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Remaining ScheduleRemaining Schedule

• Friday April 27Friday April 27– Review for Quiz 4Review for Quiz 4

• Monday April 30Monday April 30– Quiz 5Quiz 5– 50 points50 points– Crib SheetCrib Sheet

• Wednesday May 2Wednesday May 2– Final ReviewFinal Review

• Wednesday May 9 noon-1:50PMWednesday May 9 noon-1:50PM– Final!Final!– 100 points100 points– Open bookOpen book

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The Final Four LessonsThe Final Four Lessons

• Chapter 11: General RotationChapter 11: General Rotation• Vectors, Torque, Angular Momentum Vectors, Torque, Angular Momentum

(10-4,11-1, 11-2, 11-3)(10-4,11-1, 11-2, 11-3)• Angular Momentum & Torque for Angular Momentum & Torque for

Systems and Rigid Bodies, (11-4, 11-5, Systems and Rigid Bodies, (11-4, 11-5, 11-6)11-6)

• Conservation of Angular Momentum and Conservation of Angular Momentum and Special Topics (11-7, 11-8, 11-9, 11-10)Special Topics (11-7, 11-8, 11-9, 11-10)

• Chapter 12: Static Equilibrium (12-1, Chapter 12: Static Equilibrium (12-1, 12-2, 12-3, 12-4)12-2, 12-3, 12-4)

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Vector Nature of Angular Vector Nature of Angular QuantitiesQuantities• As we’ve discussed in the As we’ve discussed in the

past rotation has a sense past rotation has a sense of direction.of direction.

• Both angular velocity and Both angular velocity and acceleration can be acceleration can be treated as vectors once treated as vectors once we define their direction.we define their direction.

• Consider a spinning wheelConsider a spinning wheel– The direction can’t be The direction can’t be

given by the linear given by the linear velocity since it points in velocity since it points in all directions.all directions.

– The only special direction The only special direction is the axis of rotation is the axis of rotation which is perpendicular to which is perpendicular to the motion. the motion.

– This is a natural choice This is a natural choice for the direction of for the direction of angular velocity and angular velocity and acceleration acceleration

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• We’ll use the axis of We’ll use the axis of rotation for rotation for direction. But direction. But there’s still an there’s still an ambiguity since ambiguity since could point either could point either way along the axis.way along the axis.

• Enter theEnter the Right-hand Right-hand rule: If your fingers rule: If your fingers curl around the axis curl around the axis and along the and along the direction of motion, direction of motion, then the thumb then the thumb points in the points in the direction of the direction of the angular velocity.angular velocity.

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• If the rotation axis is fixedIf the rotation axis is fixed– Only the magnitude of Only the magnitude of

can changecan change– Since Since =d=d/dt, the angular /dt, the angular

acceleration also points acceleration also points along the axis of rotation. along the axis of rotation. But not necessarily in the But not necessarily in the same direction as same direction as ::• For instance if the For instance if the

rotation in the figure is rotation in the figure is CCW and CCW and is increasing is increasing is increasing and in is increasing and in the same direction or the same direction or upwardupward

• However, if However, if is is decreasing decreasing points in points in the opposite direction the opposite direction or downwardor downward

• If the axis is not fixedIf the axis is not fixed– still points along the still points along the

axisaxis– But But cannot point along cannot point along

the axisthe axis

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Vectorial Treatment of Vectorial Treatment of and and ..

• As you might expect as with linear motion a As you might expect as with linear motion a more precise vector treatment of angular more precise vector treatment of angular motion will be helpful.motion will be helpful.

• We already have a hint that something We already have a hint that something more is needed since we don’t have any more is needed since we don’t have any machinery to deal with axes of rotation that machinery to deal with axes of rotation that change direction with time.change direction with time.

• The motion of objects not constrained about The motion of objects not constrained about a fixed axis is very complicated but a fixed axis is very complicated but immensely rich and rewarding…spinning immensely rich and rewarding…spinning tops, the earth, hurricanes…tops, the earth, hurricanes…

• We’ll look at a few illustrative examples but We’ll look at a few illustrative examples but first we need to define a new vector first we need to define a new vector quantity to deal with angular momentum quantity to deal with angular momentum and torque.and torque.

