03/25

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1l
rl
I
In
-l.r
genrral
,i orc relqteå
1r
L*
Ly
Le
ri
,i
I
ri
.
rt
i
't
Ol'
wrtl
-t!
t
-
fl,e ve,ilocs È
qnå
a\
h,
Iìntnch¡ q s
Irr
Iy*
tx
htr * I-*v wy + I¡o6.
W* + fr, -t t IyeW¿
Jzfhl + Xr¿h/.
t{¡ìrx ZHy 3 âZ
G
wr
+
3
L; =
¡!j¡¡uri
ìn l4ctcix Êr-
[t,l
fr., r*, r,J
ltrl = lr)' "v rr"
f
lr
'1
l
'i
i
ri
LL"J Lx." J.r r'aJ
tS
-'\
H
L = I.Lr)
-
Wh"te I ts *he Ïn.;Ìiq Jbnsoc
,, Tor ròt*tions qbo.tï Somg !.hi1 C
C*y or noy rul bc Co Ì1) '
,
iA
'1
a body wtth mcrlãs denar'fY Pci),
ohoke _dF oflglq ' Ò {hen
parttøular
".
cb'nponsrrts ùF
g
qre:
Ìo, = J¿l¡ pc!(yaz)
!y e Jj'rpci) (2"+!
Jr
= S¿'Ê pcñ) (r"*!
Iy" =t), = - Jd'Fpcf) yz
For
q
åtscr¿te søt'
df
pr^{r S + 2
urrìfbrrn dens
f4= Pot
tvlo^ad'
f
].e.Jr^ dF c'\ C.rbe. .bo.îl
Ca(her,
I"x = SJSa3i p(y"n!
p $"J"ay S"¿"y" = àpo'
p (ll¿x)(9loyfl(Jl¿") = po (å o')q
f Sl¿ vsln\¿yt = åp"'
L
eâfl
q
3,o =¿1-=3:** = â p"u
&y =-$SalF fxy
= .- fslr.çig*v)(sa) P
Jry = -t po' -- elt¡qqt :Ï7,
th
= lq, 2/g -tíq1Mq*
-V,
I
-Vn
VUq-yqzls
I
)
:
Riå
Wrîflen
t/t*r^î
in
qyì qrbitrafy
or fnurqe â t
¡
ï^= [u
'lt"'
ç
[r'*
arcrdinal¿
2
t
.["
J*y Jr.
ïyy Jy.
T,Y
J¿z
I'
),
llq¡ 6 ir,Je p^4".1 conponeals ì ,,
lencførl
s
passlble {" fi"J
ì3 qlwqy
fx"À 'w.r.{. *he b"4y, So¿tn
f
di ogorrøl |
¡
;;lol
ï=lr,o 2L
-f
Lo o ^"J
Ar,A",\
diogot*rlíze
rne*hoås
!
I
z
3
qrE prlncìprl turrnl5,
wi'ttr us,^^l r.^tril
No* tf
eypcsss ñ in *erg5, of
2 ê,, ê",ê"1 (*he pclndple qxts) we frnd
we
¡-L
2,',â + 2r- *.â
[=
+2g,¡.â
In
portt,lhr, m the ro*dtio", ls
qboot q prt;cìp al oìx I S ãooh J+,^'t
..^
t!
-ih"t,
Ë
U,i ê¡
whrrl i = t, z, 0ß 3
-\
L =2;Ë
Êr rotcflo", qbod* cr "f ln cipql oxìs
{he dtrec,'tion ôf i *d ù cf Ìn.i,le.
Thls ls ô'le wqy. +o ide"trf) /drút"e
'q
prlneìp^l
o\x¡s
d
T"
?eneral a, #2"t 23 qnd. *he
cholce ùF tê,, ôr,á.1 is uvrio,ue.
,,rf L
r.o-er{g qrs ejr.^l 2, = 2z / 2"
-\
e;
*hen
,,
qncl
ê. crrh'b" ony 5ù
^
iâC dìreetìons n,rtuoll7 orlh'¡*ú\ +o e;
, onol
!rnch o{htr s*..
A
,/ä0. ì>Â
orÃf
'1
i
.,le
^
e2
ri
r1
c^,
qrls df s7r.r.o{r7 ìS
EF3
'ilt*r,
P.
A. + Z
or}to¡or".l
rnona'ù org
2ê,,
t,, âJ
t
''lc.
'i
4
.:
Q
^¡
l
I
rl I
:
I
Jt
:jKot"lis" ¿f Rìåee.l BoJles.
't
i
i
rl
,lKinøtlc gneey
i
Stn
ce
l- e ?,dr à; + 1zwz éz +1gw, ã"
,i qn¿ t= ,,4 +tr;24 +wr{
l
ì
rt
i
.
tt
'i
I
I
'l
l
J= å ô.È
J = å t 2,*," + 7.¿d.. + âr,^1.)
I
l
rt
I
I
foc cr rid,ge.,\ bo.tr7 wrth
2,,4r,a"
: If rúe cqn expfess (ilr.'se+'
tù., td3
: in *erns oF q 'so ìfoìble
of
enerc^lizeå coocåt"ates w¿ qîe
.-rer
4y *o wrtle clow n -The L"! rc^n$øüt
S.;tofio" aF Kîâg!d Beoli¿s,
Erler A'gles
We hqve shown +hdl
íf a b"dv has
pql momeRTs Í7, ,4r,27J th,í" the
k¡net;" energ) of r;t"tio'n t"kes -the simple!rt
pcincì
J= àça,,¡,"+ 1rwï 1?rw"
r,uhcre ô= d,â * ht¿å.*,¡rê.
on¿
lê,,, #..,
whi¿h
åJ
arre
-
{fie pcincip"l
qre fixe"l 'l% +he boåy
¡-ty
4
ox$sr
r¡td¡e,
T" wrÌte down *he K E f"',d 't{t¡s ü. Lgtonglô
ìn qn ine,fti.¡l ff.rrne we ros-t exofigss
suìtahle genero\lìz¿clt ro'or¿inalrs
Ìn
^.) whtoh dìc¿¿flan +hÊ body
*hot ensode
t t^,, , d",
is
ocìerteâ
P lsco ss .'
