!
" #$!
%&'()*+ , -.0/1
243
56879/12)
: 2*;
<= >2?@AB7C/12)+ : 2*5DEB@(F7
/12)
GH
'2AIJ/F8K987L/12)
>M3@A7N/O2A
: 2<2*
P 24Q5+7C/12)
R;*<ST'ED! U/12)+ GH2A;V;
FW XZY\[+[
]_^a`\b[@cd`e[fhgjie[akVgj^dlm[anOop[+qAX;`\`\[$[riesp[alpop^
lp^$tu^dlOie^Cvxw4yzvx{'|~}aF;y
a^dlpoX
WhXY\[a^asie`\[ ]_^d`\bV[ cd`e[fhgjie[akVgj^dlm[ag opX;fhgjop^
[XtpXV`\lpX k;^dX;lplObie`e[[op[8^LWF^d[ nB X[Nomsp[lpgj4ieasp`\[g[
*umXV
kV^aL^[ ]_^d`b[ cd`\[fhg~iJ[ak;g^dlm[rg opX;fhgjop^[a^
tpn[alX;iJ[
Fasptpg~ieXV`;
@X;i\splp^p
1XVl1s X;iek@1spX@glOieX;`]_XV`XVlp^L^fhgXVlOi\^p
OopX kV`gjiJ[
lp^5X;]_X;`\X;lpkVg[anZlpX;`\kVg[an?opX^d`gcdX; (O) lm[!]_^d`[
¡ 6¢
£¤opX n^hk[aXVlOi\^Tom[$[ \ [TsplgjiJr`\g[ m k;^dn^hk[rom[+XV (P ) tu^FoXV`eT[rglpom[$X ie[a`
F¥ ]_^d`b[ opg gtm[ri\g~¨
sppX;i\gop^¥ ]_^d`\bV[ opX;fhgjop^¦[cd`e[ropgX;l1i\X opX5tp`\X ^
fa [ o^L^fFgjL X;lOie^ ~τ X[¦]_^d`\bV[!kV^d©1 spX[![ \\\ § [Lª«P~ Pom[!) qAX;`\`e[[ri\`e [ag?[!\ L[ \ [
splpgjiea`\g[ m k;^dlp¬pX;kVgop[zk;^d^z]_^a`\b[ (P~ )
WFXZY\[+[+oX k;`\gjb § ^$o^+^6fhgjLX;l1i\^@]_X;gjie[
lp^
`\X;]_X;`\X;lpkVg[anFZlXV`\k;g[anhopXkV^h^d`\opX;lm[aom[
opX^d`gcdX; (O) WFXY\[ (C) ^kVX;lOie`\^om[q)XV``e[ (C )
(C ) X (C ) ^ kVXVlOi\`\^ om[
4sp[
FWF^dnmX5^dsi\`\^ tpn[alpX;ie[ 1spX
XE®FXV`\k;XV.[ab § ^cd`e[fhgjie[akVgj^dlm[anFlm[L[ [ m WFXZY\[
`X;]_X;`\go^C[a^9`X;]_XV`XVlpk;g[anAZlpXV`kVg[an ie`XVn[
−→ ^9fdX(ie^d`+t¯^ gb § ^om[9tp[a`i\°kVsn[
OP
³ g~®[ ¢ ´ ]_^d`\bV[C`\X spnjie[alOieXom[ ^dmL[9fdX;i\^d`\g[an'om[ ]_^d`b[ 1spXL[riesp[a²± XV (P )
[akVX;nXV`\[!^faX;ie^a`@tu^ gb § ^op[zL[ [!splpg~iJa`g[ m XTtu^hopX XV`5XE®t`\X [lp[!]_^a`\L[9
N
X
F~i(P )
i=1
(
(P )
eso
L
^dlopXa
m.