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The “Vector Cross Product”The “Vector Cross Product”

• Also called the “vector Also called the “vector product” or the “cross product” or the “cross product”.product”.

• An operation between two An operation between two vectors A & B that creates a vectors A & B that creates a 33rdrd vector C. vector C.

• The The magnitude magnitude of the new of the new vector is given by C=ABsinvector is given by C=ABsin where where is the smallest is the smallest angle between A and Bangle between A and B

• The The direction direction is is perpendicular to the plane perpendicular to the plane of A and B and given by of A and B and given by applying the RHR:applying the RHR:– Start w/ fingers along A.Start w/ fingers along A.– Curve fingers toward B.Curve fingers toward B.– Thumb in direction of C.Thumb in direction of C. Rule. HandRight se U

B andA lar toPerpendicu :Direction

sin :Magnitude

:ProductVector

ABC

BAC

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Properties of the Cross ProductProperties of the Cross Product

directions oppositein point they RHR by theBut

sin magnitude haveboth Since

00sin magnitude theSince 0

AB

ABBA

AAAA

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kjijikikj

ikji

kj

kkjjii

)())((ii

ˆˆˆ,ˆ ˆˆ,ˆˆˆ :rsunit vecto threeallfor trueisrelation similar A

ˆ ˆˆ Thus .ˆdirection has is RHR by the And

190)(1)(1)sin( magnitude has ˆˆ handother On the

0ˆˆ,0ˆˆ,0ˆˆ :rsunit vecto threeallfor trueis This

00sin11 magnitude has ˆˆ

1length have rsunit vectoRember

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In Vector NotationIn Vector Notation

kBABAjBABAiBABABA

kjijikikj

jiBABAikBABAkjBABABA

jkkjikkiijji

jkBAikBAkjBAijBAkiBAjiBA

jBiBkAkBiBjAkBjBiABA

kkjjii

kBjBiBkAkBjBiBjAkBjBiBiA

kBjBiBkAjAiABA

xyyxzxxzyzzy

xyyxzxxzyzzy

yzxzzyxyzxyx

yxzzxyzyx

zyxzzyxyzyxx

zyxzyx

ˆ)(ˆˆ)(

ngsubstituti ,ˆˆˆ,ˆ ˆˆ,ˆˆˆ RHR theFrom

)ˆˆ)(()ˆˆ()ˆˆ)((

ngsubstituti ,ˆˆˆˆ,ˆˆˆˆ,ˆˆˆˆ RHR theFrom

)ˆˆ()ˆˆ()ˆˆ()ˆˆ()ˆˆ()ˆˆ(

)ˆˆ(ˆ)ˆˆ(ˆ)ˆˆ(ˆ

thus,0ˆˆ,0ˆˆ,0ˆˆ

zero is vectorsparallel ofproduct cross theSince

)ˆˆˆ(ˆ)ˆˆˆ(ˆ)ˆˆˆ(ˆ

)ˆˆˆ()ˆˆˆ(

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1111

kBABAjBABAiBABA

B BB

A AA

k j i

BA

xyyxzxxzyzzy

zyx

zyx

ˆ)(ˆˆ)(

:rsunit vectoth product wi cross theof structure the

remember onotation tt determinan theusecan We

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kBABAjBABAiBABA

B BB

A AA

k j i

BA

xyyxzxxzyzzy

zyx

zyx

ˆ)(ˆˆ)(

:rsunit vectoth product wi cross theof structure the

remember onotation tt determinan theusecan We

Need a negative sign here!

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kBABAjBABAiBABA

B BB

A AA

k j i

BA

xyyxzxxzyzzy

zyx

zyx

ˆ)(ˆˆ)(

:rsunit vectoth product wi cross theof structure the

remember onotation tt determinan theusecan We

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More PropertiesMore Properties

dt

BdAB

dt

Ad

dt

BAd

CABACBA

)(

rs)unit vecto usingproven becan (which

)(

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Torque is a Vector Product!Torque is a Vector Product!

