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Unformatted text preview: CHAPTER 7 Coordinate Systems 1 COORDINATE RDTATIONS Theory Frequently, mere than {the enm'dinate system is needed in erder 1n
deserihe the tnnlien ul' u purtiele. lni'ermetidn may be given in time
ederdihale s}‘ﬁ1e111. hill net in the nne that yen wish 1e Use. it may he
preferable it] use a refitted entirdinute system heetntse mes: nf‘ the threes
in yen: i'ree tied}; diagram line up along its uses, er heeause the metiun is
direeteti tilting tine ui‘ its uses. 1When Ihis happens, it beeemes advanta—
gemis In transfnrm your quantities frnm the urigina] enurdinide system
title the preferred entirdinate system. When yet: have finished. it may lie desirable in transform your quantities heel; tn the erigina! enerdinate
systeiit. Referring In Fig. 1H. the linsitinn H}
temis ed" the eeurdieates X and F
1.t'ith 11m: ennrtliuete system.
in terms ui' the quantities .
enurdinute system. We write etnr ni'a paint is expressed in
and 1he unit veelurs 1' and J nssneinled
The same pnsitieri s'eetnr is aisn espresseti
t. _t_. i. and j tissueiated with the retated r  .11 'r _'I'j [ii—Ia] 1H"
r ..rt+r.l. LII—Ila] I31 I34 MARKS' MECHANICS FROBLEHEDLVING CUMFANIDN Figure 3'. IJ Figure 3. fE {Jur interest lies in expressing X and 1" in terms of x and y, and 1r'iee
versa. In order In ﬁt} this. lets ﬁrst express I and J in terms et'i and j. and
viee verse. The retstienships between the unit veetnrs are easy In see if
they are celleeted end pltteetI in :I eirele. as shnwn in Fig. ’HE. We get
the unit veeter transfnrmstiens ‘ i=eesI’J‘I—sinUJ, l=eesUiIsin[tj. ‘
j; sintt l+eestJJ. .l —. .. sintiiteesﬂj, an”? Next, suhstitut'tnh,Ir Eqs. [1 1 2b] inter Hq. (T. l1h} yields .‘ri +_t=j XEeest’i i + sin ﬂj}+ Yt— sinﬂ i +eestlj]
= {Xensti — I“ sin HJirt {X sin H t Fees tJJj. It lhllews that .'t' ees I'i' .lt' sin t} T.
_ {ll—3a]
.t' 2 smth teestJ 1". Equations [ll3s} express xsnrlr in terms at" X and tr’. Similarly. X and
Y are expressed in terms ef I and _1" by substituting Hqs. [it23} inte Eq.
{ll—ls} to yield X _ ‘u'tt.t'——."tntl '.
J L 3 “ J {11—3b}
} ~ snﬂ.rteesﬂ_t'. i Coordlnato Systems I 35 Notice that the transfonnation equations of the unit vectors and the
transformation equations of the coordinates. are of the same form. The
coordinates .r. y, X. and l" ore interchangeable with the unit vectors i, j. l,
and J. respectively. liy interchanging these quantities. Ens. {Tl23]
become Eqs. [ll—3:1}. and Eels. Ell—2h} become Eqs. {'i‘. l—3h}. Further—
more, notice that this entire derivation. rather than having started out
with Eq. {I 1—lj expressing the position vector 1* in two ways. could have
started out with F : ﬁt t F rj and F = FNI t FirJ. espressing the force
vector P in two ways. or could have started out with expressing any other
vector in two ways. In other words, the trans’rormation of coordinates
given above is vaIid not only for the components oi" the position vector,
but also for the components of any vector. For example. the transforma
tions of the components of force are fr; = {mg [11 FX — 5i“ Ff. [ll45C!
£1. = sinti' F} + cos iilr‘r.
[—. _
 F = costi'F + sin [it“s
” ‘ ‘ t?.l4hi FF : siniii’ft +costiF_... Once you know how the tmit vectors translornt. you transform the
components of any vector in the same way. to short. the components of
all of the vectors transform like the unit vectors. Examples 't‘.ll Detennine the X and 1’ components of
the position vector shown. Solution
Let's lirst determine the unit vector transforma—
tions. Referring to Fig. 112. we get Example; 1L; 1: cos 3G" i + sirtJti"' j. J =  sin ﬂit" i t cos 30" j.
