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Class #3

Vectors and Dot productsRetarding forces! Stokes Law (viscous drag)! Newton’s Law (inertial drag)! Reynolds number! Plausibility of Stokes law! Projectile motions with viscous drag! Plausibility of Newton’s Law! Projectile motions with inertial drag

Worked ProblemsHomework Review

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Vectors and Central forces

Vectors! Many forces are of

form! Remove dependence

of result on choice of origin

1 2r r−! !

1r!

2r!

Origin 1Origin 2

1 2( )F r r−! ! !

2r ′!

1r ′!

1 2r r′ ′−! !

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Vector relationships

Vectors! Allow ready

representation of 3 (or more!) components at once." Equations written in

vector notation are more compact

zdtdzy

dtdyx

dtdx

dtrd ˆˆˆ ++=!

xxx!

=ˆ rrrr !!! ⋅≡≡

∑=

=

≡⋅3

1

)cos(

iii sr

srsr θ!!

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Dot product is a “projection” operator

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O

m

h α

β

βα

ˆ ' cos

ˆ ' cos siny

x

W W y W

W W x W W

α

β α′

′

= =

= = =

!i!i

y

x

ˆ 'y ˆ 'xα

β

Block on ramp with gravity

Choose coordinates consistent with “constraints”

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Defining Viscosity

Two planes of Area “A” separated by gap Top plane moves at relative velocity

defines viscosity (“eta”)

MKS Units of are Pascal-seconds Only CGS units (poise) are actually used1 poise=0.1 =0.1 Pouseille

y

x

xu ˆ∆y∆

y∆xu ˆ∆

A

yuAF

∆∆= η

!

yuAFdrag ∆

∆= η!

η

dragF!

η 2/ msN ⋅

2/ msN ⋅ :30

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Viscosity of Common Substances

0.018Air (STP)

1E15Window Glass

~250,000Peanut Butter

~10,000Honey (20 C)

~20030 Weight motor oil

~84Olive Oil

~20Antifreeze

~1.00.28

Water (@20C)Water (@99C)

Viscosity (CentiPoise)

Substance

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Viscous Drag I

An object moved through a fluid is surrounded by a “flow-field” (red).

Fluid at the surface of the object moves along with the object. Fluid a large distance away does not move at all.

We say there is a “velocity gradient” or “shear field”near the object.

Molecules (particularly long molecules) intertwine with their faster moving neighbors and hold them back. This “intermolecular friction” causes viscosity.

xu ˆA x

dyduAFdrag ˆη−=

!dragF!

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Viscous Drag II

“k” is a “form-factor” which depends on the shape of the object and how that affects the gradient field of the fluid.

“D” is a “characteristic length” of the objectThe higher the velocity of the object, the

larger the velocity gradient around it.Thus drag is proportional to velocity

xu ˆ

DxuDkFdrag ˆη−=

!dragF!

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Viscous Drag III – Stokes Law

Form-factor k becomes “D” is diameter of sphereViscous drag on walls of

sphere is responsible for retarding force.

George Stokes [1819-1903] #(Navier-Stokes equations/ Stokes’ theorem)

xu ˆD

xuDFdrag ˆ3 ηπ−=!dragF

!

π3

rbFdrag#!!

=

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Falling raindrops I

Problems:

A small raindrop falls through a cloud. At time t=0 its velocity is purely horizontal.

Describe it’s velocity vs. time.

Raindrop is 10 µm diameter, density is 1 g/cc, viscosity of air is 180 µPoise

Work the same problem with a 100 µm drop.

dragF!

gm !

z

x

ρη ,

20 /ˆ3 smxv =!

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Falling raindrops II

1) Newton

2) On z-axis

3) Rewrite in terms of v

4) Variable substitution

5) Solve by inspection

zz

zz

vmbuv

mbguDefine

vmbg

dtdv

zbmgzm

rbzmgrm

##

###

#!##!

−=⇒−=

−=

−=

−=motionverticalAssume

ˆ

0

b tmbu u u u e

m−

= − ⇒ =#

dragF!

gm !

z

x

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Falling raindrops III

1) Our solution

2) Substitute original variable

3) Apply boundary conditions (v0=0)

4) Expand “b”

5) Define vterminal

ubm

bmgvv

mbgu zz −=⇒−≡0

b tmu u e

−=

0( )b tm

zmg m bv g v eb b m

−= − −

−=

− tmb

z eb

mgv 1

−=

− tmD

z eD

mgvηπ

ηπ

3

13

ηπDmgvt 3

≡

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1g tv

zv v e ττ

− = −

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Velocity Dependent Force

$Forces are generally dependent on velocity and time as well as position

$Fluid drag force can be approximated with a linear and a quadratic term

= Linear drag factor(Stokes Law, Viscous or “skin” drag)

= Quadratic drag factor( Newton’s Law, Inertial or “form” drag)

2)( rcrbrFr#!#!#!! +=

),,( trrFF #$$!!=

b

c

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quad

lin

fRatio

fis important

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The Reynolds Number

R < 10 – Linear drag1000< R < 300,000 – Quadratic R > 300,000 – Turbulent

( )( )

inertial quad dragRviscous linear drag

=densityviscosity

ρη

D

v

ηρ DvR =

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Reynolds Number Regimes

R < 10 – Linear drag1000< R < 300,000 – Quadratic R > 300,000 – Turbulent

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The Reynolds Number II

( )( )

inertial quad dragRviscous linear drag

=

DR OR

DR wherev

v

ρη

ηρ

≡

≡

=

v

v

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2

2

163

163

inertial

viscous

inertial

viscous

F D

F D

DF DK K RF D

π ρ

π ηπ ρ ρ

π η η

=

=

= = × ≡ ×

2

2

v

v

v vv

vD

densityviscosity

ρη

“D”= “characteristic” length

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The Reynolds Number III

R < 10 – Linear drag1000< R < 300,000 – Quadratic R > 300,000 – Turbulent

1 22

1 222

1 22

1 22

1 (1 / #)R

dD

Linear Regime

D

Quadratic Regime

D

FCv A

kD v DCvDv A

Reynolds

kA vC k

v A

ρ

η ηρρ

ρρ

≡

=

=

=

∼

∼

D

v

ηρ DvR =

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Vector Relationships -- Problem #3-1“The dot-product trick”

Given vectors A and B which correspond to symmetry axes of a crystal:

Calculate:

Where theta is angle between A and B

xA ˆ2=!

zyxB ˆ3ˆ3ˆ3 ++=!

θ,, BA!!

A

B

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Falling raindrops L3-2

A small raindrop falls through a cloud. It has a 10 µm radius. The density of water is 1 g/cc. The viscosity of air is 180 µPoise.

a) Quantify the force on the drop for a velocity of 10 mm/sec.

b) What should be the terminal velocity of the raindrop?

c) What is the Reynolds number of this raindrop?

Work the same problem with a 100 µm drop.

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Pool Ball L3-3

A pool ball 6 cm in diameter falls through a graduated cylinder. The density of the pool ball is 1.57 g/cc. The viscosity of air is 180 µPoise.

a) Quantify the force on the ball for a velocity of 100 mm/sec.

b) What should be the terminal velocity of the ball?

c) Quantify the force if we assume quadratic drag

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2 2

16dragF D vπ ρ=!

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Stokes Dynamics

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