Mechanics of Solids and Fluids

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Mechanics of Solids ec a cs o So dsand Fluids

Physics Rapid Learning Series

Rapid Learning Centerwww.RapidLearningCenter.com/ Rapid Learning Inc. All rights reserved.

Wayne Huang, Ph.D.Keith Duda, M.Ed.

Peddi Prasad, Ph.D.Gary Zhou, Ph.D.

Michelle Wedemeyer, Ph.D.Sarah Hedges, Ph.D.



Learning ObjectivesBy completing this tutorial, you will learn the properties of two basic states of matter solids and fluids:

Classification of states of matter

Elasticity of solids Pressure variation of

fluids and buoyancy

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y y

Fluids in motion Surface tension and

viscosity.

Concept MapPrevious content

New content

States of Matter

Includes ViscousFlow

Solids Fluids

Elasticity

Pressure

Static fluids

Surface

Viscosity

Non viscous Non-viscous Flow

By

In motion

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Variation

Buoyancy and Buoyancy and Archimedes

Principle

Leads to

Bernoullis Equation

Work and EnergyLaw of

Work and EnergyTension

YieldsCapillary Action

Leads to

Density

ExpansionThermal

Expansion



States of Matter

Low temperature High temperature

Change of state with temperature

Solids GasesFluids (Liquid)

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Strong inter-molecular force

Loose inter-molecular force

Fixed molecule

Definite shape

Wandering molecule

Indefinite shape

Solids and Elasticity

Lets start out by discussing solids.

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Classification of Solids - 1

Crystalline Ordered atomic

Two categories:

Ordered atomic structure

Existence of melting point

Amorphous solid Randomly arranged

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y gatomic structure

No melting pointExample:

Nature Crystal



Two categories:




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y gatomic structure

No melting pointOrdered atomic structure





Two categories:




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y gatomic structure

No melting pointMelting curve



Two categories




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y gatomic structure

No melting pointExample:Candle





Two categories:




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y gatomic structure

No melting point Randomly arranged structure



Two categories:




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y gatomic structure

No melting point Melting curve



Density and Thermal Expansion - 1

DensityMass per unit volume,i ld h hi hi.e. gold has a higher density than iron.

Thermal expansion effect

Volume or length increases with temperature

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temperature.

Thermal expansion effect


DensityMass per unit volume, Thermal

expansioni.e. gold has a higher density than iron.

Thermal expansion effect Volume or length

increases with temperature

( )00 TTLL =High Temp.

Low

Length athigh Temp.

Length at

expansion coefficient

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Thermal expansion coefficients

LowTemp.

Length atlow temp.




DensityMass per unit volume,i e gold has a higher densityi.e. gold has a higher density than iron

Thermal expansion effect Volume increases

with temperature

Thermal expansion ffi i t

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coefficients

Applications Used in Thermostat

Bimetallic strips

Deformation of Solids

Stress Force causing deformation

StrainDegree of deformation

Elastic behavior Proportional limit Elastic limit

Proportional limitElastic

limit

Elastic behavior

Plastic behavior

Breakingpoint

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Elastic limit Plastic behavior Breaking point

Stress-strain curve



Three Types of Elastic Modulus - 1

Youngs modulus

Definitions:

stresstensileY AF

Force

Cross area

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straintensileY

0LL

Length change

Original length


Shear modulus

Definitions:

AF

Tangent force

Cross area

stressShearG =

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hx

area

Tangentdisplacement

Original thickness

strainShearG =




Bulk modulus

Definitions:

AF

Normal force

Normal areastressVolumeB =

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VV

Volume change

Original Volume

strainVolumeB =

Example: Squeezing a SphereA sphere, with volume of 0.50 m3 and bulk modulus of 8 GPa, is immersed in water deeply. The pressure on it is 20 MPa. What is the change in volume of the sphere?

Solution:Step 1: By the definition of bulk modulus

Step 2: Finally we get

( )( )V/V

F/AB =BV

BV

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Step 2: Finally we get ( )BF/AV

V =Step 3: Substituting the parameters, we get

Pressure

33m101.25V =



Melting process

Question: States of Solids

What are the physical differences between crystalline and amorphous?

Melting process

Definite shape and volume

Micro-structures

Transparency

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Transparency

Conductivity

Melting process

Answer: States of Solids

What are the physical differences between crystalline and amorphous?

Melting process

Definite shape and volume

Micro-structures

Transparency

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Transparency

Conductivity

Crystalline solids have a fixed melting point and defined micro-structures. Amorphous solids have neither.



Static Fluids

The mechanical properties of static fluids

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Pressure in Fluids

Equilibrium equation for arbitrary portion in fluidsy p

Pressure in fluids

0APMgPA 0 =AhM =

ghPP 0 +=

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The forces on a water volume



Pascals Principle

Pascals principle: The pressure at a depth of h below the surface of a fluid open to the atmosphere is greater than atmospheric pressure by the amount gh

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Same depth will have the same pressurePA = PB = PC = PD

Pressure is the same at all points that have the

same elevation

Pressure Transmission - 1Application of Pascal principle Hydraulic jack

Fluid inside the jack is compressed through a pistol.

This pressure is transmitted to the larger output piston that creates a great force to

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Hydraulic jackcreates a great force to lift objects.



Pressure Transmission -2

Pressure can be transmitted in fluid.

Application of Pascal principle Hydraulic jack

Pressures at A1 and A2are the same.PA1 = F1/A1 = F2/A2 = PA2

Force can be amplifiedIf A1 F1

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Stroke is reduced by energy conservation law and continuity law

Illustration of operating Mechanism forhydraulic jack

Buoyancy - 1

What is buoyancy?

The lifting force for objects in fluids

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They are lifted by buoyancy



Buoyancy - 2

What is buoyancy?The lifting force for objects in fluids

What causes buoyancy?Pressure difference on objects(1) Upper pressure(2) Lower pressure

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(2) Lower pressure(3) Resultant force

imgVB =Buoyancy of an objectImmersed in fluids

Buoyant force, N

Archimedes Principle

Archimedes principle:Any body completely orAny body 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 body.

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imgVB = Archimedes 287-212B.C.

Volume immersed m3

Acceleration of gravity 9.8m/s2

Density of material

Kg/m3



Question: Hot Air BalloonWhy can a hot air balloon fly in the sky?

XX ___________

X

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X ___________

Answer: Hot Air BalloonWhy can a hot air balloon fly in the sky?

X Density of inner air X ___________

X

decreases

Buoyancy generated

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X ___________y y g



Surface Tension

Questions:

Why insects can walk on water?

Why can razor blades float on water?

Why can water raise in

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Why can water raise in thin tubes?

Why does this happen?

Surface Tension- 2

Questions-Why can insects walk on water? -Why can razor blades float on water?-Why can water raise in thin tubes?

Surface tensionThe tendency for the Reason

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surface of a liquid to contract in area.

Why does this happen?



Surface Tension- 3

Questions-Why can insects walk on water? Why can razor blades-Why can razor blades

float on water?-Why can water raise in thin tubes?

Surface tensionThe tendency for the surface of a liquid to

t t i

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Mathematical definition

2LF

=Apparatus for testing

Surface tensionInside edge and outside edge

contract in area.

Capillary Action - 1

Two types of fluid-surface interactions:

Mercury vs Glass

Water vs Glass

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Left: unwettableRight: wettable




Two types of fluid-surface interactions: Unwettable Wettable

Unwettable(water vs paraffin)

Contactangle

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Wettable(water vs glass)


Two types of fluid-surface interactions: Unwettable Unwettable Wettable

Water rise for wettable fluid-surface interaction( )cosr 2Fv =

Balanced by:

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hgrw 2=Balanced by:

Yields:

cos gr2h =

Vertical component of surface forceWeight of

the watervolume

Water rise due to capillary action



Question: Water Goes Up or Down?If a thin tube made of paraffin is inserted in water, the level of water inside the tube will be than the outside water. (water-paraffin is unwettable)

higher

lower

same

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uncertain

Answer: Water goes up or down?If a thin tube made of paraffin made is inserted in water, the level of water inside the tube will be than the outside water. (water-paraffin is unwettable)

higher

lower

same

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uncertain



Fluids in Motion

Now well study the properties of fluids that are moving

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fluids that are moving.

Fluid in Motion: A Preview

Properties of fluids:Density, gravity, viscosity

Continuity Laminar and turbulent

flow

Bernoullis Equation for steady ideal flow

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steady ideal flow



Density and Gravity

Definition of densityMass per unit volume (fluids take indefinite shape but definite volume)p )

GravityThe force between the earth and the matter. It is governed by Newtons law of gravitation.

This force is pointing downwards (toward the earth) and is proportional to the mass

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) p pof the matter by the scaling factor of 9.8 m/s2, which is denoted by g.

For fluids, the force per volume is used instead of the force itself.

Viscosity

DefinitionInternal friction of a fluid; damping mechanism of fluids

Coefficient of viscosity

SvFl

ratestrainshearstressshear

==

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Viscous flow



Viscosity Applications

DefinitionInternal friction of a fluid; damping mechanism of fluids

Coefficient of viscosity

SvFl

ratestrainshearstressshear

==

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ApplicationsDashpot or damper

Configuration and products

The Equation of Continuity

ContinuityThe net rate of flow of mass inward across any closed surface is equal to the rate of increase of the mass within the surface

Mathematical expression222111 vAvA =

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For non-compression fluid222111

constant 21 =Yields:

2211 vAvA =

Flow into and out of a portion of a tube



Laminar and Turbulent Flow - 1

Laminar or streamline Every particle y p

passing a particular point moves exactly along the smooth path followed by particles passing that point earlier.

Laminar(Streamline)

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Different streamlines cannot intercross.

Velocity is low.Smoke from a cigarette

Cigarette

Laminar and Turbulent Flow - 2

Turbulent flow Motion is highly

Turbulent flow

g yirregular.

Vortices are developed within the fluid.

Resistance to the flow is increased

Laminar(Streamline)

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flow is increased.

Velocity is high.

Smoke from a cigarette

Cigarette



Steady Ideal Flow

Ideal flow:Non-viscous & non-compressible

The work at lower end

The work at upper end

VPxFW 1111 ==

VPW 22 = Ideal flow in a pipe

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Work and energy law

( )12212221 gVygVyVv21

Vv21WW +

=+

Kinetic energy Potential energyWork

Bernoullis Equation

Mathematical expression

Bernoullis Equation

Physical meaning:

constantgyv21P 2 =++

For an ideal flow, the sum of the pressure (P),

Ideal flow in a pipe

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, p ( ),

the kinetic energy per unit volume (1/2v2) ,

and the potential energy per unit volume(gy)

has the same value at all points along a stream line.



Application for Bernoullis Equation

Velocity at 2 is zero

Velocity at 1 is to be measured

Pitot tube

Illustration of pitot tube

Velocity at 1 is to be measured

Heights of 1 and 2 are the same

By Bernoullis equation

12112

222 gyv2

1Pgyv21P ++=++ =0

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111222 gy2gy

2Same

3l1221 ghPPv2

1 ==Density of the liquid in the tube

3l

1 gh2v =

yields

Example: Speed at a Drain HoleConsider a tank with a drain hole at its bottom. The cross-sectional area of the tank is large relative to that of the hole. The water level drops slowly so that we assume V2=0. Find the speed at which the water leaves the hole 2 pwhen the water level is 0.5m above the hole.

Solution:Step 1: By Bernoullis Equation


22201

210 gyv2

1Pgyv21P ++=++

=0

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Step 3: Substituting the parameters, we get

( )121 yy2gv =

3.13m/sv 1 =



Question: Aerodynamic Force on Wings

Why can a fixed-wing aircraft fly?Tip: Consider and apply Bernoullis equation

Air flow near the upper surface of the wing is moving faster than that near its lower surfaceAn air flow blowing toward the wing

The pressure on the upper surface of the wing is smaller than that near its lower surface

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Other mechanism

Answer: Aerodynamic Force on Wings

Why can a fixed-wing aircraft fly?Tip: Consider and apply Bernoullis equation

Air flow near the upper surface of the wing is moving faster than that near its lower surfaceAn air flow blowing toward the wing

The pressure on the upper surface of the wing is smaller than that near its lower surface

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Other mechanism



Ideal flow: BernoullisIdeal flow: Bernoullis

States of matter: Solid,

States of matter: Solid, Applications Applications

Learning Summary

Static behavior ofStatic behavior of

Bernoullis equation

Bernoullis equation

Liquid and Gas

Liquid and Gas

of fluid flowof fluid flow

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Static behavior of fluid: pressure, buoyancy, and surface tension

Static behavior of fluid: pressure, buoyancy, and surface tension

Solids and elasticity

Solids and elasticity

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