The coordinates transform like the unit vectors, so X = cos 3D” x + sin 3U" y = Uﬁﬁﬁtﬂﬁt + 0.3104} = Union],
1" = — sinltﬂ‘ s; + C0530" _t' = —ﬂ.5{ﬂ.3] + U.Hﬁti~l[U.4] : tilt} in. Coordinate Systems I 45 F; = {3: + Eli + {42‘ id El]. and particle 3 moves along the path
rytr} : {IEI — 4H + [2:11]. Partiele A is initially at rest1 and rnr, = 1.1? Find the path rift]! and veloeityr rift} ot‘a small partiele Flying in
the air. The particle has mass m = 0.125 kg and is being inﬂuenced by
the resultant three P111] : {H.313 — “.4! + ilﬁli + U. 1!] + [15 eostﬁhk N.
The pnrtiele is initially loenied at the origin and at rest. TJlﬂ Find the relative position vector rmmti‘} and the relative ~.r:;:]:;ir;it:,r veetor vR Mfr} of a model rocket as seen Ii}r Ed. lid is traveling on a
Inerrygorronntl following the path rapt!) = 5t] sini‘i + ﬁll eosr j fl
and the roeket blasts oft“ following the path rﬁill = Ellill + it] +
[see — 3a: — 51.31:; a. 1.1l I Find the pntli ofn srnnil rocket rpmAI} ns seen by Ed. Ed is now
on a train traveling with a veloeit}.r of‘vhJfr} — Tﬂi + 4tlj ﬁfs. The rocket.
which was initially loealetl at rﬂrjtl} = zoom IL is ﬁred upward when
Ed is at rid{ll} ; —ltltltli it. For a short time, the rocket has mass
m3 = Ilﬂtl sing and a eonslnnt thrust of FREE] 2 ll {lﬂtllt lb. 1.3 POLAR CDOHDINATES Theory We have seen that the standard way of“ expressing the position vector of
a point is r =xi l _t'j  :lt, in whieii i r" {I ti [l], j : [If] 1 U}, and
It = {ll {l I] denote standard unit vectors and 1nrhere tar. and :. are
called rectangular coordinates. Another convenient was,r ofexp ressing the position vector ot'a point is
by writing it as a magnitude times a direetion. This is ealled the polar
Fonn otthe position vector. As shown in Fig. 134. the polar form of the
position vector is r: rn,.. {Till in which 1': r dennles the magnitude oi" r and nr denotes the unit
vector in the direction of r. Polar coordinates are used when the motion MEI HARKS' MECHANICS PROBLEMSOLVING COMPANION ut‘ the puint is restrieted tu :1 flat plane. They are
particularly cunvenient tu use when the puint
reeves in a circular ur clreulat‘wlike path. When pelsr cuurdinhtes are used, the vecturs
are expressed in terms uf the unit vectur n,.. culled
the radial unit vector. and the unit vectur n“. Figure 13!
called the circuml’erential unit vector. The
circmrtierentiel unit vectur is perpendicular Lu tt,.. Netiee that the unit vectors It... and n“ are merely rotated unit vecturs.
like these we srnv in Seetiun 't'. 1. The I end .I in that sectiun are the same
as the n, and n" here. Theref'ure. 11F and n” are related tu i and j by 114!) = eus till} i '1 sin iltr] j. {13—24:}
ﬂair] = — sinil'lt‘} i+ eus tilt} j. t?.32h] Netiee that the radial and circumferential unit vecturs are functiuus ut"
time. Unlike the standard unit vectors, the rudiul end circumferential unit
vecturs depend en the angle l3. whieh in turn depends uu time t.
