"2006-2007 Physics Olympiad Preparation Program
– University of Toronto –
We continue the competition for the best problems created by our POPTOR contestants. Name your self-made problem as “My Problem”, and mail this problem and its solution on a separate sheet of paper along with solutions of POPTOR problems. You may send us any number of problems, but only self-made and unique will be considered. Authors of the best problems will be awarded during the POPTOR Weekend.
Problem Set 3: Thermodynamics
Due January 8, 2007
Problem 1 The tank in Figure 1.1 is filled with water 2.00 m deep. At the bottom of one side wall is a rectangular hatch 1.00 m high and 2.00 m wide, which is hinged at the top of the hatch. (a) Determine the force the water exerts on the hatch. (b) Find the torque exerted by the water about the hinges.
Fig.1.1 Problem 2 For this problem you will probably need to get familiar with solution of the Problem #1 in the Problem Set 3 of our last year Program. One mole of an ideal monatomic gas undergoes a linear process 1 - 2, in which its pressure P and its volume V change as shown in Fig.2.1. (a) Find the maximum temperature of the gas during this process. (b) In some range of volume the gas expands with absorbing the heat (the endothermic process); in the other range the gas emits the heat (the exothermic process). Find the boundaries for these ranges of the volume of the gas. P P0 1
2 0 Fig.2.1 V0 V
Problem 3 Brief Theory: When we use an indoor heater, we think that the air in the room becomes warmer because of the molecules of the air come in contact with the hot surface molecules of the device and exchange kinetic energy with them. This process is called heat transfer. The other thermal process – convection (conduction for solids) makes “hot” and “cool” molecules to mix and spread heat in surroundings. However, one more process can heat the room even if the space is evacuated (without molecules of the air). This is possible due to radiation, or emission of electromagnetic waves by the hot surfaces. The greater the temperature of the body, the greater power P of radiant energy it emits. The power emitted by a unit surface area is called intensity I. If the intensity of radiation, absorbed by an object, equals the intensity that the object emits, such object is called the black body. The black body is an ideal absorber, as it absorbs all incident energy. However it also emits the same amount of energy at the same rate. Therefore, the black
body is also an ideal radiator. Numerically, the relationship between the emitted radiation intensity I and the temperature T of the black body is expressed by a Stefan’s law: I = σT4. The constant σ = 5.67x10-8 W/ (m2K4) is one of the main fundamental constants. -- -----------Your home heater has three identical parallel planes, which can be treated as black bodies. You may set up any temperature for each plane individually and turn on any combination of planes. The distance between each pair of planes is much less than the dimensions of the planes. (a) Find the equilibrium temperature of the outside planes TO if the temperature of the inside plane is maintained at T K. (b) Find the temperature of the inside plane TI if the left plane has a stable temperature T K and the right plane has a temperature 2T K. Problem 4 A heat engine is a device that uses heat to perform work. The goal of an inventor of the engine is to increase the efficiency and reduce the fuel consumption. The ideal heat engine with greatest possible efficiency was predicted theoretically to use an ideal gas, which performs work according to the Carnot cycle: isothermal expansion → adiabatic expansion → isothermal compression → adiabatic compression with returning to the initial state. Then the cycle starts again and may be repeated any number of times. To solve the problem, you should get familiar with the theoretical explanation of functioning of the Carnot engine. Different real heat engines run different cycles. Fig. 4.1a shows the idealized Otto cycle, which approximately represents the processes in the conventional PBC internal combustion gasoline engine. Fig. 4.1b shows an airstandard Diesel cycle for an idealized Diesel engine. For two shown cycles (a) Determine the process that corresponds to the useful work produced by the VAD Fig.4.1a Fig.4.1b engine; (b) Find the efficiency of the engine; (c) Find the temperature of the hot and of the cold reservoir and calculate the efficiency of the Carnot engine operating between these two temperatures. (d) Compare the efficiency of two engines with the efficiency of the Carnot engine and choose between the Otto engine and the Diesel engine to get the higher efficiency. Problem 5 (experimental) Preparing for the Holidays, buy a couple of balloons that tend to fly in the sky. Manage to attach the known masses to the free end of the string in such a way that makes the string stretched and vertical and the balloon motionless. Try to find what gas is inside the balloon, designing the remaining part of the experiment on your own.
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