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ELASTIC PROPERTIES OF SOLIDS We shall discuss the deformation of solids in terms of the concepts of stress and strain. Stress is the external force acting on an object per unit cross-sectional area. Strain is proportional to stress; the constant of proportionality depends on the material being deformed and on the nature of the deformation. Strain is a measure of the degree of deformation strain is proportional to stress Prestressed Concrete The tension in the cable was 940 N. What diameter should a 10-m-long steel wire have if we do not want it to stretch more than 0.5 cm under these conditions? Solution: From the definition of Young’s modulus, we can solve for the required crosssectional area. Assuming that the cross section is circular, we can determine the diameter of the wire. From Equation of Young Module, we have The radius of the wire can be found from To provide a large margin of safety, we would probably use a flexible cable made up of many smaller wires having a total cross-sectional area substantially greater than our calculated value. Squeezing a Brass Sphere A solid brass sphere is initially surrounded by air, and the air pressure exerted on it is 1.0 105 N/m2 (normal atmospheric pressure). The sphere is lowered into the ocean to a depth at which the pressure is 2.0 107 N/m2. The volume of the sphere in air is 0.50 m3. By how much does this volume change once the sphere is submerged? Solution From the definition of bulk modulus, we have Because the final pressure is so much greater than the initial pressure, we can neglect the initial pressure and state that The negative sign indicates a decrease in volume. TEMPERATURE AND THE ZEROTH LAW OF THERMODYNAMICS Heat is the transfer of energy from one object to another object as a result of a difference in temperature between the two Thermal equilibrium is a situation in which two objects in thermal contact with each other cease to exchange energy by the process of heat. Thermal equilibrium is a situation in which two objects in thermal contact with each other cease to exchange energy by the process of heat. If objects A and B are separately in thermal equilibrium with a third object C, then objects A and B are in thermal equilibrium with each other. Celsius temperature scale, this mixture is defined to have a temperature of zero degrees Celsius, which is written as 0°C; this temperature is called the ice point of water. Another commonly used system is a mixture of water and steam in thermal equilibrium at atmospheric pressure; its temperature is 100°C, which is the steam point of water. THE CONSTANT-VOLUME GAS THERMOMETER AND THE ABSOLUTE TEMPERATURE SCALE Fahrenheit scale. This scale sets the temperature of the ice point at 32°F and the temperature of the steam point at 212°F. The relationship between the Celsius and Fahrenheit temperature scales is THERMAL EXPANSION OF SOLIDS AND LIQUIDS At ordinary temperatures, the atoms in a solid oscillate about their equilibrium positions with an amplitude of approximately 10-11 [m] and a frequency of approximately 1013 [Hz]. The average spacing between the atoms is about 10-10 [m]. As the temperature of the solid increases, the atoms oscillate with greater amplitudes; as a result, the average separation between them increases. The average coefficient of linear expansion: Expansion of a Railroad Track A steel railroad track has a length of 30.000 m when the temperature is 0.0°C. (a) What is its length when the temperature is 40.0°C? Solution Making use of Table 19.2 and noting that the change in temperature is 40.0°C, we find that the increase in length is If the track is 30.000 m long at 0.0°C, its length at 40.0°C is: 30.013 m. (b) Suppose that the ends of the rail are rigidly clamped at 0.0°C so that expansion is prevented. What is the thermal stress set up in the rail if its temperature is raised to 40.0°C? From the definition of Young’s modulus for a solid, we have Because Y for steel is 20 1010 N/m2 (see Table 12.1), we have If the rail has a cross-sectional area of 30.0 cm2, what is the force of compression in the rail? One mole of any substance is that amount of the substance that contains Avogadro’s number The number of moles n of a substance is related to its mass m through the expression where M is the molar mass of the substance (see Section 1.3), which is usually expressed in units of grams per mole (g/mol). For example, the molar mass of oxygen (O2) is 32.0 g/mol. Therefore, the mass of one mole of oxygen is 32.0 g. One mole (mol) of a substance is that amount of the substance that contains as many particles (atoms, molecules, or other particles) as there are atoms in 12 g of the carbon-12 isotope. One mole of substance A contains the same number of particles as there are in 1 mol of any other substance B. For example, 1 mol of aluminum contains the same number of atoms as 1 mol of lead. Size of each atom: A solid cube of aluminum (density 2.7 g/cm3) has a volume of 0.20 cm3. How many aluminum atoms are contained in the cube? Solution Since density equals mass per unit volume, the mass m of the cube is To find the number of atoms N in this mass of aluminum, we can set up a proportion using the fact that one mole of aluminum (27 g) contains 6.02x1023 atoms: What is the size of an aluminum atom: mAl=13 amu, 13x1.66x10-27 =2.158x10-26 kg Size of each atom: You are designing apparatus to support an actor of mass 65 kg who is to “fly” down to the stage during the performance of a play. You decide to attach the actor’s harness to a 130-kg sandbag by means of a lightweight steel cable running smoothly over two frictionless pulleys, as shown in Figure. You need 3.0 m of cable between the harness and the nearest pulley so that the pulley can be hidden behind a curtain. For the apparatus to work successfully, the sandbag must never lift above the floor as the actor swings from above the stage to the floor. Let us call the angle that the actor’s cable makes with the vertical . What is the maximum value can have before the sandbag lifts off the floor? Sol.: