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Chapter 12:

Mechanics 2: Linear & Rotational

Dynamics

Ian ParberryUniversity of North Texas

Fletcher DunnValve Software

3D Math Primer for Graphics & Game Development

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What Youll !ee in "his Chapter

"his chapter consi#ers the cause o$ motion% its orientation% an# ho e

mi'ht 'o about simulatin' it on a computer( It is #i)i#e# into si* sections(

!ection 12(1 'i)es an o)er)ie o$ +etons , las(

!ection 12(2 tal-s about the cause o$ motion: the $orce(

!ection 12(, intro#uces momentum(

!ection 12(. loo-s at collisions an# impulse(

!ection 12(/ is about rotational #ynamics(

!ection 12(0 #iscusses #i'ital simulation o$ mechanics(

Chapter 12 +otes ,D Math Primer $orraphics & ame De)

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Wor# Clou#


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!ection 12(1:

+etons , Las


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!ir Isaac +eton

!ir Isaac +eton establishe# three

simple las that pro)i#e a $rameor-%

hich e call Newtonianor classical

mechanics(

It #oesnt hol# at hi'h spee#s or small#istances% but its 'oo# enou'h $or

e)ery#ay li$e% an# )i#eo 'ames(

Ima'e $rom Wi-ime#ia Commons(3


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+etons First La

4)ery bo#y persists in its state o$ bein' at

rest or o$ mo)in' uni$ormly strai'ht $orar#%e*cept inso$ar as it is compelle# to chan'e

its state by $orce impresse#(


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+etons !econ# La

"he acceleration o$ a bo#y is proportional to

an# in the same #irection as3 the net

e*ternal $orce actin' on the bo#y% an#

in)ersely proportional to the mass o$ the

bo#y:

(


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"he Force

Force is a )ector( It has units li-e

-'(m6sec2% also calle# a Newton(

7Duct tape is li-e the $orce( It has a li'ht

si#e% a #ar- si#e% an# it hol#s the uni)erse

to'ether(8 Carl 9ani'3


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Free se +eton?s 2n# la to compute the

acceleration o$ the ob=ect(.( Inte'rate the acceleration to #etermine the

motion o$ the ob=ect(


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Di$$erential 4Auations

When sol)in' problems analytically% this means

sol)in' #i$$erential eAuations( We #on?t use any

#i$$erential eAuations in this boo- because there

are only a $e simple cases that e ill loo- atanalytically( +umerical metho#s o$ inte'ration must

be use#( Later% e e*amine 4uler inte'ration%

hich is the most simple metho# ima'inable% but

also the one use# by most realBtime ri'i# bo#ysimulators(


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Inertial Re$erence Frames

"his only or-s in a re$erence $rame that is not

acceleratin'(

You ha)e to in)ent $ictional $orces to e*plain

hy ob=ects are not acceleratin' accor#in' to

+etons 1st an# 2n# las(

robot in a $allin' ele)ator is in a noninertial

$rame( Ee must in)ent a $ictitious upar# $orce

to counteract 'ra)ity to e*plain hy his herrin'

san#ich #oesnt $all(


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12

"o a passin' alien ho is not acceleratin'%

+etons las or- =ust $ine% an# there is nonee# to in)ent a $ictional $orce(

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+etons "hir# La

"o e)ery action there is alays an eAual

an# opposite reaction( r% the $orces o$ tobo#ies on each other are alays eAual an#

are #irecte# in opposite #irections(


1,

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4*ample

"here are $our $orces here(

1( Moe pushin' the bo*(

2( "he bo* pushin' Moe(

,( Moe pushin' the 4arth(.( "he 4arth pushin' Moe(

+ote that an# cancel out(


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!ection 12(2:

!ome !imple Force Las


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ra)ity in the Real Worl#

+etons La o$ >ni)ersal ra)itation:

here is the ma'nitu#e o$ the $orce% an#are the masses o$ the to ob=ects% an# is the

#istance beteen their centers o$ mass% an#

(


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Gi#eo ame ra)ity

% here (

"his is not physically accurate% but then

a'ain% neither is bein' able to =ump to orthree times your on hei'ht% steer in mi#air%

or #ouble =ump( When it comes to =umpin' in

)i#eo 'ames% reality is not =ust o)errate#% it?s

completely i'nore#( It =ust #oesn?t $eel ri'ht(


1@

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Gi#eo ame ra)ity

In most $irstBperson shooters% hen you =ump% you are 'i)en

an initial burst o$ upar# )elocity% an# then your position is

simulate# =ust li-e e)ery other airborne ob=ect in the orl#(

In most thir#Bperson 'ames your character ill sprin' up

almost instantaneously an# reach a ma*imum hei'ht )eryAuic-ly( In many 'ames the character ill ho)er there% then

slam bac- #on on the 'roun# as Auic-ly as it rose up%

perhaps lea)in' a crater behin#(


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Gi#eo ame ra)ity

!imulatin' a =ump mechanic usin' a )alue o$ may be e)en orse% because

most players e*pect a =ump to ta-e a certain amount o$ time but also e*pect

to be capable o$ =umpin' to unrealistic hei'hts(

When realBorl# 'ra)ity is use# to attain these hei'hts% the player is in the air

too lon'% an# it $eels 7$loaty8(

Many arca#e racin' 'ames increase 'ra)ity to 'et the car bac- on the 'roun#

Auic-ly( "he player ants to be in $ull control a'ain as Auic-ly as possible% an#

aitin' $or realBorl# 'ra)ity to 'et them bac- #on usually ta-es too lon'(

"here are other racin' 'ames that use a )alue o$ 'ra)ity that is less than the

real orl# )alue% to $acilitate unrealistic =umps at realistic )ehicle spee#s(


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Gi#eo ame ra)ity

"here are also reasons to $i##le ith 'ra)ity $or nonBplayerBcharacter

ob=ects as ell( !ometimes realBorl# 'ra)ity can create an 7ob=ects

ma#e o$ styro$oam8 $eelin'% so 'ra)ity is increase# to 'et an ob=ect to

tip o)er an# come to rest more Auic-ly(

In other situations% an arti$icially lo )alue o$ 'ra)ity can ma-e alar'e ob=ect seem e)en more massi)e especially hen

accompanie# by the ri'ht soun# e$$ects3% because acceleration on

4arth is constant an# is one o$ a $e cues humans instincti)ely use

to establish an absolute scale $or ob=ects in the #istance(


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Realism )ersus 4ntertainment

What 7$eels ri'htH is a sub=ecti)e matter( It is base# more on

player e*pectation than physical reality(

In the en#% hat matters most in a )i#eo 'ame is not hat?s

'oin' on in the CP> or e)en on the screen% but hat is 'oin'

on in the player?s min#( "he human min# is hi'hlysusceptible to su''estion(

"he Auest $or realism shoul# ne)er be an en# unto itsel$(

success$ul )i#eo 'ame ill harness realism only here it

ser)es the ultimate 'oal% hich is entertainment(


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Friction

"he stan#ar# #ry

$riction mo#el is

sometimes calle#

"o#lom! friction(

CharlesBu'ustin #e

Coulomb 15,0B1;03(

Ima'e $rom

Wi-ime#ia Commons(3


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!tatic Friction

When an ob=ect is at rest on top o$ another

ob=ect% a certain amount o$ $orce is

reAuire# to 'et it unstuc- an# set it in

motion(

I$ any less $orce is applie#% the $orce o$

$riction ill push bac- ith a counteractin'

$orce up to some ma*imum amount(

"his is calle# static friction(


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!tatic Friction

"he $olloin' eAuation is a 'oo#

appro*imation $or the ma*imum ma'nitu#e

o$ static $riction: (

is a constant calle# the coefficient of static

frictionthat #epen#s on the type o$

sur$aces rubbin' to'ether( ust loo- it up in

a table(

is the ma'nitu#e o$ the normal $orce(


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"he +ormal Force

"he normal force is the $orce actin' perpen#icular to the

sur$aces that pre)ent them $rom o)erlappin'(

For e*ample% hen an ob=ect such as a bol o$

petunias3 is restin' on top o$ another ob=ect such as atable3% the normal $orce is the $orce reAuire# to

counteract 'ra)ity(

It is the $orce reAuire# to counteract the component o$

'ra)ity that acts perpen#icular to the sur$aces(


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,

+ot !li#in' n the

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Jinetic Friction

nce static $riction is o)ercome an# the ob=ect is

mo)in'% $riction continues to push a'ainst the

relati)e motion o$ the to sur$aces(

"his is calle# $inetic friction(

"he ma'nitu#e -inetic $riction is 'enerally less

than that o$ static $riction(

Its compute# the same ay o$ static $riction: (


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Coulombs La o$ Friction

"he #irection o$ the $orce o$ -inetic $riction is alays oppose# to the

relati)e motion o$ the sur$aces(

s e sai# earlier% the coe$$icient o$ -inetic $riction is usually less

than the coe$$icient o$ static $riction(

"hus% i$ e increase the an'le o$ the table sloly so that static$riction is =ust o)ercome% the petunias ill be'in to accelerate(

Coulomb?s primary contribution to the theory% sometimes calle#

"o#lom!%s law of friction% as that the $orce o$ -inetic $riction #oes

not #epen# on the relati)e )elocities o$ the sur$aces(


,,

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Control !ystems

"here are to types o$ sprin' motion% ampe oscillation

an# #nampe oscillation(

)irtual sprin' o$ten in the $orm o$ a sprin'B#amper

system3 is a type o$ control system(

"here are certain a#)anta'es to be ha# hen the

physical nature o$ the problem is #roppe# an# e thin- o$

it purely in mathematical terms(

In#ee#% many times the problem as ne)er reallyphysical to be'in ith% an# as only recast in physical

terms so that the sprin'B#amper apparatus coul# be

applie#(


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"he Rest Len'th

Consi#er a sprin' ith one en# $i*e# an# the other

en# $ree to mo)e in one #imension(

When the sprin' is at eAuilibrium ith no e*ternal

$orces on it% it has a natural len'th% calle# the restlength(

I$ e stretch the sprin'% then it ill pull bac- to try to

re'ain its rest len'th(

Li-eise% i$ e compress the sprin'% it ill push bac-(


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Restlen'th

Compress

!tretc

h

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Eoo-es La

Robert Eoo-e 10,/ K15,3(

Ima'e $rom Wi-ime#ia Commons(3

"he ma'nitu#e o$ the restorati)e $orce is

proportional to the #istance $rom the rest

len'th pro)i#e# the $orce #oes note*cee# the elastic limito$ the material

use# to construct the sprin'3(


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Eoo-es La

Where is the spring constant

that #escribes ho sti$$ thesprin' is% is the sprin's rest

len'th% an# is the len'th that

the sprin' has been stretche#

or compresse# to(


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Reritin' Eoo-es La

With those notational chan'es% e can

rerite Eoo-es La as % here is

acceleration as a $unction o$ time an# is

position as a $unction o$ time(

"his is calle# a ifferential e#ation% since

it is an eAuation in both position an# its

secon# #eri)ati)e% acceleration (


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!ol)in' Di$$erential 4Auations

We #ont ha)e the tools to sol)e

'eneral #i$$erential eAuations% but

this one is not too har#(

I$ e 'rab a sprin' an#

e*perimentally 'raph the positiono$ its en# as a $unction o$ time

a$ter compression% e 'et a

'raph li-e this:


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!ol)in' ur Di$$erential 4Auation

"his $unction ou'ht to loo- $amiliar to you: it?s the

'raph o$ the cosine $unction(

Let?s see hat happens i$ e =ust try as our position

$unction( Di$$erentiatin' tice to 'et the )elocity an#acceleration $unctions% e 'et:

hich is close% but e?re missin' the $actor o$ (


.,

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Eo Much Does Matter

Furthermore% e obser)e that the

$reAuency is proportional to the sAuare

root o$ ( For e*ample% hen e increase

by a $actor o$ $our% the $reAuency #oubles(

"his 'i)es us a hint as to here shoul#

appear% since all e are #oin' is scalin'

the time a*is(


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"hree More De'rees o$

Free#om

"here are some #e'rees o$ $ree#om inherent in the motion o$ the

sprin' that e ha)e not accounte# $or(

1( We are not accountin' $or the ma*imum #isplacement% -non as

the amplit#eo$ the oscillations an# #enote# ( ur eAuation

alays has an amplitu#e o$ 1(2( We are assumin' that % meanin' the sprin' as initially

stretche# to the ma*imum #isplacement an# release# ith ero

initial )elocity( Eoe)er% in 'eneral% e coul# ha)e pulle# it to

#isplacement an# then 'i)en it a sho)e so it has initial )elocity (


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"he "hree are "o

It oul# appear that e ha)e three more )ariables that nee# to be

accounte# $or in our eAuation i$ it is 'oin' to be completely 'eneral(

s it turns out% the three )ariables e ha)e =ust #iscusse# the

amplitu#e% initial position% an# initial )elocity3 are interrelate#(

I$ e pic- any to% the )alue $or the thir# is $i*e#( We?ll -eep as is% but e?ll replace an# ith thephase offset % hich

#escribes here in the cycle the sprin' is at (

#=ustments to the phase o$$set ha)e the simple e$$ect o$ shi$tin' the

'raph horiontally on the time a*is(


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!imple Earmonic Motion

##in' these to )ariables% e arri)e at the

'eneral solution% the eAuations o$ simple

harmonic oscillation:


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Dampin' Forces

!o $ar% e ha)e been stu#yin' a physically none*istent situation in

hich the sprin' ill oscillate $ore)er(

In reality% there are usually at least to more interestin' $orces% riving

force an# friction(

"he riving force is an e*ternal $orce% that acts as the input to thesystem an# causes the motion to be'in( 'rictione ha)e alrea#y met(

"he 'eneral term use# to #escribe any e$$ect that ten#s to re#uce the

amplitu#e o$ an oscillatory system is amping% an# e call oscillation

here the amplitu#e #ecays o)er time ampe oscillation(

Dampin' $orces are use$ul in )i#eo 'ames% so let?s #iscuss them inmore #etail(


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Nualitati)e bser)ations

"he #ampin' $orce has an e*tremely simple $orm% but thin's 'et interestin'

hen e stu#y motion o)er time(

Nualitati)ely% e can ma-e some basic pre#ictions about ho #ampe#

oscillation o$ a sprin' oul# #i$$er $rom un#ampe# oscillation o$ the same

sprin'(

"he more ob)ious pre#iction is that e oul# e*pect the amplitu#e o$

oscillation to #ecay o)er time( Li-e the $orce o$ $riction% #ampin' remo)es

ener'y $rom the system(

"he secon# obser)ation is only sli'htly less ob)ious: !ince #ampin' in

'eneral slos the )elocity o$ the mass on the en# o$ the sprin'% e oul#

e*pect the $reAuency o$ oscillation to be re#uce# compare# to un#ampe#oscillation(

"hose to intuiti)e pre#ictions turn out to be correct% althou'h% o$ course%

to be more speci$ic e ill nee# to analye the math(


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Earmonic Motion ith Dampin' Forces

Combinin' the restorati)e an# #ampin'

$orces% the net $orce can be ritten as

(

"o #eri)e the eAuation o$ motion% e ill

nee# accelerations% not $orces( pplyin'

+eton?s !econ# La an# #i)i#in' bothsi#es by the mass% e ha)e:

Chapter 12 +otes ,D Math Primer $orraphics & ame De) /,

! i D ! t i Gi#

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!prin'BDamper !ystems in Gi#eo

ames

!prin'B#amper systems are use# in )i#eo 'ames as control systems(

control system ta-es as input a $unction o$ time that represents some tar'et

)alue( For e*ample:

1( Camera co#e mi'ht compute a #esire# camera position base# on the

player?s position each $rameO

2( I co#e mi'ht #etermine an e*act tar'etin' an'le $or an enemyO

,( We may ha)e a #esire# player character )elocity base# on the

instantaneous amount o$ control stic- #e$lectionO

.( We mi'ht ha)e a #esire# screenBspace position $or some hi'hli'ht e$$ect%

base# on the currently selecte# choice in a menu(

Chapter 12 +otes ,D Math Primer $orraphics & ame De) /.

