C}$. It is placed horizontally on a smooth floor, wedged between two fixed rigid blocks. The faces of the blocks in contact with the rod are smooth inclined planes making an angle of $30^\circ$ with the horizontal. The ends of the rod are cut to perfectly match the $30^\circ$ incline. The rod is constrained by a smooth horizontal channel so that it cannot move vertically or bend. The ends of the rod are then maintained at steady temperatures of $90^\circ\text{C}$ and $50^\circ\text{C}$, while its lateral surface is thermally insulated. Assuming the blocks do not deform, what is the magnitude of the normal force $N$ exerted by each inclined plane on the rod in the steady state?