s
i
−→
N
X
d2o OP
~ (P ) + P~eso (P ) +
=
P
F~(P ) + ~τ (P )
dt2
i=1
xµd¢
z
ω
Referencial Horizontal
(Nao Inercial)
Cs
Sol
Cl
Jupiter
P
r
C
z
x
Lua
y
Terra
Venus
x
O
Referencial do
centro da Terra
(Nao Inercial)
Referencial Inercial
(Estrelas Fixas)
Urano
y
³ gjcdsp`e[
i\XV`L^@[]_^a`\b[T1spX
[akVX;nX;`e[T^TfdX;i\^d`Bt¯^ gb § ^ −OP→ om[L[ [ m X; `\X;n[ab § ^
1
[a^z`\X;]_X;`\X;lpkVg[an?ZlpXV`kVg[an £ ¢
i\XV`L^L[N]_^d`b[NopX;fhgjop^[a^8cd`\[aopgX;lOieXzopXtp`\X ^ lm[8[ [ kV^an^hk[aom[
2
XV (P ) lp^8`\X(]_XV`XVlpk;g[an (h) 1spX z^X L^N1spX\^e§ `X;]_PXV~`XVlpk;g[an
¥a^NmZlpXV`kVg[an'op^
XVlOie`^om[!q)XV`\`\[
[ ¢~ XE®FXV`\k;gom[CtuXVn[C[rie`\[ab § ^Ncd`\[fFg~iJ[ak;g^dlp[anAop[NL[ [ om[
e
i
;
X
\
`
^
z[
9
_
]
d
^
`
b
3
qAX;`\`e[ ^ap`\X[ztm[a`i\°k;Pspn[ m kV^dnj^FkV[aom[¦lp^tu^dlOie^ (P ) l^`X;]_X;`\XVlkVg[rn (h) M
ieX;`\^ T[]_^d`b[¦opX(fFgjop^z¥ ]_^d`b[ cd`e[fhgjie[akVgj^dlm[ag F~
pg
j~
&om[!?sm[F
mWF^dn
4
X^asie`^ kV^d`t¯^ k;XVnjX i\X [riesp[alpop^XV m kV^dnj^FkV[aom[¦X; (P ) lp^z`\X(]_XV`XVlpk;g[an (h)
i\XV`\^ e§ ^[ ]_^d`b[ opg \ gjtm[ri\gjfr[ ~τ op^^fFgjLX;lOie^ ~u hsX$[L[ [ m [aop1spgj`\X
5
^dX;faX([L¨ [rX¦b § lp^z^aieom[a[ `@ h]_s^dX`blm[ [ 1XVsp1smX[a[ab cdX;^ lp^tusp^dt lO
FieXE^ ¨ (PX[z) fdopXV^znj^F`\k;X(g]_omXV[r`opXVlpXk;[ag[alpn cd(h)
om[qAX;`\`e[
p
s
n
a
[
#
`
§
ω
~
xµd¢
^d`^asie`^n[rop^p
h[]_^d`b[hsXT[akVX;nX;`e[^fdX(ie^d`t¯^ gb § ^ −
→ om[L[ [slpgjiea`\g[
OC
*1sm[alpo^kV^dnj^FkV[aom[Nlp^8kVXVlOi\`\^om[8qAX;`\`\[ ¢ op[aom[
)lp^`X;]_X;`\XVlkVg[rnBZlXV`\k;g[an
m£
mtu^d`V
¢
−→
X
d OC
m.
= P~ (C) + P~ (C) +
F~ (C) + ~τ (C)
O¢
dt
^dlopXa
o
o
o
eso
o
i
o
N
2
o
2
eso
i
µ
i=1
i\XV`L^ @[]_^a`\b[T1spX@[ak;XVnX;`e[^TfdX;i\^d`t¯^ gb § ^ −OC→ om[L[ [ m X; `\X;n[ab § ^
[a^z`\1X;]_X;`\X;lpkVg[an?ZlpXV`kVg[an £ ¢
i\XV`\^ $[]_^a`\b[ P~ XV m k;^dn^hk[aop[X; (C) opX(fhgop^[a^cd`\[aopgjXVlOieX@opX+t`\X e§ ^
2
ieX;`\^ [!]_^a`\b[ P~ XV m kV^an^hk[aom[XV (C) opX;fhgo^L[r^!t¯X ^zom[!tm[a`¡ie°k;spn[
3
i\XV`L^ [!]_^d`b[ ~ g
µ
~j &XV m kV^dnj^FkV[aom[zXV (C) opX(fFgjop^9¥z]_^d`\bV[ cd`e[6¨
4fFg~iJ[ak;g^dlp[ag op[?sm[F
uF
opX;fhgo^T[ WF]_^d^an`\bX[ ^dsopi\g `\^ gj tmk;[r^di\gj`\frtu[ ^ kVX;on^$X LieX ^ fhg XVlOi\^+om[+L[ [ k;^dn^hk[aop[
\
i
V
X
\
`
^
\
~τ
m
5
XV (C)
@^aieXE¨ Xz1spX P~ = 0
P~ (C) = 0 X ~τ = 0
4tu^dg lp^ XVlOi\`\^op[8q)XV``e[
^cd`\[aopgjXVlOieXopXtp`\X ^
A[N]_^d`b[Nt¯X ^X[ ]_^d`b[ opg gjtm[ri\gjfr[ ^8i\^Fom[
lh sn¢ [
^ai\X5¨ X@iJ[a¦ V\e§ 1spX
XV d¢ spt FXE¨ X
l1 spn[[fdX;n^h\kV gjom[aopX ω~ ToeX+§ `\^riJ[ab § ^
om[q)XV``e[ ´ tm[a`¡ieg`#oX O¢ tu^FopXE¨ XTXVlOi § ^X kV`X;fdX;`V
o
o
o
eso
o
i
o
(C)
(C)
eso
−→
N
d2o OC X ~
m.
