Review. A house roof is a perfectly flat plane that makes an angle $\theta$ with the horizontal. When its temperature changes, between $T_{c}$ before dawn each day and $T_{h}$ in the middle of each afternoon, the roof expands and contracts uniformly with a coefficient of thermal expansion $\alpha_{1}$ . Resting on the roof is a flat, rectangular metal plate with expansion coefficient $\alpha_{2},$ greater than $\alpha_{1}$ . The length of the plate is $L,$ measured along the slope of the roof. The component of the plate's weight perpendicular to the roof is supported by a normal force uniformly distributed over the area of the plate. The coefficient of kinetic friction between the plate and the roof is $\mu_{k}$ . The plate is always at the same temperature as the roof, so we assume its temperature is continuously changing. Because of the difference in expansion coefficients, each bit of the plate is moving relative to the roof below it, except for points along a certain horizontal line running across the plate called the stationary line. If the temperature is rising, parts of the plate below the stationary line are moving down relative to the roof and feel a force of kinetic friction acting up the roof. Elements of area above the stationary line are sliding up the roof, and on them kinetic friction acts downward parallel to the roof. The stationary line occupies no area, so we assume no force of static friction acts on the plate while the temperature is changing. The plate as a whole is very nearly in equilibrium, so the net friction force on it must be equal to the component of its weight acting down the incline. (a) Prove that the stationary line is at a distance of
$$\frac{L}{2}\left(1-\frac{\tan \theta}{\mu_{k}}\right)$$
below the top edge of the plate. (b) Analyze the forces that act on the plate when the temperature is falling and prove that the stationary line is at that same distance above the bottom edge of the plate. (c) Show that the plate steps down the roof like an inchworm, moving each day by the distance
$$\frac{L}{\mu_{k}}\left(\alpha_{2}-\alpha_{1}\right)\left(T_{h}-T_{c}\right) \tan \theta$$
(d) Evaluate the distance an aluminum plate moves each day if its length is 1.20 $\mathrm{m}$ , the temperature cycles between $4.00^{\circ} \mathrm{C}$ and $36.0^{\circ} \mathrm{C}$ , and if the roof has slope $18.5^{\circ},$ coefficient of linear expansion $1.50 \times 10^{-5}(\mathrm{C})^{-1}$ , and coefficient of friction 0.420 with the plate. (e) What If? What if the expansion coefficient of the plate is less than that of the roof? Will the plate creep up the roof?