Chapter 9Chapter 9

FluidsFluids

Objectives Objectives for Todayfor Today

Hydrostatic Pressure; P = Hydrostatic Pressure; P = ghgh Buoyancy; Archimedes’ PrincipleBuoyancy; Archimedes’ Principle

FFbuoyancybuoyancy = = g(Volume displaced)g(Volume displaced) Pascal’s EquationPascal’s Equation

P=F/A = f/aP=F/A = f/a Continuity EquationContinuity Equation

AA11VV11=A=A22VV22

Bernoulli’s EquationBernoulli’s Equation P +1/2 P +1/2 vv2 2 + + gh = constantgh = constant

DensityDensity

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

Units are kg/mUnits are kg/m33 (SI) or g/cm (SI) or g/cm33 (cgs) (cgs) 1 g/cm1 g/cm33 = 1000 kg/m = 1000 kg/m33

V

m obj

fluid

specific gravity

PressurePressure

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

2m

NPain

A

FP

Variation of Variation of Pressure with DepthPressure 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 portions of the fluid must be in static equilibriumstatic equilibrium

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

equilibrium (Think weather)equilibrium (Think weather)

Pressure and Pressure and DepthDepth

Examine the darker Examine the darker region, assumed to region, assumed to be a fluidbe a fluid It has a cross-It has a cross-

sectional area Asectional area A Extends to a depth Extends to a depth

h below the surfaceh below the surface Three external Three external

forces act on the forces act on the regionregion

Pressure and Pressure and Depth equationDepth equation

PPoo is normal is normal atmospheric atmospheric pressurepressure = 101.3 kPa= 101.3 kPa = 14.7 lb/in= 14.7 lb/in22

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

ghPP o

Pressure UnitsPressure Units

One atmosphere (1 atm) =One atmosphere (1 atm) = 760 mm of mercury760 mm of mercury 101.3 kPa101.3 kPa 14.7 lb/in14.7 lb/in22

Pressure Pressure CalculationCalculation

Hoover DamHoover Dam Average HeadAverage Head

158.5 meters of 158.5 meters of waterwater

Max Pressure;Max Pressure; ??????

Worksheet #1

Pressure CalculationPressure Calculation

P = Po + P = Po + ghgh

h=158.4 metersh=158.4 meters = 1000 kg/m= 1000 kg/m33

Pressure:Pressure: Po + Po + gh = 101.3KPa + 1000 x 9.8 x 158.5 Pagh = 101.3KPa + 1000 x 9.8 x 158.5 Pa

= 101.3 KPa + 1,553,300 Pa= 101.3 KPa + 1,553,300 Pa = 1655 KPa= 1655 KPa

Why Black and Why Black and White?White?

Power turbinesPower turbines

DownstreamDownstream

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Archimedes' Archimedes' PrinciplePrinciple

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

Buoyant ForceBuoyant Force

The upward force is The upward force is called the called the buoyant buoyant forceforce

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

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

The upward buoyant force is The upward buoyant force is B=ρB=ρfluidfluidVVobjobjgg

The downward gravitational force The downward gravitational force is w=mg=ρis w=mg=ρobjobjVVobjobjgg

The net force is B-w=(ρThe net force is B-w=(ρfluidfluid-ρ-ρobjobj)gV)gVobjobj

Totally Submerged Totally Submerged ObjectObject

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 Totally Submerged ObjectObject

The object is The object is more dense than more dense than the fluidthe fluid

The net force is The net force is downwarddownward

The object The object accelerates accelerates downwarddownward

Archimedes’ Principle:Archimedes’ Principle:Floating ObjectFloating Object

FFbuoyancybuoyancy = = g(Volume displaced) g(Volume displaced) The object is in static equilibrium.The object is in static equilibrium. The upward buoyant force is balanced The upward buoyant force is balanced

by the downward force of gravity.by the downward force of gravity. Volume of the fluid displaced Volume of the fluid displaced

corresponds to the volume of the object corresponds to the volume of the object beneath the fluid level.beneath the fluid level.

Buoyancy in actionBuoyancy in action

Ship displacement810 million N!

332 meters long

How many cubic meters are displaced?

Worksheet #2

Got milk?Got milk?

