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Chapter 9: Solids and Fluids States of Matter Three states of matter • Normally matter is classified into one of three (four) states: solid, liquid, gas (, plasma). solid : crystalline solid (salt etc.) amorphous solid (glass etc.) ordered structure atoms arranged at almost at random States of Matter Three (four) states of matter (cont’d) • Normally matter is classified into one of three (four) states: solid, liquid, gas (, plasma). liquid : A molecule in a liquid does random-walk through a series of interactions with other molecules. - For any given substance, the liquid state exists at a higher temperature than the solid state. -The inter-molecular forces in a liquid are not strong enough to hold molecules together in fixed position. -The molecules wander around in random fashion. States of Matter Three (four) states of matter (cont’d) • Normally matter is classified into one of three (four) states: solid, liquid, gas (, plasma). gas : In gaseous state, molecules are in constant random motion and exert only weak forces on each other. -The average distance between the molecules of a gas is quite large compared with the size of molecules. - Occasionally the molecules collide with each other, but most of them move freely. - Unlike solids and liquids, gases can be easily compressed. plasma : At high temperature, electrons of atoms are free from nucleus. Such a collection of ionized atoms with equal amounts of positive (nucleus) and negative charges (electrons) forms a state called plasma. Deformation of Solids Stress, strain and elastic modulus • Until external force becomes strong enough to deform permanently or break a solid object, the effect of deformation by the external force goes back to zero when the force is removed – Elastic behavior. • Stress : the force per unit area causing a deformation Strain : a measure of the amount of the deformation Elastic modulus : proportionality constant, similar to a spring constant stress = elastic modulus x strain Deformation of Solids Young’s modulus: elasticity in length • Consider a long bar of cross-sectional area A and length L0, clamped at one end. When an external force F is applied along the bar, perpendicular to the cross section, internal forces in the bar resist the distortion that F tends to produce. • Eventually the bar attains an equilibrium in which: (1) its length is greater than L0 (2) the external force is balanced by internal forces. The bar is said to be stressed. F L Y A L0 tensile stress Young’s modulus SI unit: Pa = 1 N/m2 tensile strain SI unit: dimensionless Deformation of Solids Young’s modulus: elasticity in length (cont’d) • Typical values • Stress vs. strain Deformation of Solids Shear modulus: Elasticity of shape • Another type of deformation occurs when an object is subjected to a force F parallel to one of its faces while the opposite face is held fixed by a second force. • The stress in this situation is called a shear stress. F x S A h shear stress Shear modulus SI unit: Pa = 1 N/m2 shear strain SI unit: dimensionless Deformation of Solids Bulk modulus: Volume elasticity • Suppose that the external forces acting on an object are all perpendicular to the surface on which the force acts and are distributed uniformly. • This situation occurs when a object is immersed in a fluid. V P B V volume stress bulk modulus SI unit: Pa = 1 N/m2 volume strain SI unit: dimensionless Deformation of Solids An example • Example 9.3 : Stressing a lead ball A solid lead sphere of volume 0.50 m3, dropped in the ocean, sinks to a depth of 2.0x103 m, where the pressure increases by 2.0x107 Pa. Lead has a bulk modulus of 4.2x1010 Pa. What is the change in volume of the sphere? P B V / V VP (0.50 m3 )( 2.0 107 Pa) 4 3 V 2 . 