Fr τamF

Fr τ

Fr

θrFrFτ

τFr

Fr

Iατ

F

and analogy

thecompleted now have weNotice .bygiven

exactly is torqueofdirection and magnitude esummary thIn

. of magnitude theas same the

is which sin is torque theof magnitude The

:magnitude Now

direction. same in thepoint all and,, , , words

otherIn page! theofout points also toapplied rule

handright The page. theofout pointing be alsomust

then Since page. theofout pointing are and

rule handright by the rotation,CCW inducing isˆ Since

:FirstDirection

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Formal Definition of TorqueFormal Definition of Torque

Fr τ

r

Fr τ

ii

)(

bygiven is particles of

system aon torque totalThe particle. theand

Obetween ector position v theis where

by give is Opoint aabout torquethe

F force subject to m mass of particle aFor

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jmNkjNmi

N

m m .

k j i

Fr

iN)(Fkm).(im).(r

1800)150)(2.1(0

00150

2.10 21

:help onotation tt determinan theuse We

object? on the torque theisWhat

.150 and 2121 Suppose

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Angular Momentum is a Vector Angular Momentum is a Vector ProductProduct

• Newton’s second law can be written as Newton’s second law can be written as FF=d=dpp/dt./dt.

• By the linear motionBy the linear motionangular motion angular motion analog we expect something similar for analog we expect something similar for =d=dLL/dt. /dt.

• That is a vector form of the scalar analogy That is a vector form of the scalar analogy we developed in Chapter 10, but we were we developed in Chapter 10, but we were limited to rotation about an axis.limited to rotation about an axis.

• Since torque is vector product then L Since torque is vector product then L should be too! should be too!

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• Let’s consider a Let’s consider a particle of mass m particle of mass m with momentum with momentum pp and position vector and position vector rr with respect to with respect to the origin O in an the origin O in an inertial reference inertial reference frame.frame.

• Let’s also assume Let’s also assume the definition of the the definition of the angular momentum angular momentum is the vector is the vector product: product: ll==r r xx p p..

• Then the magnitude Then the magnitude is given by rpsinis given by rpsin and the direction by and the direction by the RHR.the RHR.

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The connection between The connection between and l.and l.

dt

ldτ

τdt

ld

Frdt

ld

dt

ld

dt

pdF

dt

pdr

dt

ldvvmvmvpvp

dt

rd

dt

pdrp

dt

rdpr

dt

d

dt

ld

prl

:net torque theequals

momentumangular of change of rate time theIn words bodies. rigid rotating

for result earlier our with consistent andfor looking were what weis This

. particle on the net torque theis side handright

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that know wehold) Laws sNewton' (where frame reference inertialan in Now

. so ,0)(

:first term heConsider t . )(

be wouldsidesboth of ederiviativ The .nsuppositioBy

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Student EvaluationsStudent Evaluations

• Now’s your chance to evaluate teaching Now’s your chance to evaluate teaching effectivenesseffectiveness

• Assessment will got to personnel Assessment will got to personnel committee and to instructor.committee and to instructor.

• More importantly an opportunity to More importantly an opportunity to improve instruction.improve instruction.

• Your comments/criticisms are very Your comments/criticisms are very welcome.welcome.

• As usual, one of your colleagues will give As usual, one of your colleagues will give you instructions and collect them.you instructions and collect them.

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Filling out the Scan-tronFilling out the Scan-tron

• ““LAST NAME”: BlazeyLAST NAME”: Blazey• ““INIT.”: GINIT.”: G• First four boxes of “ID Number”: First four boxes of “ID Number”:

31303130• ““SEC”: 1SEC”: 1• ““DEPT.” : PHYSDEPT.” : PHYS• ““COURSE”: 253COURSE”: 253• ““DATE”: 4/20/07DATE”: 4/20/07