=
Fi(C)
dt2
i=1
¢
^dlopXa
e
i
XV`L^ [¦]_^d`\bV[1spX[akVX;nX;`e[!^faX;i\^d`5t¯^ gb § ^ −OC→ op[L[ \ [ m kV^dnj^FkV[aom[!XV
1
¢i\XVXV`©L^ `XV+[n[ab § ^a^[a[T^ fd gX( iei\^dXV`Lg[a[!nmomZlp[ XV `]_kV^dg`[rbn [ £ cd¢ `\ [fhg~iJ[ak;g^dlm[rg ~ opX kV^d`t¯^ kVX;nX ieX
2
W F^dnx
m?sm[
ptn[alpX(iJ[
mX;i\k ¢
m1spX[ri\sm[a©lp[ztp[a`i\°kVsn[ mFk;^dn^hk[rom[¦¢ XV ¢
^ai\X(¨ XT1spXXV ¢ e§ ^!kV^dl gopX;`e[aop[ ie^hom[ [ ]_^a`\b[ cd`e[fhgjie[akVgj^dlm[ag 1spXr
[riesF¨
[alpop^zXV
tp`\^hops;XV©^ ^fhgXVlOie^ om[z4sp[zXop^LWF^anXV `\XVn[ab ^z¥zq)XV``e[
¯X;,kV^d^z ^d¢ sie`\[ ]_^d`\bV[ opXXVl^d`5[aca lpgji\spopXT`\X;n[ri\gjfr[ [r^ tpn[alX;iJ§ [ astpgji\XV`V
`e[rlp^LXT^asie`^
X \ []_^d`[
Fom[ ³ gcasp`e[ lp^aie[¨ XhsXa
−→ −→ −→
CP = OP − OC
a¢
Xtu^d`ie[alOie^p
o
o
i
−→
−→
−→
d2o CP
d2o OP
d2o OC
=
−
.
dt2
dt2
dt2
WFX;cdspXE¨ X1spX ps Fie`e[rglpop^¨ X ¢ po X µd¢ ^aieX;z¨ X
O¢
−→
−→ #
i
Xh
d2o OP
d2o OC
~
~
~
~
Fi (P ) − Fi (C) + ~τ (P ) m.
= P (P ) + Peso (P ) +
−
dt2
dt2
d¢
"
^dlopXa
ieXV`L^ []_^d`\bV[¦`\X spnjie[alOieX$lp[¦[ [ m kV^dnj^FkV[aom[X; (P ) opXplpgom[F
kV^alF¨
1
]_^d`\X![9XV1sm[rb § ^ (6)
¯t¯X;n[C[ak;XVnX;`e[ab § ^9op^CfaX;ie^a` −OP→ − −OC→ ~r
X;R`\X;n[ab § ^C[
£ ie¢ XV `L^ @[$]_^d`b[ ~ [riesp[alpop^XV opX(fFgjop^[a^cd`e[ropgX;l1i\X5opX#tp`\X \\§ ^X;
2
P ~¢ [riesm[rlpop^mX; opX;fhgjop^[[rie`\[ab § ^cd`e[fhgjie[akVgj^dlm[a(Pn*om)[
\
i
V
X
\
`
^
9[
8
_
]
a
^
\
`
b
[
3
m
[ [zom[ M op[q)XV`(`eP[zXV ) (P )
ieX;`\^ $ogj]_XV`XVlpbV[XVlOi\`\XT[ ]_^d`\bV[ cd`\[fFg~iJ[ak;g^dlp[ag X; ¢ X ¢ hsXT[riesp[XV
4
kV^an^hk[aom[¦X; ¢
m
i\XV`\^ []_^a`\b[opg gjtm[rieg~fr[ ~τ ¢ op^ ^fhgXVlOie^ ~u
01spX[ L[ [ L[ m
5
kV^dnj^FkV[aom[!XV (P ) [aop1spg`X ^d[z[ab § ^op[ ^dsie`\[ ]_^d`\bV[ 1spXlpXVn[[iesm[a
£$lpopX!