4 10 m B 4.2 1010 Pa Density and Pressure Density • The density r of an object is defined as: r M V M: mass, V: volume • The specific gravity of a substance is the ratio of its density to the density of water at 4oC, which is 1.0x103 kg/m3, and it is dimensionless. SI unit: kg/m3 (cgs unit: g/cm3 ) Density and Pressure Pressure • Fluids do not sustain shearing stresses, so the only stress that a fluid can exert on a submerged object is one that tends to compress it, which is a bulk stress. • The force F exerted by the fluid on the object is always perpendicular to the surfaces of the object. • If F is the magnitude of a force exerted perpendicular to a given surface of area A, then the pressure P is defined as: P F A F: force, A: area SI unit: Pa = N/m2 Density and Pressure Variation of pressure with depth • When a fluid is at rest in a container, all portions of the fluid must be in static equilibrium – at rest with respect to the observer. • All points at the same depth must be at the same pressure. If this were not the case, fluid would flow from the higher pressure region to the lower pressure region. • Consider an object at rest with area A and height h in a fluid. P2 A P1 A Mg 0 M rV rA( y1 y2 ) P2 P1 rg y1 y2 • Effect of atmospheric pressure: P P0 rgh P1 P2 for y1 y2 0 P0 : atmospheric pressure, P: pressure at depth h Density and Pressure Examples • Example 9.5 : Oil and water P1 P0 rgh1 1.01105 Pa (7.00 10 2 kg/m 3 )(9.80 m/s 2 ) (8.00 m) 1.56 10 Pa 5 Pbot P1 rgh2 2.06 105 Pa r=0.700 g/cm3 h1=8.00 m r=1025 kg/m3 h2=5.00 m Density and Pressure Pascal’s principle • A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and to the walls of the container. Hydraulic press P1 P2 F1 F2 A1 A2 F2 F1 A2 A1 F2 > F1 if A2 > A1 Density and Pressure Car lift • Example 9.7 : Car lift (a) Find necessary force by compressed air at piston 1. 2 A1 r1 F1 F2 2 F2 A2 r2 weight=13,300 N 1.48 103 N (b) Find air pressure. F1 P 1.88 105 Pa A1 circular x-sec (c) Show the work done by pistons is the same. V1 V2 A1x1 A2 x2 A2 / A1 x1 / x2 F1 / F2 A1 / A2 r1=5.00 cm r2=15.0 cm W1 F1x1 A1 A2 1 W2 F2 x2 A2 A1 Pressure Measurements Absolute and gauge pressure • An open tube manometer (Fig.(a)) measures the gauge pressure P-P0 P : absolute pressure P P0 rgh P P0 rgh P=PA=PB P0 : atmospheric pressure • A mercury barometer (Fig.(b)) measures the atmospheric pressure P0 P P0 rgh • One atmospheric pressure defined as the pressure equivalent of a column of mercury that is exactly 0.76 m in height. P0 rgh 1.013 105 Pa 1 atm vacuum Pressure Measurements Blood pressure measurement • A specialized manometer (sphygmomanometer) -A rubber bulb forces air into a cuff wrap. -A manometer is attached under cuff and is under pressure. -The pressure in the cuff is increased until the flow of blood through brachial artery is stopped. -Then a valve on the bulb is opened, and measurer listens with a stethoscope to the artery at a point just below the cuff. -When the pressure at the cuff and the artery is just below the max. value produced by heart (the systolic pressure), the artery opens momentarily on each beat. -At this point, the velocity of the blood is high, and the flow is noisy and can be heard… Buoyant Forces and Archimedes’s Principle Archimedes’s principle Any object completely or partially submerged in a fluid is buoyed up by a force with magnitude equal to the weight of the fluid displaced by the object. Upward force (buoyant force) : ( P2 P1 ) A r fluidhAg r fluidVg r fluidV fluid g B Downward force: M obj g r obj ghA r objVg Buoyant Forces and Archimedes’s Principle Archimedes’s principle and a floating object Upward force (buoyant force) : B r fluidV fluid g Vobj Downward force: M obj g r objVobj g r obj V fluid r fluidV fluid g r objVobj g r fluid Vobj Vfluid Buoyant Forces and Archimedes’s Principle Examples • Example 9.8 : A fake or pure gold crown? Is the crown made of pure gold? Tair =7.84 N T mg 0 air Twater =6.86 N Twater mg B 0 Twater Tair B 0 B Tair