Question #42080

A rigid, massless plank is hinged to a horizontal, non-expanding floor. The plank is supported by a vertical cylindrical pillar of height $H$ at a horizontal distance $L$ from the hinge, forming an incline of angle $\alpha = 30^\circ$. The pillar is made of a material with coefficient of linear thermal expansion $\beta$ and Poisson's ratio $\nu$. It is tightly fitted inside a rigid, unexpandable cylindrical sleeve that prevents any lateral expansion. A small block of mass $m$ is launched from the hinge up the incline by a spring of stiffness $k$ compressed by $x_0$. The block travels as a projectile in the vertical plane, making an angle $\theta = 30^\circ$ with the incline. If the temperature of the pillar is increased by a small amount $\Delta T$, find the fractional change in the range of the projectile along the incline, $\frac{\Delta R}{R}$. Assume the spring and floor are unaffected by the temperature change.