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UT Arlington PHYS 1441 - Lecture Notes

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Wednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 1 PHYS 1441 – Section 002 Lecture #24 Wednesday, Dec. 8, 2010 Dr. Jaehoon Yu • Elastic Properties of Solids • Density and Specific Gravity • Fluid and Pressure • Depth Dependence of Pressure • Absolute and Relative Pressure • Pascal’s Principle and Hydraulics • Buoyant Forces and Archimedes’ PrincipleAnnouncements • The Final Exam – Date and time: 11am – 1:30pm, Monday Dec. 13 – Place: SH103 – Comprehensive exam • Covers from CH1.1 – CH10.7 • Plus appendices A.1 – A.8 • Combination of multiple choice and free response problems • Submit your Planetarium extra credit sheet Wednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 2Wednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 3 Young’s Modulus Let’s consider a long bar with cross sectional area A and initial length Li. Fex=Fin Young’s Modulus is defined as What is the unit of Young’s Modulus? Experimental Observations 1. For a fixed external force, the change in length is proportional to the original length 2. The necessary force to produce the given strain is proportional to the cross sectional area Li A:cross sectional area!Tensile stress Lf=Li+DL Fex After the stretch Fex Fin Tensile strain Force per unit area Used to characterize a rod or wire stressed under tension or compression Elastic limit: Maximum stress that can be applied to the substance before it becomes permanently deformedWednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 4 Bulk Modulus Bulk Modulus characterizes the response of a substance to uniform squeezing or reduction of pressure. Bulk Modulus is defined as Volume stress =pressure After the pressure change If the pressure on an object changes by ΔP=ΔF/A, the object will undergo a volume change ΔV. V V’ F F F F Compressibility is the reciprocal of Bulk Modulus Because the change of volume is reverse to change of pressure.Wednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 5 Example for Solid’s Elastic Property A solid brass sphere is initially under normal atmospheric pressure of 1.0x105N/m2. The sphere is lowered into the ocean to a depth at which the pressures is 2.0x107N/m2. The volume of the sphere in air is 0.5m3. By how much its volume change once the sphere is submerged? The bulk modulus of brass is 6.1x1010 N/m2 The pressure change ΔP is Since bulk modulus is The amount of volume change is Therefore the resulting volume change ΔV is The volume has decreased.Wednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 6 Density and Specific Gravity Density, ρ(rho), of an object is defined as mass per unit volume Unit? Dimension? Specific Gravity of a substance is defined as the ratio of the density of the substance to that of water at 4.0 oC (ρH2O=1.00g/cm3). Unit? Dimension? None None What do you think would happen of a substance in the water dependent on SG? Sink in the water Float on the surfaceWednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 7 Fluid and Pressure What are the three states of matter? Solid, Liquid and Gas Fluid cannot exert shearing or tensile stress. Thus, the only force the fluid exerts on an object immersed in it is the force perpendicular to the surface of the object. How do you distinguish them? Using the time it takes for the particular substance to change its shape in reaction to external forces. What is a fluid? A collection of molecules that are randomly arranged and loosely bound by forces between them or by an external container. We will first learn about mechanics of fluid at rest, fluid statics. In what ways do you think fluid exerts stress on the object submerged in it? This force by the fluid on an object usually is expressed in the form of the force per unit area at the given depth, the pressure, defined as Note that pressure is a scalar quantity because it’s the magnitude of the force on a surface area A. What is the unit and the dimension of pressure? Expression of pressure for an infinitesimal area δA by the force δF is P =δFδAUnit:N/m2 Dim.: [M][L-1][T-2] Special SI unit for pressure is PascalWednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 8 Example for Pressure The mattress of a water bed is 2.00m long by 2.00m wide and 30.0cm deep. a) Find the weight of the water in the mattress. The volume density of water at the normal condition (0oC and 1 atm) is 1000kg/m3. So the total mass of the water in the mattress is Since the surface area of the mattress is 4.00 m2, the pressure exerted on the floor is Therefore the weight of the water in the mattress is b) Find the pressure exerted by the water on the floor when the bed rests in its normal position, assuming the entire lower surface of the mattress makes contact with the floor.Wednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 9 Variation of Pressure and Depth Water pressure increases as a function of depth, and the air pressure decreases as a function of altitude. Why? If the liquid in the cylinder is the same substance as the fluid, the mass of the liquid in the cylinder is It seems that the pressure has a lot to do with the total mass of the fluid above the object that puts weight on the object. Let’s imagine the liquid contained in a cylinder with height h and the cross sectional area A immersed in a fluid of density ρ at rest, as shown in the figure, and the system is in its equilibrium. The pressure at the depth h below the surface of the fluid open to the atmosphere is greater than the atmospheric pressure by ρgh. Therefore, we obtain Atmospheric pressure P0 is P0A PA Mg h Since the system is in its equilibriumWednesday, Dec. 8, 2010 PHYS 1441-002, Fall 2010 Dr. Jaehoon Yu 10 Absolute and Relative Pressure How can one measure pressure? One can measure the pressure using an open-tube manometer, where one end is connected to the system with unknown pressure P and the other open to air with pressure P0. This is called the absolute pressure, because it is the actual value of the system’s pressure. In many cases we measure the pressure difference with respect to the atmospheric pressure to avoid the effect of the changes in P0 that depends on the environment. This is called gauge or relative pressure. The common barometer which consists of a mercury column with one end closed at vacuum and the other


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