Question: A perfectly plane house roof makes an angle
A perfectly plane house roof makes an angle - with the horizontal. When its temperature changes, between Tc before dawn each day to Th in the middle of each afternoon, the roof expands and contracts uniformly with a coefficient of thermal expansion α1. Resting on the roof is a flat rectangular metal plate with expansion coefficient α2, greater than α1. The length of the plate is L, measured up 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 μk. 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. We call this 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 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 (d) Evaluate the distance an aluminum plate moves each day if its length is 1.20 m, if the temperature cycles between 4.00°C and 36.0°C, and if the roof has slope 18.5°, coefficient of linear expansion 1.50 X 10-5 (°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?
Answer to relevant QuestionsIs it possible for two objects to be in thermal equilibrium if they are not in contact with each other? ExplainA 50.0-g copper calorimeter contains 250 g of water at 20.0°C. How much steam must be condensed into the water if the final temperature of the system is to reach 50.0°C?A sample of ideal gas is expanded to twice its original volume of 1.00 m3 in a quasi-static process for which P =αV 2, with α = 5.00 atm/m6, as shown in Figure P20.23. How much work is done on the expanding gas?Consider the cyclic process depicted in Figure P20.30. If Q is negative for the process BC and ΔEint is negative for the process CA, what are the signs of Q, W, and ΔEint that are associated with each process?In Figure P20.40, the change in internal energy of a gas that is taken from A to C is "800 + J. The work done on the gas along path ABC is - 500 J. (a) How much energy must be added to the system by heat as it goes from A ...
Post your question