Chapter 11,12Chapter 11,12

Matter, Fluid MechanicsMatter, Fluid Mechanics

States of MatterStates of Matter

SolidSolid LiquidLiquid GasGas PlasmaPlasma

SolidsSolids

Has definite volumeHas definite volume Has definite shapeHas definite shape Molecules are held in Molecules are held in

specific locationsspecific locations• by electrical forcesby electrical forces

vibrate about vibrate about equilibrium positionsequilibrium positions

Can be modeled as Can be modeled as springs connecting springs connecting moleculesmolecules

More About SolidsMore About Solids

External forces can be applied to External forces can be applied to the solid and compress the the solid and compress the materialmaterial• In the model, the springs would be In the model, the springs would be

compressedcompressed When the force is removed, the When the force is removed, the

solid returns to its original shape solid returns to its original shape and sizeand size• This property is called This property is called elasticityelasticity

Crystalline SolidCrystalline Solid

Atoms have an Atoms have an ordered structure ordered structure

This example is This example is saltsalt• Gray spheres Gray spheres

represent Narepresent Na++ ions ions• Green spheres Green spheres

represent Clrepresent Cl-- ions ions

Amorphous SolidAmorphous Solid

Atoms are Atoms are arranged almost arranged almost randomlyrandomly

Examples include Examples include glassglass

LiquidLiquid

Has a definite volumeHas a definite volume No definite shapeNo definite shape Exists at a higher Exists at a higher

temperature than solidstemperature than solids The molecules “wander” The molecules “wander”

through the liquid in a through the liquid in a random fashionrandom fashion• The intermolecular forces The intermolecular forces

are not strong enough to are not strong enough to keep the molecules in a keep the molecules in a fixed positionfixed position

GasGas

Has no definite volumeHas no definite volume Has no definite shapeHas no definite shape Molecules are in constant random Molecules are in constant random

motionmotion The molecules exert only weak The molecules exert only weak

forces on each otherforces on each other Average distance between Average distance between

molecules is large compared to the molecules is large compared to the size of the moleculessize of the molecules

PlasmaPlasma

Matter heated to a very high Matter heated to a very high temperaturetemperature

Many of the electrons are freed from Many of the electrons are freed from the nucleusthe nucleus

Result is a collection of free, Result is a collection of free, electrically charged ionselectrically charged ions

Plasmas exist inside starsPlasmas exist inside stars

DensityDensity

The density of a substance of The density of a substance of uniform composition is defined as its uniform composition is defined as its mass per unit volume:mass per unit volume:

Units are kg/mUnits are kg/m33 (SI) (SI)

mV

Iron(steel) 7,800 kg/mIron(steel) 7,800 kg/m33 Water 1,000 kg/mWater 1,000 kg/m33 Air 1.3 kg/mAir 1.3 kg/m33

Density, cont.Density, cont.

The densities of most liquids and The densities of most liquids and solids vary slightly with changes in solids vary slightly with changes in temperature and pressuretemperature and pressure

Densities of gases vary greatly with Densities of gases vary greatly with changes in temperature and pressurechanges in temperature and pressure

Specific GravitySpecific Gravity

The The specific gravityspecific gravity of a substance is of a substance is the ratio of its density to the density the ratio of its density to the density of water at 4° Cof water at 4° C• The density of water at 4° C is 1000 The density of water at 4° C is 1000

kg/mkg/m33

Specific gravity is a unitless ratioSpecific gravity is a unitless ratio

water

Gravity Specific Iron: 7.8Water: 1.0Air: 0.0013

FluidsFluids

Liquids and gases do not maintain a Liquids and gases do not maintain a fixed shape, have ability to flowfixed shape, have ability to flow

Liquids and gases are called fluidsLiquids and gases are called fluids Fluids statics: study of fluids at restFluids statics: study of fluids at rest Fluids dynamics: study of fluids in Fluids dynamics: study of fluids in

motionmotion

PressurePressure

Pressure is force Pressure is force per unit areaper unit area 2m

NPain

A

FP

Ex: 60kg person standing on oneFoot (10cm by 25cm). The force exerted The force exerted

by a fluid on a by a fluid on a submerged object submerged object at any point if at any point if perpendicular to perpendicular to the surface of the the surface of the objectobject

Measuring PressureMeasuring Pressure

The spring is The spring is calibrated by a calibrated by a known forceknown force

The force the fluid The force the fluid exerts on the exerts on the piston is then piston is then measuredmeasured

Variation of Pressure with DepthVariation of Pressure with Depth

If a fluid is at rest in a container, all If a fluid is at rest in a container, all portions of the fluid must be in static portions of the fluid must be in static equilibriumequilibrium

All points at the same depth must be at All points at the same depth must be at the same pressurethe same pressure• Otherwise, the fluid would not be in Otherwise, the fluid would not be in

equilibriumequilibrium• The fluid would flow from the higher The fluid would flow from the higher

pressure region to the lower pressure regionpressure region to the lower pressure region

