States of Matter. Chapter 9 Solids and Fluids. Solids: Stress and Strain. Solids: Stress and Strain. Stress = Force Area. Strain =!


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1 Elasticity Chapter 9 Solids and Fluids Archimedes Principle Bernoulli s Equation Solid Liquid Gas Plasmas States of Matter 1 2 Solids: Stress and Strain Solids: Stress and Strain Stress = Measure of force felt by material Stress = Force Area Strain = Measure of deformation Strain =!L L A F!L SI units are Pascals, 1 Pa = 1 N/m 2 (same as pressure) dimensionless L 3 4 Young s Modulus (Tension) Example 9.1 Y = ( F A ) ( )!L L tensile stress tensile strain A F!L King Kong (a 8.0x10 4 kg monkey) swings from a 320 m cable from the Empire State building. If the 3.0 cm diameter cable is made of steel (Y=1.8x10 11 Pa), by how much will the cable stretch? Measure of stiffness Tensile refers to tension L 1.97 m 5 6
2 Shear Modulus S = ( F A ) ( )!x h Sheer Stress Sheer Strain "F B =! A =! "P "V "V V V ( ) Bulk Modulus Change in Pressure Volume Strain B = Y Example 9.2 Pascals as units for Pressure A large solid steel (Y=1.8x10 11 Pa) block (L 5 m, W=4 m, H=3 m) is submerged in the Mariana Trench where the pressure is 7.5x10 7 Pa. P = F A 1 Pa = 1 N/m 2 a) By what percentage does the length change? % b) What are the changes in the length, width and height? mm, mm, mm c) By what percentage does the volume change? % 9 10 Solids and Liquids Solids have Young s, Bulk, and Shear moduli Liquids have only bulk moduli Ultimate Strength Maximum F/A before fracture or crumbling Different for compression and tension 11 12
3 Densities Density and Specific Gravity! = M V Densities depend on temperature, pressure... Specific gravity = ratio of density to density of H 2 O at 4 C Example 9.3 Pressure & Pascal s Principle The specific gravity of gold is What is the mass (in kg) and weight (in lbs.) of 1 cubic meter of gold? P = F A Pressure applied to any part of an enclosed fluid is transmitted undimished to every point of the fluid and to the walls of the container 19,300 kg lbs Each face feels same force Transmitting force P = F 1 A 1 = F 2 A 2 An applied force F 1 can be amplified : F 2 = F 1 A 2 A 1 Hydraulic press Examples: hydraulic brakes, forklifts, car lifts, etc. Pressure and Depth w is weight w = Mg =!Vg =!Ahg Sum forces to zero, PA! P 0 A! w = 0 Factor A P = P 0 +!gh 17 18
4 Example 9.5 (skip) Find the pressure at 10,000 m of water. DATA: Atmospheric pressure = 1.015x10 5 Pa. Example 9.6 Assume the ultimate strength of legos is 4.0x10 4 Pa. If the density of legos is 150 kg/m 3, what is the maximum possible height for a lego tower? 27.2 m 9.82x10 7 Pa Example 9.7 Estimate the mass of the Earth s atmosphere given that atmospheric pressure is 1.015x10 5 Pa. Data: R earth =6.36x10 6 m Archimedes Principle Any object completely or partially submerged in a fluid is buoyed up by a force whose magnitude is equal to the weight of the fluid displaced by the object. 5.26x10 18 kg Example 9.8 A helicopter lowers a probe into Lake Michigan which is suspended on a cable. The probe has a mass of 500 kg and its average density is 1400 kg/m 3. What is the tension in the cable? 1401 N Example 9.9a A wooden ball of mass M and volume V floats on a swimming pool. The density of the wood is " wood <" H20. The buoyant force acting on the ball is: a) Mg upward b) " H20 gv upward c) (" H20 " wood )gv upward 23 24
5 Example 9.9b A steel ball of mass M and volume V rests on the bottom of a swimming pool. The density of the steel is " steel >" H20. The buoyant force acting on the ball is: a) Mg upward b) " H20 gv upward c) (" steel " H20 )gv upward Example 9.10 A small swimming pool has an area of 10 square meters. A wooden 4000kg statue of density 500 kg/m 3 is then floated on top of the pool. How far does the water rise? Data: Density of water = 1000 kg/m 3 40 cm Floating Coke Demo (SKIP) The can will Paint Thinner Demo (SKIP) When I pour in the paint thinner, the cylinder will: a) Float b) Sink a) Rise b) Fall Equation of Continuity What goes in must come out! mass density!m = "A!x = "Av!t Example 9.11 Water flows through a 4.0 cm diameter pipe at 5 cm/s. The pipe then narrows downstream and has a diameter of of 2.0 cm. What is the velocity of the water through the smaller pipe? Mass that passes a point in pipe during time!t Eq. of Continuity! 1 A 1 v 1 =! 2 A 2 v 2 20 cm/s 29 30
6 Laminar or Streamline Flow Fluid elements move along smooth paths Friction in laminar flow is called viscosity Turbulence Fluid elements move along irregular paths Sets in for high velocity gradients (small pipes) or instabilities Ideal Fluids Laminar Flow > No turbulence Nonviscous > No friction between fluid layers Incompressible > Density is same everywhere Bernoulli s Equation P + 1 2!v2 +!gy = constant Sum of P, KE/V and PE/V is constant How can we derive this? Bernoulli s Equation: derivation Consider a volume!v of mass!m of incompressible fluid,!ke = 1 2 Mv 2 2 " 1 2 Mv 2 1 = 1 2 #!Vv 2 2 " 1 2 #!Vv 2 1!PE = Mgy 2 " Mgy 1 = #!Vgy 2 " #!Vgy 1 W = F 1!x 1 " F 2!x 2 = P 1 A 1!x 1 " P 2 A 2!x 2 = P 1!V " P 2!V P 1 +!gh !v 1 2 = P 2 +!gh !v 2 2 Example 9.12 A very large pipe carries water with a very slow velocity and empties into a small pipe with a high velocity. If P 2 is 7000 Pa lower than P 1, what is the velocity of the water in the small pipe? 3.74 m/s Venturi Meter 35 36
7 Applications of Bernoulli s Equation Venturi meter Curve balls Airplanes Example 9.13a Consider an ideal incompressible fluid, choose >, < or = Beach Ball & Straws Demos " 1 " 2 a) = b) < c) > Example 9.13b Example 9.13c Consider an ideal incompressible fluid, choose >, < or = Mass that passes 1 in one second mass that passes 2 in one second a) = b) < c) > Consider an ideal incompressible fluid, choose >, < or = v 1 v 2 a) = b) < c) > Example 9.13d Example 9.14 Water drains out of the bottom of a cooler at 3 m/s, what is the depth of the water above the valve? a b Consider an ideal incompressible fluid, choose >, < or = P 1 P 2 a) = b) < c) > 45.9 cm 41 42
8 Viscosity Three Vocabulary Words F =!A v d Viscosity Diffusion Osmosis Friction between the layers Pressure drop required to force water through pipes (Poiselle s Law) At high enough v/d, turbulence sets in Diffusion Molecules move from region of high concentration to region of low concentration Fick s Law: Diffusion rate = Mass time = DA " C 2! C 1 % # $ L & ' Osmosis Movement of water through a boundary while denying passage to specific molecules, e.g. salts D = diffusion coefficient 45 46
States of Matter. Chapter 9 Solids and Fluids. Solids: Stress and Strain. Solids: Stress and Strain. Stress = Force Area. Strain =!L L. Example 9.
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