Ship weighs 810 x 10Ship weighs 810 x 1066 N = B N = B Density of water = 1000 kg/mDensity of water = 1000 kg/m33

Volume of water displaced isVolume of water displaced is B=(810 x 10B=(810 x 1066 )=V )=Vdispdisp x (1000 x 9.8) x (1000 x 9.8)

VVdispdisp = 82600 cubic meters or = 82600 cubic meters or22 million gallons!22 million gallons!

B=B=fluidfluidgVgVdispdisp

VVdispdisp=W=Wshipship//waterwatergg

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 undimished to every point of the fluid and to the walls of the fluid and to the walls of the container.container.

Pascal’s PrinciplePascal’s Principle

The hydraulic press The hydraulic press is an important is an important application of application of Pascal’s PrinciplePascal’s Principle

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

2

2

1

1

A

F

A

FP

ApplicationApplication

Worksheet #3a

Fluids in Motion:Fluids in Motion:Streamline FlowStreamline Flow

Streamline flow Streamline flow every particle that passes a particular point every particle that passes a particular point

moves exactly along the smooth path moves exactly along the smooth path followed by particles that passed the point followed by particles that passed the point earlierearlier

also called laminar flowalso called laminar flow Streamline is the pathStreamline is the path

different streamlines cannot cross each different streamlines cannot cross each otherother

the streamline at any point coincides with the streamline at any point coincides with the direction of fluid velocity at that pointthe direction of fluid velocity at that point

Characteristics of an Characteristics of an Ideal FluidIdeal Fluid

The fluid is nonviscousThe fluid is nonviscous There is no internal friction between There is no internal friction between

adjacent layersadjacent layers The fluid is incompressibleThe fluid is incompressible

Its density is constantIts density is constant The fluid is steadyThe fluid is steady

Its velocity, density and pressure do not Its velocity, density and pressure do not change in timechange in time

The fluid moves without turbulenceThe fluid moves without turbulence No eddy currents are presentNo eddy currents are present

Equation of Equation of ContinuityContinuity

AA11vv11 = A = A22vv22

The product of the The product of the cross-sectional area cross-sectional area of a pipe and the fluid of a pipe and the fluid speed is a constantspeed is a constant Speed is high where Speed is high where

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

Av is called the Av is called the flow flow rate – what are its rate – what are its units?units?

ApplicationApplication

Worksheet #3b

Bernoulli’s EquationBernoulli’s Equation

Let’s take a minute to show how much you already know about this equation!Do a dimensional analysis -

21

2P v gh constant

Bernoulli’s EquationBernoulli’s Equation

21

2P v gh constant

What do the second and third terms look like?

What happens we multiply by Volume?

Conservation of Conservation of energyenergy

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

21

2P v gh constant

ApplicationApplication

Worksheet #4

Applications of Applications of Bernoulli’s Principle: Bernoulli’s Principle: Venturi MeterVenturi Meter

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 measure the speed of the fluid flowof the fluid flow

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

Prairie DogsPrairie Dogs

Build burrows Build burrows with two openingswith two openings

One is even with One is even with ground, the other ground, the other built up, why?built up, why?

Prairie DogsPrairie Dogs

He wants his He wants his family to have family to have fresh air.fresh air.

Apply Bernoulli’s Apply Bernoulli’s Eq’n to a breeze Eq’n to a breeze over both holes.over both holes.

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2P v gh constant

Breeze

Prairie DogsPrairie Dogs

How will the How will the pressures over pressures over each hole each hole compare?compare?

What will this do What will this do the air in the the air in the tunnel?tunnel?

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2P v gh constant

Breeze

Questions?Questions? Hydrostatic Pressure; P = Hydrostatic Pressure; P = ghgh Buoyancy; Archimedes’ PrincipleBuoyancy; Archimedes’ Principle

FFbuoyancybuoyancy = = g(Volume displaced) g(Volume displaced)

Pascal; F/A=f/aPascal; F/A=f/a Continuity EquationContinuity Equation

AA11VV11=A=A22VV22

Bernoulli’s EquationBernoulli’s Equation P + 1/2 P + 1/2 vv2 2 ++ gh = constantgh = constant

Greek or Geek?Greek or Geek?

Greek or Geek?Greek or Geek?

Greek or Geek?Greek or Geek?

Greek or Geek?Greek or Geek?

Greek or Geek?Greek or Geek?

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