Differentiating Eq. [1323] with respect tu time using the chain rule yields
I'tr : —iisintl l+ ileustl] = lit" sinti i + eusttj} —. tin”
Similarly, differentiating liq. {1321:} with respect in time yields
[tutu ;_ itn,.. Thus1 the derivatives til the ttnit vecturs are given by sr = huh—l {13th m, = 41H tieset We are ready tu diﬁerentiete r with respect tu time in uhtsin the velucity
veetur v, and tu differentiate v tu uhtitin the ueeelemtiun veetur e. We Coordinate System: HT diﬁ‘erenliale Eq. {131} with rereel to time using the prhduel rule for
diﬂ'eretttiat'ton and [it]. {13—311} tc get V _ fir“, i Pun”, “1'11ch Fr. _ j'. :1” = frilj Differentiating Eq. [13—4) with reepcet to time yields
a . rrrnr. + mitt”. where H... = i" — rtiz. H” ...' r'i'} + 2H}. {135} Notice that the radial ccrnpencnt of acceleration a, is made up of twn
terms and that the circuttll'crctttialchinphnenl ﬂi'acccicratinlt is made up
t'lr twn terms. The radial term “1'”? [which is always negative} is called
the centrifugal term. The circumferential tenn 2H} is caiicrl the Cerinlis
term. A graphical rcpresentatien at" each cf the acceleration terms is
Hhewn in Fig. 132ta—ej. The ﬁgures chew hew each hf the terms arises. 1h) % —t.', i? I15: firing in; Muir] Sﬁ=uﬁﬂ l"It1h Figure ﬁﬁEfa—tt') Coordinate Systems IE] 1.4 CYLINDRICAL CDORDINATES Theelr'gulr {Iylindl'ieal eeerdinates represent an exteneien of polar eenrdinates tn
three dimensientt. The extension is directed perpendieulnr tn the pulat
eeerdinttte plane in the tilteetien of the :: ttxie. The eemmen error that
One makes When thiittg cylindrical eeettttnnles is: te let r dennte the
distance between the erigin and the tip of the pesttien veeter, as in pelar
eenrdinates. In eyltndrieal enerdinates, hnwever, r denetes the distance
between the might and the pmjeeﬁnn at" the pettitinn tweeter elite the r—ti'
plane. as Shown in Fig. 7".41. In ey1int1rieal enerdinuteﬁ. r 7E r.
Deneting the prejeetien of r unto the :—t} plane by rp = r11," then
rP = r. As shnwn, the pneitten vector in eylindriettl eeerdinatee is. r = rn, + 211;. {14—1} where n: denotes the unit vector In the 3 directinn. Differentiating L‘q.
{144} with respeet te time yieldﬁ the veleeity veetert and dillerentlatlng
the veleeity veetnr with respect tn time yields the acceleration veelnt‘,
written as ' 2 WE + Hallie + “:“r l.— [143) Figure 341 I54 MARKS' MECHANICS FRUBLEHSDLVING COMPANION Netice that the radial and circumferential eempeiients ef pesitien,
velecily, and aeceleratien in cylindrical eeerdinates and the eerrespend—
ing einnpnnents in JJUIELT eunrdinates are the same [eee 17.1.15. H.341 and
(135)]. The differences lit: in the : directive. Examples 'MI Determine the unit veeters 11],, tint, and er, and the cylindrieal
centpnnents el' the pettitititt veeler r = 4i + 3] l [Eli l'l'l. Salutlan The prejeelidn of the familiar. vector nnle the r—t'} plane ili r“ = 4i —l— 3].
The magnitude eF the prejeetinn at" r is. r = J43 + 33 = 5. The radial
unit veeter is then nr : rpfi' : {4i l 3]}!5 : {Lilli i llﬁj. Frem the unit
vector tmnathi‘tnatiens, nr 2 [131+ [Hi] T ces ti i + sin ti j, se eestl ;
[iii and sini’i : {1.15. Thiis1 n“ = ~511th i + eretil] : —fi_tii —l— {1.Hj. The
unit ‘v'ectttr n‘. is the same as the unit 1tweeter k. We new vedfy the
cylindrical cempenents til" the pesitien veeter. Taking dat preduets. r— rnr = tili+ fij + lEk‘J {llEi+ﬂ.ﬁj} : 4tﬂ.$i +3{ﬂ.tii = 5m.
ezrn:=[4i+3j+ lEItJltz 2m_ Fiﬁ r— 5nr+ lln_.m_ 1.44 A head is sliding, dawn a cylindrical Spiral having an incline
angle til" ‘3 = iii"~ as shewn. Neglect the Frictien between the head and the Spiral. and assume that the head starts t‘rem rest at the tap at the
spiral. Detennine the pesitien 1t'eeter ef the head as a ihnetien at time. Salutian As shewn in the tree tied}; diagram. the head is subjected Le a
gi'avitalienal Ierce and a nermal three. 1which lies heen hrekcn Liewri
intn lwe eenipenenta that are perpendicular te the tangent til" the spiral.