^!hsp[a`i\^zi\XV`L^!opX TkJ¬m[r[ao^ ³ ^a`\b[¦q)`e[iegjfr[oXª[a` T~ ¢
^ds
iJ[a¦ V ³ ^a`\b[ TXV`\[aop^d`\[!opXª [ad` ¢
o
o
o
eso
o
o
T~ =
N h
X
i
F~i (P ) − F~i (C) .
^FopXE¨ XX;lOi § ^X kV`\X(fdX;`$[!XV1sm[ab § ^ d¢ lm[!]_^d`[
i=1
−→
d2o~r
d2o CP
m. 2 = m.
= P~ + P~eso + T~ + ~τ .
dt
dt2
O¢
d¢
XE®t`\X e§ ^ d¢ 5[ ± 1sm[rb § ^ ³ splpom[aXVlOie[anpop^ £$k;X[alp^ X k;`\g~iJ[TXV i\XV`L^
´
op^ `X;]_X;`\XVlkVg[rn$ZlpXV`kVg[an$om[ ± i\`\X;n[ ³ g ®[ £ ¢ ^a k;^dlpopgjb § ^ opXfaXVn^hkVgjom[aopX
[alpcdspn[a`¦opXN`^aiJ[ab § ^ ω~ om[q)XV``e[
[ kV^alpopgb FX opXCfdX;n^hkVgop[aopXN[alpcaspn[a`
opX`^aiJ[rb[a^zl1spn[z^`\X;]_X;`\X;lpkVg[an?k[a`¡ieX Vg[rlp^ ¢ op^ X;lOie`\^Lom[zqAX;`\`e[!ieX;©^ @ X;s
XVg~®F^ ®F^ lm[ ± ie`\X;n[ ³ g ®[
W 0@
@£ £ ª£5+¡ª
q@£ £ ³ ´ @£ ´ ±
±
¢
ª± ± ± ´ @£ ´ £ ± ±± ± ´ £ ¢
XE®Ftp`\X \e§ ^ O¢ ]_^d`lpXVk;X[]_^d`\bV[¦1spXTtp`\^hops $[![akVX;nXV`\[ab § ^¦op^¦fdX(ie^d`#tu^ gb § ^
´
om[L[ \ [ m k;^dn^hk[rom[8X; ¢
−CP→ = ~r
*`X;]_X;`\go^[a^`\X(]_XV`\X;lpkVg[anZlpXV`kVg[an £ ¢
cdsp`etupk^dVlp^ op^¨T XL[9[ag `^alpiJ[r[aieb sp§ `\^C[anom
F[CXVlOq)ieXV``\X;`\ie[9[alOl1iesp^pn
m[ ]² [ ; XV[a`5`\[¦[C]_[N^d`\[abVlm[!ahnjg s XXopt^`\^h op]_s X;ld[z[aXVkVlpX;^ nXV `\[a^hbkVXV[a^!lpop^r^ ¨
fdX;i\^d`Bt¯ ^ gb § ^ XV`B`\X(]_XV`\gjom[[a^`\X;]_X;`\X;lpkVg[anu § ^ZlpXV`kVg[anmop^ XVlOie`^om[qAX;`\`e[ § (C)
X sptu^d`¡¨ X[!`\^aie[ab § ^zom[!q)XV``e[!l § ^zl1spn[F
pt¯^ag X [ [zk;^dlpopgjb § ^z`X[an fhg ie[!op^
`\X;]_X;`\X;lpkVg[an £ ¢\¢
)kV^a 1spXie^hop^ ^ ^fFgjLX;lOie^ lm[ sptuXV`]_°jkVgjXom[8qAX;`\`e[ e§ ^p
z
ω
∆θ
r(t+∆t)
Referencial
Inercial
∆r
P
r(t)
γ
x
C
ο
y
Referencial Nao Inercial
³ gjcdsp`e[ µ
XV©cdX;`e[anx
popX kV`gji\^
g i\XVL[opX!`X;]_XV` Vlpk;g[k;^dn^hk[rop^9lm[ spt¯X;`]_°jkVgXom[9q)XV`\`\[ ¦kJ¬m[aL[aop^CoX