Twater r water gVwater 0.980 N 4 Vwater Vcrown 1.00 10 m m Tair / g 0.800 kg 3 rgold=19.3x103 kg/m3 r crown m / Vcrown 8.00 103 kg/m 3 Buoyant Forces and Archimedes’s Principle Examples • Example 9.9 : Floating down the river What depth h is the bottom of the raft submerged? rwood=6.00x102 kg/m3 B mraft g 0 B mraft g mraft g ( r raftVraft ) g B mwater g ( r waterVwater ) g ( r water Ah) g ( r water Ah) g ( r waterVraft ) g r raftVraft h 0.0632 m r raft A A=5.70 m2 Fluid in Motion Some terminology • When a fluid is in motion: (1) if every particle that passes a particular point moves along exactly the same smooth path followed by previous particles passing the point, this path is called streamline. If this happens, this flow is said to be streamline or laminar. (2) the flow of a fluid becomes irregular, or turbulent, above a certain velocity or under any conditions that can cause abrupt changes in velocity. • Ideal fluid : 1. The fluid is non-viscous : There is no internal friction force between adjacent layers. 2. The fluid is incompressible : Its density is constant. 2. The fluid motion is steady : The velocity, density, and pressure at each point in the fluid do not change with time. 3. The fluid moves without turbulence : Each element of the fluid has zero angular velocity about its center. Fluid in Motion Equation of continuity • Consider a fluid flowing through a pipe of non-uniform size. The particles in the fluid move along the streamlines in steady-state flow. In a small time interval t, the fluid entering the bottom end of the pipe moves a distance: x1 v1t The mass contained in the bottom blue region : M1 r1 A1x1 r1 A1v1t From a similar argument : M 2 r2 A2x2 r2 A2v2t Since M1=M2 (flow is steady): M1 M 2 r1 A1v1 r2 A2v2 A1v1 A2v2 if incompress ible Equation of continuity Fluid in Motion An example • Example 9.12 : Water garden Fluid in Motion Bernoulli’s equation • Consider an ideal fluid flowing through a pipe of non-uniform size. Work done to the fluid at Point 1 during the time interval t: W1 F1x1 P1 A1x1 P1V Work done to the fluid at Point 2 during the time interval t: W2 P2 A2x2 P2V Work done to the fluid : W fluid P1V P2V Fluid in Motion Bernoulli’s equation (cont’d) If m is the mass of the fluid passing through the pipe in t , the change in kinetic energy is: 1 2 1 2 KE mv2 mv1 2 2 The change in gravitational potential energy in t is: PE mgy2 mgy1 From conservation of energy: W fluid KE PE Fluid in Motion Bernoulli’s equation (cont’d) From conservation of energy: W fluid KE PE ( P1 P2 )V 1 2 1 2 mv2 mv1 mgy2 mgy1 2 2 r m /V 1 2 1 2 P1 rv1 rgy1 P2 rv2 rgy2 const . 2 2 1 2 P rv rgy const. Bernoulli’s equation 2 Fluid in Motion Venturi tube Consider a water flow through a horizontal constricted pipe. A1v1 A2v2 A1 A2 v2 v1 y1 y2 1 2 1 2 P1 rv1 P2 rv2 2 2 Fluid in Motion Examples • Example 9.13 : A water tank Consider a water tank with a hole. (a) Find the speed of the water leaving through the hole. P0 h =0.500 m y1 =3.00 m 1 2 rv1 rgy1 P0 rgy2 2 v1 2 g ( y2 y1 ) 2 gh 3.13 m/s (b) Find where the stream hits the ground. 1 2 y 0 y1 gt v0 y t t 0.782 s 2 x v0 xt v1t 2.45 m y x Fluid in Motion Examples • Example 9.14 : Fluid flow in a pipe Find the speed at Point 1. A1v1 A2v2 v2 A2=1.00 m2 A1=0.500 m2 h =5.00 m A1 v1 A2 1 2 1 2 P0 rv1 rgy1 P0 rv2 rgy2 2 2 2 P0 1 2 1 A rv1 rgy1 P0 r 1 v1 rgy2 2 2 A2 A 2 2 gh v12 1 1 2 g ( y2 y1 ) 2 gh v1 11.4 m/s A2 1 ( A1 / A2 ) 2 Surface Tension, Capillary Action, and Viscous Fluid Flow Surface tension • The net force on a molecule at A is zero because such a molecule is completely surrounded by other molecules. • The net force on a molecule at B is downward because it is not completely surrounded by other molecules. There are no molecules above it to exert upward force. this asymmetry makes the surface of the liquid contract and the surface area as small as possible. • The surface