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"he !et Point

"he current )alue o$ the input si'nal is -non as

the set point in control system terminolo'y(

"he set point is essentially the rest position o$ the

sprin'% an# the input si'nal is li-e somebo#y ta-in'the other en# o$ the sprin' an# yan-in' it aroun#(

It is similar to a #ri)in' $orce% but e are 'i)en a

$unction #escribin' a position rather than a $orce or

acceleration(

Chapter 12 +otes ,D Math Primer $orraphics & ame De) //

Wh t D C t l ! t

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What Does a Control !ystem

Do

"he =ob o$ any control system is to ta-e this input

si'nal an# pro#uce an output si'nal( >sin' our

e*amples $rom 2 sli#es a'o% the output si'nal

mi'ht be respecti)ely3:1( "he camera position to use $or each $rame

2( "he animate# tar'etin' an'le the enemy ill use

to aim the eapon%

,( "he player character )elocity(

.( "he screenBspace position o$ the hi'hli'ht(

Chapter 12 +otes ,D Math Primer $orraphics & ame De) /0

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+o er-s lloe#

For many control systems% the actual position an# set point are

not use#O rather% only the error is nee#e#(

$ course% an ob)ious Auestion is% i$ e -no the #esire# )alue%

hy #on?t e =ust use that #irectly

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PD Controllers

"he camera or screenBspace hi'hli'ht are nonphysical e*amples

in hich the Auantity o$ mass is not really appropriate an# is

#roppe#(

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Dont Rein)ent the Wheel r the !prin'3

PD controllers an# their more robust cousin% the PID

controller% here the I stan#s $or inte'ral an# is use# to

remo)e stea#yBstate error3 are broa#ly applicable tools(

"hey ha)e been stan#ar# en'ineerin' tools $or #eca#es

centuries3 an# are ell un#erstoo#(

+e)ertheless% they are one o$ the most $reAuently

rein)ente# heels in )i#eo 'ame pro'rammin'(

Chapter 12 +otes ,D Math Primer $orraphics & ame De) /@

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Chapter 12 +otes ,D Math Primer $orraphics & ame De) 0

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"unin'

Di$$erent cars ha)e suspensions that are tune# #i$$erentlyO sports cars areti'hter an# the cars retirees li-e to #ri)e are smoother( In the same ay% e

tune our control systems to 'et the response e li-e(

+otice that the co#e uses the an# $rom our earlier eAuations( Eoe)er% most

people #on?t $in# those to be the most intuiti)e #ials to ha)e $or tea-in'(

Instea#% the #ampin' ratio an# $reAuency o$ oscillation are use# $or the#esi'ner inter$ace% hile an# are compute# as #eri)e# Auantities(

"o tune the $reAuency% e mi'ht a#=ust either the #ampe# or un#ampe#

)ersion% usin' either an'ular $reAuency or simply EertO the units an# absolute

)alue are o$ten not important because the )alue that $eels 'oo# ill be

#etermine# e*perimentally anyay(


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lternati)es an# eneraliations

"he secon#Bor#er systems e ha)e #escribe# here are certainly not the only type o$control system% nor e)en the simplest% but they #o beha)e nicely un#er a )ery broa# set

o$ circumstances an# are easy to implement an# tune(

nother commonly use# control system is a simple $irst or#er la'% % un#er hich the error

#ecays e*ponentially(

"his is similar to a critically #ampe# secon#Bor#er system% but ith a bit =er-ier response

to a su##en chan'e in the set point(

nother common techniAue is to chase the set point at a $i*e# )elocity(

$ilter is another broa# class o$ control system in hich the output is compute# by ta-in'

some linear combination o$ set points or )alues on pre)ious $rames(


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!ection 12(,:

Momentum

Chapter 12 +otes ,D Math Primer $orraphics & ame De) 0,

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Moes

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!ection 12(.:

Impulsi)e Forces an# Collisions

Chapter 12 +otes ,D Math Primer $orraphics & ame De) 0/

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!ection 12(/:

Rotational Dynamics


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!ection 12(0:

RealB"ime Ri'i#

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"hat conclu#es Chapter 12( +e*t% Chapter 1,:

Cur)es in ,D

12 Mechanics 2

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