`\X;]_X;`\X;lpkVg[an @^d`\g V^dlOie[an ¬ ¢ ³ gc 6¢
tu^dg ieX;,^¬p^d`g ;^dlOieXn^hk[an?k;^dL^Ltpn[rlp^LoX
`\X;]_X;` ;lpkVg[X#^ X;g~®F^ k[r`ieX Vg[alp^ tm[r`e[anjXVn^ [a^ o^`X;]_XV`XVlpk;g[ankV^dnj^FkV[aop^Tlp^ XVlh¨
ie`\^Lom[Lq)XV``e[ ¢ +g i\^Lo^`\X(]_XV`XVlpk;g[an)ZlpXV`kVg[an4opX¦£$`gcdX; (O) ^LL^fhgXVlOi\^
om[tm[r`ie°jkVspn[ kV^dnj^FkV[aom[lp^tu^dlOie^
1lp^`\X;]_X;`\X;lpkVg[an @^d`\g V^dlOie[an ¬
hi\XVHXV
`\XVn[ab § ^[a^TXVg ®m^opX#`^aiJ[rb § ^Tom[$q)XV``e[(P
1sp)L[
faXVn^hkVgjom[aopX5[rlpcdspn[a`V
ω~ 6= F
O¢ kV^dl]_^a`\X
glpopgjk[ao^!lm[ ³ gc
[ fdX;n^hkVgop[aopX[alpcdspn[a` ω~ [ LX L[ tm[a`\[i\^Fop^ ^ t¯^dlOi\^ om[ qAX;`\`e[F
\
±
glpk;nspgjlpop^^ tu^dlOie^ (P ) X8^ `X;]_XV`XVlpk;g[an @^d`\g V^dlOiJ[rn ¬ ¢ opX8kV^h^d`\oXVlm[aop[
0tu^dg
XVnXCie[a¦V cdgj`e[kV^a fdXVnj^Fk;gom[ropXN[alpcdsn[a` ω~
XV `XVn[ab § ^ [a^`\X(]_XV`XVlpk;g[an ®F^
lm[ ± ie`XVn[ ³ g ®[ £ ¢ @X [ kV^dlpogb hX [TfdXVnj^Fk;gom[ropX#iJ[alpcaXVlpk;g[an ~u op^faX;i\^d`
t¯^ gb § ^ ~r = −CP→
pX;©`\X;n[ab § ^[a^ XVlOie`^Lom[zq)XV`\`\[ ¢ om[aom[zt¯^a`V
−→
dCP
d~r
~u =
=
=ω
~ ∧ ~r,
¡
¢
dt
dt
tu^dg s[tm[a`¡ie°k;spn[$oX
L[ \ [ m kV^dnj^FkV[aom[Tlm[ sptuXV`¡]_°kVgjX5om[q)XV``e[
1lp^t¯^al1i\^
XL^fhgXVlOiJ[XV ie^d`lp^ op^ X;g~®F^ opX`^aiJ[ab § ^ om[ qAX;`\`e[ k;^d faXVnj^Fk;gom[aoX
(P )
[alpcdspn[a` ω~ kV^al]_^d`\XTX 1spXVL[rieg [ao^Llm[vx{|~}ah Vy
[vx{|~}ah Vyf (¨ X1spXlp^znggji\X+tm[r`e[ ∆t ⇒ 0 ^aieX;z¨ Xa
T
T
lim
∆~r
∆t
njXV¦p`\[alpop^1spXa
¡ d 6¢ ^aieX;z¨ Xa
∆t→0
=
d~r
dt
=
|~r| senγ
|~ω ∧ ~r| = |~ω ||~r|senγ
~uT =
XhsXa
d~r
=ω
~ ∧ ~r
dt
|~ω | =
¡ d 6¢
s i\gjiesglpop^¨ XXV
ω
~ ∧r
|~ω ∧ ~r|
dθ
dt
dθ
dt
¡ µd¢
1 spX $[faXVnj^Fk;gom[aoX@ie[alpcdX;lpkVg[an
OXE®t`\X [!tuXVn[XV1sm[rb § ^ ¢
hk;^dH1spX
i\^Fom[