Pressure and DepthPressure and Depth

Examine the area at Examine the area at the bottom of fluidthe bottom of fluid• It has a cross-sectional It has a cross-sectional

area Aarea A• Extends to a depth h Extends to a depth h

below the surfacebelow the surface Force act on the region Force act on the region

is the weight of fluidis the weight of fluid

ghA

Ahg

A

Vg

A

mgP

ghP

Pressure and Depth equationPressure and Depth equation

PPatmatm is normal is normal

atmospheric atmospheric pressurepressure• 1.013 x 101.013 x 105 5 Pa = Pa =

14.7 lb/in14.7 lb/in22

The pressure does The pressure does not depend upon not depend upon the shape of the the shape of the containercontainer

ghPP atm

ExamplesExamples

1.1. Two levels in a fluid.Two levels in a fluid.

2.2. Pressure exerted by 10 m of water.Pressure exerted by 10 m of water.

3.3. Pressure exerted on a diver 10 m Pressure exerted on a diver 10 m under water.under water.

Pressure Measurements:Pressure Measurements:ManometerManometer

One end of the U-One end of the U-shaped tube is open shaped tube is open to the atmosphereto the atmosphere

The other end is The other end is connected to the connected to the pressure to be pressure to be measuredmeasured

Pressure at A is Pressure at A is P=PP=Poo+ρgh+ρgh

Pressure Measurements: Pressure Measurements: BarometerBarometer

Invented by Invented by Torricelli (1608 – Torricelli (1608 – 1647)1647)

A long closed tube A long closed tube is filled with is filled with mercury and mercury and inverted in a dish inverted in a dish of mercuryof mercury

Measures Measures atmospheric atmospheric pressure as ρghpressure as ρgh

Pascal’s PrinciplePascal’s Principle

A change in pressure applied to an A change in pressure applied to an enclosed fluid is transmitted enclosed fluid is transmitted undimished to every point of the fluid undimished to every point of the fluid and to the walls of the container.and to the walls of the container.• First recognized by Blaise Pascal, a First recognized by Blaise Pascal, a

French scientist (1623 – 1662)French scientist (1623 – 1662)

Pascal’s Principle, contPascal’s Principle, cont

The hydraulic press is The hydraulic press is an important an important application of Pascal’s application of Pascal’s PrinciplePrinciple

Also used in hydraulic Also used in hydraulic brakes, forklifts, car brakes, forklifts, car lifts, etc.lifts, etc.

2

2

1

1

A

F

A

FP

ExampleExample

Consider AConsider A11=5 A=5 A22, F, F22=2000N. Find F=2000N. Find F1.1.

ArchimedesArchimedes

287 – 212 BC287 – 212 BC Greek Greek

mathematician, mathematician, physicist, and physicist, and engineerengineer

Buoyant forceBuoyant force InventorInventor

Archimedes' PrincipleArchimedes' Principle

Any object completely or partially Any object completely or partially submerged in a fluid is buoyed up by submerged in a fluid is buoyed up by a force whose magnitude is equal to a force whose magnitude is equal to the weight of the fluid displaced by the weight of the fluid displaced by the object.the object.

Buoyant ForceBuoyant Force

The upward force The upward force is called the is called the buoyant forcebuoyant force

The physical cause The physical cause of the buoyant of the buoyant force is the force is the pressure difference pressure difference between the top between the top and the bottom of and the bottom of the objectthe object

Buoyant Force, cont.Buoyant Force, cont.

The magnitude of the buoyant force The magnitude of the buoyant force always equals the weight of the always equals the weight of the displaced fluiddisplaced fluid

The buoyant force is the same for a The buoyant force is the same for a totally submerged object of any size, totally submerged object of any size, shape, or densityshape, or density

fluidfluid gVFB

Buoyant Force, finalBuoyant Force, final

The buoyant force is exerted by the The buoyant force is exerted by the fluidfluid

Whether an object sinks or floats Whether an object sinks or floats depends on the relationship between depends on the relationship between the buoyant force and the weightthe buoyant force and the weight

Archimedes’ Principle:Archimedes’ Principle:Totally Submerged ObjectTotally Submerged Object

The upward buoyant force is The upward buoyant force is FFBB=ρ=ρfluidfluidgVgVobjobj

The downward gravitational force is The downward gravitational force is w=mg=ρw=mg=ρobjobjgVgVobjobj

The net force is FThe net force is FBB-w=(ρ-w=(ρfluidfluid-ρ-ρobjobj)gV)gVobjobj

ρρfluidfluid>ρ>ρobj obj floatsfloats ρρfluidfluid<ρ<ρobj obj sinkssinks

ExampleExample

A block of brass with mass 0.5 kg and A block of brass with mass 0.5 kg and specific gravity 8 is suspended from specific gravity 8 is suspended from a string. Find the tension in the string a string. Find the tension in the string if the block is in air, and if it is if the block is in air, and if it is completely immersed in water.completely immersed in water.

