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Lecture #5 of 25

Moment of inertiaRetarding 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

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2

Moment of inertia L5-1

Given a solid quarter disk with uniform mass-density and radius R: Calculate I total Write r in polar coords Write out double integral, both r and phi

components Solve integral

rO1

R

Calculate

Given that CM is located at (2R/3, Calculate ICM

1OI

22 ( )CMI r dm r r dA 22222222222222

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3

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

f

is important

4

The Reynolds Number

R < 10 – Linear drag1000< R < 300,000 –

Quadratic R > 300,000 – Turbulent

( )

( )

inertial quad dragR

viscous linear drag

density

viscosity

D

v

Dv

R

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5

The Reynolds Number II

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

Dv

R

1 22

1 222

1 22

1 22

1

Re

dD

Linear Regime

D

Quadratic Regime

D

FC

v

kD v DC

vDv A

kA vC k

v A

density

viscosity

Dv

Dv

R

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6

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

y

x

xu ˆy

yxu ˆ

A

y

uAF

y

uAFdrag

dragF

2/msN

2/msN :30

7

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.

We are changing the momentum of the nearby fluid.This dp/dt creates a force which we call the viscous

drag.

xu ˆA x

dy

duAFdrag ˆ

dragF

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8

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 object

The higher the velocity of the object, the larger the velocity gradient around it.

Thus drag is proportional to velocity

xu ˆ

DxuDkFdrag ˆ

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 dragF

3

rbFdrag

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Falling raindrops L5-2

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

a) Draw the free-body diagram.b) Quantify the force on the drop for a velocity

of 10 mm/sec.c) What is the Reynolds number of this raindropd) What should be the terminal velocity of the

raindrop?

Work the same problem with a 100 m drop.

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11

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 m diameter, density is 1 g/cc, viscosity of air is 180 Poise

Work the same problem with a 100 m drop.

dragF

gm

z

x

,

20 /ˆ3 smxv

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12

Falling raindrops II

1) Newton

2) On z-axis

3) Rewrite in terms of v

4) Variable substitution

5) Solve by inspection

zz

zz

vm

buv

m

bguDefine

vm

bg

dt

dv

zbmgzm

rbzmgrm

motionverticalAssume

ˆ

tm

b

euuum

bu

)0(

dragF

gm

z

x

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Falling raindrops III

1) Our solution

2) Substitute original variable

3) Apply boundary conditions

4) Expand “b”

5) Define vterminal

ub

m

b

mgvv

m

bgu zz

tm

b

euu

)0(

tm

b

z eub

m

b

mgv

)0(

tm

b

z eb

mgv 1

tm

D

z eD

mgv

3

13

Dmg

vt 3

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1gt

vzv v e

14

Stokes Dynamics

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Lecture #5 Wind-up

.

.

.

Read sections Taylor 2.1-2.4Office hours today 3-5Registration closes Friday

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Dv

R

1gt

vzv v e

Dmg

vt 3