[ kV^dg [ ¥9opg i alpk;g[ |~r|senγ ³ gc µa¢
op^CX;g~®F^CopX!`^aiJ[ab § ^9om[9q)XV``e[Ctu^ \ spXV8
[5XVX lop^$[5fa^ XVrln^hpkVcdgjspomnj[a^ opγXA[@iJ[ak;lp^dcan[rXVlpi\gjk;i\gsp[aopnx
Xt¯opX;X n^0 ]²[r¢ i\ ^#q)op^FXBop[0^ qA¬mX;[r`\`ep[#gjiecd[agjlO`e[aieX ` kVom^d[5qAfdX;XV`\nj`\^F[
k;gX om [ri §op^ X[aslpYcdX;sgji\n^[a`
\ XVie^a`\lp^
op^+X;g~®F^$@^d`i\X(¨¡Whspn
XV`XVn[rb ^
[a^+`X;]_X;`\XVlkVg[rnhop[ i\`\X;n[ ³ g~®[
§
±
ω
~ 6= 0
± q d ´ ± B~ $£ Wh¡WFq ± ª ´ d@ a
q ´ ¬ ¢
WFXZY\[^9fdX(ie^d` B~ lm[ sptuXV`¡]_°k;gXop[qAX;`\`\[9LX;opgop^XV `\XVn[ab § ^C[a^9`X;]_XV`XVlpk;g[an
^d`g V^al1ie[an<opXz^d`gcdX; ¬ ¢
B~ ∀ | B~ = B ~i + B ~j + B ~k
4^dlpopX B
B X
^[ k;^dLtu^dlpX;lOieX kV[a`i\X g[alm[ ^d`¡ie^dca^dlm[ag XVcasplpop^^ fdXV` ^d`X
oX
B \§
hopspnj^ |i~ |
|j~ |
|k~ | = 1
mk; ^d i~ , j~ , k~ tp`\X ' ^ [a^`\X(]_XV`\ X;lpkVg[a n ^a `\i~g ;,^dj~lOiJk~[an?oX
kV^h^d`\opX;lm[aom[ ie[alpcdX;lOieX¥ sptuXV`]_°jkVgjXTom[!qAX;`\`e[
WFX XL^faXX;`\X;n[ab § ^8[a^8`\X;]_X;`\X;lpkVg[an ^d`g ;^dlOiJ[an<oX^a`\gcaXV (h)
)kV^d
fdXVnj^Fk;gomB~[ropX u~ 6= u~
[¦fr[a`\g[ab § ^!op^ B~ k;^d©^!ieX;Ltu^ op[aom[!t¯^a`V
h
h
h
h
x
y
z
x
z
h
h
h
h
h
h
y
h h
h
h
h
T
h
¡ O¢
@X X g i\XVL[TopX
`X;]_X;` VlkVg[^kV^dlYsl1i\^
X F®^lp^`\X;]_X;`\X;lpkVg[an ^r¨
`\g ;^dlOiJ[a\n ¬ ¢ X0cdg`\[
k;^d X;nX k;^d faXVnj^Fk;gom[aoX0[ai~lpcaspj~n[a` kω~~ XV `\X;n[ab § ^T[a^$`X;]_X;`\XVlkVg[rn
ZlpXV`kVg[rn?kVXVlOi\`e[aop^lm[!^d`gcdX; (O)
£ g i\XVL[LZlpX;`\k;g[anx
utu^d`@^dsi\^n[rop^p
l § ^`\^riJ[ri d`\gj^LXlpXVnjXa
usp ^d X;`fr[aop^a`
f a
l § ^ ^aLX;l1i\XT[ k;^dt¯^dlXVlOieX k[a`¡ieX Vg[rlm[ X kV[an[a`X oX B~ fr[a`\g[a`\X;.k;^dH^
ieXVtu^p
pL[ iJ[a ; ^ fdX(ie^d`X splpg~iJa`g^ i~
j~ X k~
u~h =
dB~h
dt
!
=
h
dBx ~
dBy ~
dBz ~
ih +
jh +
kh
dt
dt
dt
h
h
h
h
h
dB~h
dt
!