Totally Submerged ObjectTotally Submerged Object

The object is less The object is less dense than the dense than the fluidfluid

The object The object experiences a net experiences a net upward forceupward force

Totally Submerged Object, 2Totally Submerged Object, 2

The object is more The object is more dense than the dense than the fluidfluid

The net force is The net force is downwarddownward

The object The object accelerates accelerates downwarddownward

Fluids in Motion: ideal fluidFluids in Motion: ideal fluid

laminar flow: path, velocitylaminar flow: path, velocity Incompressible fluid Incompressible fluid No internal friction (no viscosity)No internal friction (no viscosity) Good approximation for liquids in Good approximation for liquids in

generalgeneral Ok for gases when pressure Ok for gases when pressure

difference is not too largedifference is not too large

Equation of ContinuityEquation of Continuity

AA11vv11 = A = A22vv22 The product of the The product of the

cross-sectional area cross-sectional area of a pipe and the of a pipe and the fluid speed is a fluid speed is a constantconstant• Speed is high where Speed is high where

the pipe is narrow and the pipe is narrow and speed is low where speed is low where the pipe has a large the pipe has a large diameterdiameter

Av is called the Av is called the flow flow raterate

Equation of Continuity, contEquation of Continuity, cont

The equation is a consequence of The equation is a consequence of conservation of mass and a steady flowconservation of mass and a steady flow

A v = constantA v = constant• This is equivalent to the fact that the volume of This is equivalent to the fact that the volume of

fluid that enters one end of the tube in a given fluid that enters one end of the tube in a given time interval equals the volume of fluid leaving time interval equals the volume of fluid leaving the tube in the same intervalthe tube in the same interval

Assumes the fluid is incompressible and there are no Assumes the fluid is incompressible and there are no leaksleaks

Daniel BernoulliDaniel Bernoulli

1700 – 17821700 – 1782 Swiss physicist Swiss physicist

and and mathematicianmathematician

Wrote Wrote HydrodynamicaHydrodynamica

Also did work that Also did work that was the beginning was the beginning of the kinetic of the kinetic theory of gasestheory of gases

Bernoulli’s EquationBernoulli’s Equation

Relates pressure to fluid speed and Relates pressure to fluid speed and elevationelevation

Bernoulli’s equation is a Bernoulli’s equation is a consequence of Work Energy consequence of Work Energy Relation applied to an ideal fluidRelation applied to an ideal fluid

Assumes the fluid is incompressible Assumes the fluid is incompressible and nonviscous, and flows in a and nonviscous, and flows in a nonturbulent, steady-state mannernonturbulent, steady-state manner

Bernoulli’s Equation, cont.Bernoulli’s Equation, cont.

States that the sum of the pressure, States that the sum of the pressure, kinetic energy per unit volume, and kinetic energy per unit volume, and the potential energy per unit volume the potential energy per unit volume has the same value at all points has the same value at all points along a streamlinealong a streamline

constant ghvP 2

2

1constant 2

2

1vP

Applications of Bernoulli’s Applications of Bernoulli’s Principle: Venturi TubePrinciple: Venturi Tube

Shows fluid flowing Shows fluid flowing through a horizontal through a horizontal constricted pipeconstricted pipe

Speed changes as Speed changes as diameter changesdiameter changes

Can be used to Can be used to measure the speed of measure the speed of the fluid flowthe fluid flow

Swiftly moving fluids Swiftly moving fluids exert less pressure exert less pressure than do slowly moving than do slowly moving fluidsfluids

An Object Moving Through a An Object Moving Through a FluidFluid

Many common phenomena can be Many common phenomena can be explained by Bernoulli’s equationexplained by Bernoulli’s equation• At least partiallyAt least partially

In general, an object moving through In general, an object moving through a fluid is acted upon by a net upward a fluid is acted upon by a net upward force as the result of any effect that force as the result of any effect that causes the fluid to change its causes the fluid to change its direction as it flows past the objectdirection as it flows past the object

Application – Golf BallApplication – Golf Ball

The dimples in the The dimples in the golf ball help move air golf ball help move air along its surfacealong its surface

The ball pushes the air The ball pushes the air downdown

Newton’s Third Law Newton’s Third Law tells us the air must tells us the air must push up on the ballpush up on the ball

The spinning ball The spinning ball travels farther than if travels farther than if it were not spinningit were not spinning

Application – Airplane WingApplication – Airplane Wing

The air speed above The air speed above the wing is greater than the wing is greater than the speed belowthe speed below

The air pressure above The air pressure above the wing is less than the wing is less than the air pressure belowthe air pressure below

There is a net upward There is a net upward forceforce• Called Called liftlift

Other factors are also Other factors are also involvedinvolved