Summing threes along the cylindrical directiants. we get N,  mar. = int—RIF].
N" cesT — mg = me: = HIE. — N" ain 1 a: mar; = mitt]: where we netiee that i‘ _ ii since r is eenstant. We also get the velneity
‘u'EtIEtJl' in the term 1* T RUan + in: : +r: ens }' n“. + eainr n2. taking l‘ Cuerdinaee Systems l5?r 15 TA NGE NTIALH CIHHAL CO DRDI H AT ES Th eery When tangential annual ceerdinates am used. the veetnr quantities are
expressed in terms at cnn‘tpnnents that act in the directien ef the velneity
vceter and in the directien perpendicular, er Henna], tn the velecity
veeter. 'l'he directien et' the veleeity vecter is alse the tangent tn the path
that tlte paint thllews. This is why the cenrdinates used ttt this sectinn are
called tangential—unmtal eeerdinates. These eeerdinates are ttsed when
tlte motion is cenlined In a plane, and they are particularly eenvenient te
use when the threes in the Free hetly diagram act in the dii'eet'ten at" the
velccity veetnr and perpendicular tn the velneity vectnr. This eccurs in
aerudynan‘tie and hydredynarnic preblems in which drag threes act. alen g
the velocity vectnr and litl threes act perpendicular te the velocity vecter.
lt alsu uecurs in certain n‘techanisnis, such as itt guidedtail prehletne, in
which nnrtnal reactinns act in the nnnnal dil'ectinn and Frietien threes act
in the tangential directinn. Netice, in contrast with pelar cnerd'tnates~ in
which the vecters are based en pnsitien, that 1.vith tangeittial—nennal
ceerdinates the vecters are Ihased un veleelty. Tangential nnrmal entirdinales are setup by ﬁrst writing the velncity
veetnr as magnitude times directien. Referring In Fig. 7.5—], we let v 21ml, 5 {ISll  . where t' dcnetes the magnitude ef' the velncity
vecter and Itr denetes the tangential unit veetnr.
The tangential unit veeter is directed aleng the
velneity vectnr. We alsu deﬁne the ttnit vectel' n”.
called the normal unit veetnr. The nnnnal unit
vecter is deﬁned in he perpendicular tn the Figure 151
tangential unit veetet‘. Just as in the case ni‘ pelar ceerdinates. the unit veeters nr and n,f are retaterl unit vcctnrs,
and thus are related to the standard unit vectors tltmugh transl'ermatien
equatiens. Frem liq. [lit—EL replacing nr and n” with i1r and n”. we get b: Hint." ﬂu = I53 MARKS” MECHANICS PROBLEMSOLVIHG COMPANION L. Netti, referring to Fig. 15in we v tH
deﬁne the radius of curvature p of ' >\ the path at time t. by intersecting the “thee "t lines extending from nxtt} and _ Niuili'tj‘” infirm”
nitlentil in which ﬁt is 11 small WWW
increment of time. Over the time nit} f * ﬂtttnn increment ﬂit. the angle changes an
incremental amount eti' and the point
mech an incremental utnount m.
Notice thttt In dﬂ' = dot, no dividing
by hi and letting the incremental
quantities become infu‘tileeitnal
yields ﬁgure 152 pit = in {15—3} in whiclt attifttr = tit and reﬁt! = 1:. Now, diITcrcntittting 1" in liq. {IS—l}
with respect to time yields :1 = urn, + ann”, where n, = L'I, an = {7154} Notice in Eq. [15—4) that n gé t}.
With tangential normal eeerdintttcit, il in useful to have it formula for
the rudiue ol'cur'trnture oftt ﬁtnetitin_t[.r]. lt cnn be shown From Fig. 153 {sec Example 15—4} that
f 2 in,
If}? {155} It in interesting to notice that when the Slope dyftit'e tent! ot‘ the
function ﬁt} in tiITIEIII [when tent? {<1 I]. the ﬁrst derivative of the
Function in uppreximatel‘j' {writ : H. From Eq. {155}, when the slope
of the function is small, the second derivative of the function is approximately e'lyfulxz = lip. Coordinate Systems IS'iI rh‘ In!“ (It “’1’ '1
—: J+ —'—
a; t [new] Examples ﬁgure 1.113 1.5] Find the radius: DI" curvature {if a particle that
IL'LI'} =3:3 — 16.x + 5E] m when .r = 4. Solution
Diffcrcntiating the functitm fur the path yields tbllnws the path ,3: . d2 J
“T: _—_ 3,11 _ m = 3+2~ Eff = m = 24.
so. f'mm Eq. {154},
1 323 “2
= ﬁlm—J _ = 131513 m. 1.5: Duturminc the pnlar Form 01' the veEocity vector v = ﬁn, mfg it"
the pnsitinu veclur is r = 5a,. winning n 5_ I2_ n 7 H a j r:—— , = _ —_.