Vhqt qre suì*",ble qenerutizeol
coordlnofes + ¿han^rltri. *he
orle¡¡h¡lion of q body with r¿s p"ot
{b c4 fxed frat're? t{o* Manyl'
.&tition of
Ri,Åe"¿
the
S irr
fui.q
!.rin cî
*he body'
hr.fÌ+".
pol qxes qre fìxe.l
we neeo\ only døsccìbe.
l) -[he orietitian"ìce,fìan of^
ony one dF ' 2ê, ,t
hr.nt q
-r=,4ì
fXu¿ frorne, '
*r.*î
ie@
no^be6.
eaglcs df
c^ ã¡ohcrl r",l
.oo
tti*úg
sysler.n.
^aI
Z)
r The oclølo*ío', df *h" o*h.r *wo ayes
in th" plone porprndra¡tnr +o +l.e *hicå.
-1
e?
da
a
Rt.rr.s
,,,, 1
-r -)
$r
)
E
y1(
more norrrLer
ob,rrs {ol^[
o,Q,t
E;t"tio" of
Riåqe,à.
.äC¡:s.
*ì'Jä
*"f;y
å;l]n
llÍ
**l"',,f1 *
lo rnwd q ciåq¿4 bdv +" qn or6Érurv..
octen*aìTioq rt",.rÏa w,th
'tq,4,4J
A. not rt" bv'tïqrr an ^le ô
,Í, tJ"
^
1
*'õ'
+r
= L7,3, B
*he Ê oxls
2â,n, ôl , Èì
c^bo.rt
¡¡2
^l
qvtqle e abq,¡t +he
(ô,',
I ê,o, â"í, ftl
B. Rotote bv
l.^ on
Jì
%
i3
,a
f; åi'
Ret"tion
of Rìd!ed &d'iet
C. K"tdte by oíì
J\
?tt,
1,
-7 !¿
5ince we Know -the dìre.tiorr df
\rúe cqh irnnedl*1{y \¡rfffe dow n
q^g.,lar velo*ry
¡-S
W=
W=
oß,,
tlo*e:
t7'
ivrl
qK +
fo
.(^
ê !r'
,\
e
gg
(- ôsìne) â'
*
Slnce
ô=
--\
[rJâ +
-
É¡
ù
R=
Cos
er¡r,;lt
roicilion
the
--¡
W,
-î l'
lt^
a
Ver
^
'- 5ìnê !,t
"âr'+
(ü. ô cÆs")å
vúe eil?rer corh wriig Q,t cnå er' i¡ ferts df
é, ci'.ot !r,, Af â¡.42 (ttÊ s7r,nøtrtc cose) *tren
ô,' o,,,â ár, orc çcrfc¿t" goodr pltncipal qxts
qhd\
R#tio", ofl K¡åed BoJ!f
In +lle sf"ofctc cnse '
-\
'
¿\-
2,r
o¡
s¡¡ e [-p6he) d,,* õ ei
Angohr
I'tæil,*
t=Iù
K¡n¿tk
1z + 7s
¡
o
n
+ (e+$coso]0"
È = (-2, [sine)â,' + ?,# {'*?'CV-E*0{
= 1$l ?rf $+ô^t)>ô=$ffi
tr
Lz = 2, Qsin"O + IocoS$ /
Encr"Y
J=
*q,Côir*e +å")+ åzs (.i,* Qøuf
We qr¿ nôw ln q poshlro^ to t
wrlte down Jtr$, L"grornX{Hvl i
,R"*"1ì0"
ofl Riånø &årq
The Syr,*etrr" Spt^nt^3lp
oe'**lly
T
l- =
E U("r 0, t!)
\r.-..-.-Za.-ra--¿
h{r*tic.l i. tctnc
For q top:
->
poertrm
!
à2, (Q"sinls + #)+
e"tdanl
$e) =
6f
i2rC'i,*ô-rri-
=Bi =_
Ù qgtion:
?LP+= ãu-
"
d
rtoes
lbRcoso
*
ta
L.CV*
+ Mg
F+
c*tr¿
= MgRcosö
2,A = 2,Ëstrro coso Ö.î,-+íærl
of ã,lr *nAt s
R'
"*")$tinO
einê
aoo
2, "sìnze + 2{9+ !cosê)core
ca.st*it = Q
aa
2, (Q + ! -s ê) =_co',sL,t
ÉLg
RS;lì0" q[ Riåg'â Boå'q
5teody Pce..sslovr
Too
R"{c¡t¿s c.bo.lt' z
Dle c¿tlør dF
rrÈfh ê fixed.
^aI
Sìrt ce
{hc",,
Sln¿e
0
-
ô=-lt =
r-.t'r-3
Lz
- L3CoSe
d strt"o
cons'fot¡f
6c
O co*e1qtíl
(.P+ Qcoso) is
r*t.rrJ
^,,
r{l + r[ cose =, t^13 = ,onç[";[
rt
The"
*he
e
e1*;ho"t
becornes
O = 2, !i = ?, Q".ìn ó coeo -ArC9*Qcosg ôtno +t'\Ra,ø
=jI
Fur tr5 larSel
2rÍco6o- ZswrJt + lAgR =o
qå
J¿rxsc
= #"
-fl¡'¡tt :
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