tension is defined as : F where the surface tension force F L is divided by the length L along which the force acts. SI unit : N/m=(N m)/m2=J/m2 Surface Tension, Capillary Action, and Viscous Fluid Flow Surface tension (cont’d) • The surface tension of liquids decreases with increasing temperature, because the faster moving molecules of a hot liquid are not bound together as strongly as are those in a cooler liquid. • Some ingredient called surfactants such as detergents and soaps decrease surface tension. • The surface tissue of the air sacs in the lungs contain a fluid that has a surface tension of about 0.050 N/m. As the lungs expand during inhalation, the body secretes into the tissue a substance to reduce the surface tension and it drops down to 0.005 N/m. Surface Tension, Capillary Action, and Viscous Fluid Flow Surface of liquid • Forces between like-molecules such as between water molecules are called cohesive forces. • Forces between unlike-molecules such as those exerted by glass on water are called adhesive forces. • Difference in strength between cohesive and adhesive forces creates the shape of a liquid at boundary with other materials. Surface Tension, Capillary Action, and Viscous Fluid Flow Viscous fluid flow • Viscosity refers to the internal friction of a fluid. It is very difficult for layers of a viscous fluid to slide past one another. • When an ideal non-viscous fluid flows through a pipe, the fluid layers slide past one another with no resistance. • If the pipe has uniform cross-section each layer has the same velocity. • The layers of a viscous fluid have different velocities. The fluid has the greatest velocity at the center of the pipe, whereas the layer next to the wall does not move because of adhesive forces between them. ideal fluid, non-viscous viscous fluid Surface Tension, Capillary Action, and Viscous Fluid Flow Viscous fluid flow • Consider a layer of liquid between two solid surfaces. The lower surface is fixed in position, and the top surface moves to the right with a velocity v under the action of an external force F. • A portion of the liquid is distorted from its original shape, ABCD, at one instance to the shape AEFD a moment later. The force required F to move the upper plate at a fixed speed v is : F h Av d where h is the coefficient of viscosity of the fluid, and A is the area in contact with fluid. h SI unit : N s/m2 cgs unit: dyne s/cm2= poise 1 poise=10-1 N s/m2 1 cp (centipoise) = 10-2 poise Surface Tension, Capillary Action, and Viscous Fluid Flow Poiseuille’s law • Consider a section of tube of length L and radius R containing a fluid under pressure P1 at the left end and a pressure P2 at the right. • Poiseuille’s law describes the flow rate of a viscous fluid under pressure difference: V R 4 ( P1 P2 ) t 8hL Surface Tension, Capillary Action, and Viscous Fluid Flow Reynolds number • At sufficiently high velocities, fluid flow changes from simple streamline flow to turbulent flow, characterized by a highly irregular motion of the fluid. Experimentally the onset of the turbulence in a tube is determined by a dimensionless factor called Reynolds number, RN, given by: rvd RN h r : density of fluid v : average speed of the fluid along the direction of flow d : diameter of tube h : viscosity of fluid • If RN is below about 2000, the flow of fluid through a tube is streamline. • If RN is above about 3000, the flow of fluid through a tube is turbulent. • If RN is between 2000 and 3000, the flow is unstable. Surface Tension, Capillary Action, and Viscous Fluid Flow Examples • Example 9.18 : A blood transfusion A patient receives a blood transfusion through a needle of radius 0.20 mm and length 2.0 cm. The density of blood is 1,050 kg/m3. The bottle supplying the blood is 0.50 m above the patient’s arm. What is the rate of the flow through the needle? P1 P2 rgh (1050 kg/m 3 )(9.80 m/s 2 )(0.50 m) 5.15 103 Pa V R 4 ( P1 P2 ) 6.0 108 m3 / s t 8hL Surface Tension, Capillary Action, and Viscous Fluid Flow Examples • Example 9.19: Turbulent flow of blood Determine the speed at which blood flowing through an artery of diameter 0.20 cm will become turbulent. h ( RN ) (2.7 103 N s/m 3 )(3.00 103 ) v 3.9 m/s 3 3 2 rd (1.05 10 kg/m )(0.20 10 m) Transport Phenomena • A fluid can move place to place as a result of difference in concentration between two points in the fluid. There are two processes in this category : diffusion and osmosis. Diffusion • In a diffusion process, molecules move from a region where their concentration is high to a region where their concentration is lower. • Consider a container in which a high concentration of molecules has been introduced into the left side (the dashed line is an imaginary barrier). All the molecules move in random direction. Since there are more molecules on the left side, more molecules migrate into the right side than otherwise. Once a concentration equilibrium is reached, there will be no net movement. Transport Phenomena Diffusion (cont’d) • Fick’s law mass M C2 C1 Diffusion rate DA time t L where D is a constant of proportion called the diffusion coefficient (unit : m2/s), A is the cross-sectional area, (…) is the change in concentration per unit distance (concentration gradient), and M/t is the mass transported per unit time. The concentrations, C1 and C2 are measured in unit of kg/m3. Transport Phenomena Size of cells and osmosis • Diffusion through cell membranes is vital in supplying oxygen to the cells of the body and in removing carbon dioxide and other waste products from them. • A fresh supply of oxygen diffuses from the blood, where its concentration is high, into the cell, where its concentration is low. • Likewise, carbon dioxide diffuses from the cell into the blood where its concentration is lower. • A membrane that allows passage of some molecules but not others is called a selectively permeable membrane. • Osmosis is the diffusion of water across a selectively permeable membrane from a high water concentration to a low water concentration. Transport Phenomena Motion through a viscous medium • The magnitude of the resistive force on a very small spherical object of radius r moving slowly through a fluid of viscosity h with speed v is given by: resistive Stokes’s law frictional Fr 6hrv force • Consider a small sphere of radius r falls through a viscous medium. 4 3 w rgV rg r 3 4 B r f gV r f g r 3 3 force of buoyant gravity force Transport Phenomena Motion through a viscous medium (cont’d) • At the instance the sphere begins to fall, the force of friction is zero because the velocity of the sphere is zero. • As the sphere accelerates, its speed increases resistive and so does Fr. frictional • When the net force goes to zero, the speed force of the sphere reaches the so-called terminal speed vt. Fr B w 4 3 4 3 6hrvt r f g r rg r 3 3 2r 2 g vt (r r f ) 9h force of buoyant gravity force Transport Phenomena Sedimentation and centrifugation • If an object is not spherical, the previous argument can still be applied except for the use of Stokes’s law. In this case, we assume that the relation Fr=kv holds where k is a coefficient. Fr B w Terminal speed condition rf B r f gV mg (V m / r ) r Fr kvt rf mg r f 1 mg mg kvt vt r k r Transport Phenomena Sedimentation and centrifugation (cont’d) • The terminal speed for particles in biological samples is usually quite small; the terminal speed for blood cells falling through plasma is about 5 cm/h in the gravitational field of Earth. • The speed at which materials fall through a fluid is called sedimentation rate. The sedimentation rate in a fluid can be increased by increasing the effective acceleration g: for example by using radial acceleration due to rotation (centrifuge). v2 ac 2r r mac r f 1 vt k r m 2 r r f vt 1 k r