13 I31 ’ «ft—:3 «tn—t3 Solutmn From the transformation cquatiuns Fur rutatcd unit mulnrs, n —12I1£' n —_E l+ ?—j
“"13 13" “m M' Newton's Laws of Motion I15 the aigdhnl and to solve the equations. suhstituting numbers [or the
parameters at the last possible stage of the manipulation. In feet. it"
possible, it is heat to express the answers as general expressions. and then
to substitute numbers for the parameters to obtain numerical answers.
Doing this enables twp important tests to he perlhruted. First. by
developing general expressions. the units in your answers can he veriﬁed
to mateh as a way to test for eareiess errors. Secondly. the general
expressions exhibit trends titat can be inspected. The answers will be
proportional to some of the parameters in the prohlem. and inverser
proportional to ether parameters in the prohiem. These trends are an
important part of your comprehensive understanding of the system. ' 6.1 GOVERNING EQUATIONS Theory The lield ot‘ Newtonian meehanies is governed by Newton‘s First.
Second, and Third Laws and by Newton‘s Law ot‘tLiraeitatioa.
Newton’s First Law states: 21‘ pttt'tt'ei’e mares iit'th constant speed and att'raett'ott when the
t'rts'ttt’t'ttttt ﬁttest! tit :et‘u, Newton‘s Seeond Law states: A partt't'i'e.'r ttc'eetettttt'uti t's ti'tteat'ﬂ' pt‘apttrtt'ttttat to the testtt'tattt
three. The ttt'ttttrtt'tt'attatttr rttttstattt ts t'ttttrra' tttit tttttttttefs mastr. Newton ‘s Third Law states: Titrt parades t'titetiart irr'ttt eartt other tiltTJttgit'fFJt‘t't’s' that are t’tﬂtr’ttr
in magnitude. apposite i'ti atteett'tm. atta‘ eati't'ttattt: Newton‘s Law of Gravitation states:
Two pat‘tt'etes ﬂt't.’ attracted to each other tttt‘attgttft'irt'ar tttat are at ttttear proportion to their masses and at tatarse proportion to the
.‘u't’jﬂﬂ'f't’ ttt'ttitt' tit.t'tttttr'tt tte'ftit’att that”. I16 MARKS’ MECHANICS FROBLEHSDLVING COMPANION Let's new examine a system nfparticlcs. Tlte number of particles is rt.
Letting particles he distinguished From one another by subscripts. we
pick out the ith particle and tttelfth particle and examine them [see Fig. h.2—t}. Newton‘s Second and Third Laws are expressed mathematically
HS l_.. _ _. __
. F. + H” i t};  . . . + I'm} 2 treat. : [tilla]
! [5,. —f_rl. {tillit] rl x t“ = r! >< f}... tel1c} where F. denotes the resultant external {to the system} force acting on
the hit particle. 1'” Phi"; t “stff“ is the resultant internal [to the
system} ﬁ'tt'ce acting on the itlt particle. Itr is the acceleration of tlte t'tlt
particle. at, is the mass of the ith particle. t}! is the three that thejth
particle exerts on the ith particle. t}. thc force that the ith particle exerts
an titcjllt particle. and I; is the position at the i111 particle. 'l‘itc lefthand
side of Eq. [n.El a] is the resultant Force [external and internal] acting on
the i'th particle. Thus. Eq. [szlﬂ} is the mathematical statement of"
Newton‘s Second Law [and the mathematical statement ot‘Newtnn '3 First
Law it" aJ = it}. Equation tell—th is the mathematical statement of
Newlnrt's Thirti Law. expressing that interacting ﬁtrccs are equal in
magnitude and oppesite in direction. Hquatinn {oldie} is the mathema
tical statement espressng that the associated interacting moments are Pilgrim tiltI Newton’s Laws ot‘ Motion I11 '1 too. Although Newton‘s gnitudc and opposite in directiot
te that interacting moments are equal in this t‘ollows directly from this law.