=
h
h
¡ ¢
dBx ~
dBy ~
dBz ~
di~h
dj~h
dk~h
ih +
jh +
kh + B x
+ By
+ Bz
dt
dt
dt
dt
dt
dt
d^ lopX#[ opX;`\g~fa[rom[ op^ fdXV` ^d`X i~
j~ X k~ XV `\X;n[ab § ^[a^+i\XVt¯^ , ,
eXV§lO^Li om^[L X [!]_^d`\L[!1spX¦[zX(®Ftp`X \\§ ^ 6µd¢
= ω~ ∧ ~r
kV^dl]_^a`\X0Y\!]_^dg?fhg ie^
§
= ω~ ∧ j~ X = ω~ ∧ k~ opX¦]_^d`[1spXzt¯^hopX(¨ X¦oX ¡ ¢
=ω
~ ∧ i~
X k;`\X(fdXV`5[opX;`\gjfr[aop[opX B~ X;`\XVn[ab § ^[a^ieX;Ltu^l^`X;]_XV`XVlpk;g[an ZlpX;`\k;g[an (O) lm[
]_^d`\L[
O
h
h
h
d~r
dt
di~h
dt
h
dj~h
dt
dk~h
dt
h
di~h
dt
dj~h
dt
dk~h
dt
h
h
dB~h
dt
!
=
O
dB~h
dt
!
+ Bx ω
~ ∧ i~h + By ω
~ ∧ j~h + Bz ω
~ ∧ k~h
h
¡ d¢
^ds X Y\[
¡ O¢
@^aieXE¨ XNX;
hsX[fr[a`g[ab ^kV^d=^i\XVt¯^op^faX;ie^a` [ ^hkVg[rop^¥
[ [ m lp^tu^dlO¡ ie^d¢ ¢ 9opX k;`\gjie[§ lp^`X;]_XV`XVlpk;g[anZlpXV`kVg[an (O)B~ ^dL[a lpo^r¨ X9[
fa[r`\g[rb § ^9oX B~ kV^d ^ieX;Ltu^Clp^C`\X(]_XV`XVlpk;g[an* § ^CZlpX;`\k;g[an (h) Whg i\XVL[ ^a`\g~¨
V^dlOiJ[rn ¥¦fdX;n^hkVgop[aopXTiJ[alcdXVlpk;g[an¯opX ~ k;^d^oXmlpgjom[!lm[zXV1sm[ab § ^
B
6µdh¢ sp [an1spX;`
^d¢ lF]_^d`\X[8X;hsp[ab § ^
C
[
d
f
;
X
n
h
^
V
k
j
g
m
o
a
[
p
o
z
X
J
i
a
[
l
d
c
V
X
p
l
;
k
g
a
[
'
n
o
8
^
a
f
;
X
e
i
a
^
`
~
u~ = ω~ ∧ B~
Olp^T`X;]_X;`\XVlkV¡g[r 6µan ¢ ^a`\g ;^dlOiJ[an ¬ ¢ 0go VlOiegjk[T¥
faXVn^hkVgjom[aopBXiJ[alpcaXVlpk;g[an
op^TfdX(ie^d`tu^ gjb ^ om[TL[ [ lp^`X;]_XV`XVlpk;g[an
Ot¯^dg [a¦u^ ^
u~ = ω
fdX;i\^d`\X ~ ∧iJ[rr~lpcdXVlkVg[rg e§ ^opXVk;§ ^d`\r`~XVlOieX om[\fa XVnjm^Fk;gom[aoX@oX+`\^riJ[ab § ^ ¢ ω~ op^ `X;]_X;`\XVlkVg[r n
tp`\X ^9[r^ X;lOie`\^om[!q)XV`\`\[ (C) X;©`\X;n[ab § ^z[ £ ¢
^d`ie[alOie^C[fr[a`g[ab § ^opX r~ faX;ie^a`$t¯^ gb § ^9opX m X; `XVn[ab § ^9[ (C) kV^d ^
ieXVtu^p
<opX(fdX X;`¦gcdsp[an ¥8fa[r`\g[rb § ^oX B~ faX;i\^d`!1sm[an1spX;`![ ^hkVg[rop^[ ¢ m XV
k;^d©^!ieX;Ltu^p
popXT]_^d`[¦1spXopX (16) tu^FoX(¨ XX k;`\X(fdXV`
glpgjkVg[anXVlOieXr
(h) ¢
!
dB~h
dt
dB~h
dt
=
O
!
+ω
~ ∧ B~h
h
h
h
h
h
T
h
T
h
h
h
h
¡ d¢
XVcaspX(¨ Xa
mlm[rnXVlOieXr
1spX
d¢ tu^FoX XV`?X k;`\gjie[
dB~h
dt
u~o =
ª [
6kV^d^
kV^d^p
dB~h
dt
h
=
!