opposite in direction1 equal in ma
Third Law does not explicitly sta magnitude and opposite in direction,
which states that the forces are equal in nutgnitudc, and colincar {see Llsatnple fitti}.
he equation of motion F : ma for a single particle to at" particles, in which at; denotes the accelera— tion of the system‘s rnass center. We start by delioing the position vector
at the mass center o!" a system of particles as the weighted average of the position vectors of the particles. written as Let‘s now extend t
to F r: mar for a systc l
t'mlr1+m3r:+...m,,rnl. m _: all + an; + . . . + mm. H! r{._. where m denotes the total mass of the particles. Using an indes. notation. we can rewrite these two equations as I... ._ _Jt_ _____H_
l rL =. I—Elnlrl. m = Em]. {ﬁLZE} HI l=1 r—__
_.I l___ Differentiating Eq. tel2‘; twice with respect to time yields the velocity and acceleration vectors of the mass center, written as 1 It!“ 1 n v  —= my. a  .—— — m a. {clBali
t. mfgi : r t “I; .' r i'
r the equations grwentlng the motion ol‘ each ol'rhc Nest. let‘s add togethe
Front the lollhand side of lids. [oi—la}. we get particles, Eqs. {6.21 at. (a, + {j = r.) + : = F. res—4}
_r=t J—.§J—i r  :il r'—l
internal forces are equal and opposite as stated in Newton's Third Law. Eq. {ﬁlth}. and thus cancel
tes the resultant external force acting on out. 11' in Ed. raga} deno
the system of particles. Substituting Eqs. tella] and [oi—4} into Eq.
[6.23hl. considering Eq. [til—4], and multiplying the result by m yields where the double sum is zero since the it: .—_ mac. i into} L__ _ _
states that the resultant external force acting on a
nass of the systetn multiplied by the center. The internal forces are not Equation {oi—5}
system of particles is equal to the I
acceleration at" the system‘s mass IIB HARHS’ HECHANFC‘S PROBLEMSDLVING COMPANION present in the etltttrtr'nn. Thus. the hehrreier' rrt‘ the niaernseepie heel}.r eatt
be analyzed withﬂtli regard tn the internal threes acting inside the herd)».
Sinee Hq. tel5] leeks se much like Nentnn’s Seeettrl Late it is ealled
Newtnn‘s Heeenr] Law for a System 01‘ Particles. E xamples e.1l :‘t hieeh is sliding tn the right en a
rettgh stirthee Iwhile heing stthjeeted tn the
three shewn. Determine the aeeeleratien til"
the hleelt. Assume that the ﬂ‘ietien between
the reuglt sadism and tlte bidetz is getter'netl
by a dry frietint'r ntndel that will he
diseusserl shrirtljr. The black has mass
tn = 2 slug, the ltinetr'e liietinrt eeei‘tieient
is flk : {1.2, F —' I5 ".11 and [J' = It't". Sututten 5'}
The talent: enn he regarded as at single particle. Let .T he pnsittve tn the right and t: Erﬂﬂtpfg (5,24 he pesitive upward. Unless otherwise stated, the .t axis is always taken pesitive tn the right, and the y axis is always
tatten pnsttite upward. Sttnnning ftnees in the x and y tlireetinrts yields in g
t —F', — Fenst‘l‘ 2 mm. N + Fsin ti — nag = es:er where F, is tlte Frietiett three. heettrtiing tn the kinetic dryr l‘rietinn
metiel. the friction three npprrses the directinn of the meﬁertt and the
magnitude trt' the three is prapartinnat tn the normal t‘uree N between the battles [the hleek and the grennd]. The Itrepertienalit}r constant is
called the kinetie errefﬁeient rrl' frietinn m. The Friet'ttrn three is F” .—
Flr = fixiht‘lj Alsut the problem states that the blue]: is sliding en the heriaentai
sur‘t'ttee; there is net mettert in the y direetintt, se the aeeeleratien in the it
tlireetitrn is stern1 thtrt is. tr". = t]. Substituting tr}. = ﬂ and lit]. {62—h} ittte ...
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 Spring '08
 Silverberg
 Vectors, Force, position vector, unit vectors, standard unit vectors

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