+ω
~ ∧ r~h
h
dr~h
dt c
¡ O¢
X [+]_^d`\L[
^$^fhgXVlOie^ opX
rfFg i\^$op^$`X;]_XV` Vlpk;g[anhZlpX;`\kVg[an £ tu^FoX
X V`@X kV\` gjie^zX; ieX;`\^ op^zL^fhug~XVlOi\B^ ~u~ XV `\XVn[ab § ^z[a^z`\X;]_X;`\X;lpkVg[an ^d`g ¢ ;^dlOiJ[an
hsX $gcdsp[anu[a^!`\X(]_XV` ;lpkVg[anuop^ XVlOie`^zom[q)XV`\`\[ ¢ ¥tm[a`i\g`op^faX;ie^a` r~
(h)
k Vs6Y^$ hopspnj^ opgj]_X;`\X;l1i\X0op^
hopspnj^
opX ~r ^$`eagj^
Vo(C)
g^+op[+qAX;`\`\[ ¢
a[aopgk;g^dlp[alpop^r¨
X
pkV^dlF]_^d`\X[!XV1sm[ab § ^ 6µd¢
^zfaX;i\^d`#fdXVnj^Fk;gom[ropX ω~ ∧ r~
´ ± ´ @£ ³ ´ ª ± @q ´ ´ Tª ´ £W£ ± ´ +£W
ª[ [XV1sm[rb ^ ¦[ hsp[ab ^ ³ splpop[aLX;lOiJ[an)om[ gjl agkV[Lop^ £Tk;X[al^
opX kV`\g~iJ[ XV `\X;n[a§ b § ^(9)[a^`X;]_X;± `\XVlkVg[r§ n ZlpX;`\kVg[an0opX^d`\gjcdXV (O) spt¯^alpop ^r¨ XN[fdXE¨
n^hkVgop[aopXN[alpcaspn[a`!op[ q)XV`\`\[ ω~ [a`\[ X kV`X;faXV`9X \ [XV1sm[ab § ^ X; `\X;n[ab § ^
[a^z`\X;]_X;`\X;lpkVg[an ^a`\g ;^dlOiJ[anx
F1spX ^dlpoXTX ie!^L
[fFgj^L£Tk;X[al^dcd`e mkV^p
^dNkV^alpopg~¨
k § ^LopX ω~ 6=
opX;faX(¨ Xi\`e[am[rn¬m[a`#k;^d^L lm[XV1sm[ab § ^ O¢ [riesp[alpop^r¨ X opX]_^a`\L[
[almanj^dcO[¦kV^aL^ ^ap`\X^!fdX(ie^d`51sm[anjhsXV` B~
mL[ [acd^d`\[¦gopX;lOieg pk[aop^zk;^dL^¦faX;i\^d`
fdXVnj^Fk;gom[ropXT1sm[an1spX;` u~ ~u £TtuXV`\[alpop^¨ XopX \ [z]_^d`[!X; ¡ d¢ ^dieX;z¨ Xr
u~o =
o
dr~h
dt
h
h
+ω
~ ∧ r~h
C
h
h
h
o
¡ O¢
WFs i\gjiesglpop^¨ XB^
fr[anj^d`4opX L^ ie`\[aop^
lm[
X;hsp[ab § ^ ¡ O¢ VX O¢
r]²[ ;XVlpop^¨
X u~ = ~u X[aopgji\ glpo^¨ X r~ u∼=~ ~r X; cdspX(¨ Xr
du~o
dt
=
O
du~o
dt
+ω
~ ∧ u~o
C
o
o
h
d~u
dt
d
=
dt
d~r
dt
+ω
~ ∧ ~r
+ω
~∧
± ]_X;i\sm[alpo^r¨ X[ ^at¯X;`e[ab FX gjlpopgjk[aom[
O
C
C
d~r
dt
+ω
~ ∧ ~r
C
µ ¢
.
C
x µF 6¢
^d^8^8faX;ie^a`fdX;n^hkVgjom[aopX[alpcaspn[a` ω~ om[q)XV``e[Ltp`\[riegjk[aXVlOi\Xk;^dl iJ[alOieXr
X VcaspX(¨ Xa
m]²[ ;XVlpo^ = 0
d~u
dt
=
O
d2~r
dt2
d~ω
∧ ~r
+
dt
C
d~r
+~ω ∧
dt
C
d~r
+~ω ∧
dt
C
+~ω ∧~ω ∧~r
C
d~
ω
dt
d~u
dt
^dieX;lpop^r¨ X mlm[anjLX;lOieXa
=
O
d2~r
dt2
µdµa¢
+ 2~ω ∧ ~u + ω
~ ∧ (~ω ∧ ~r)
C
µ d¢
hsX L[8fr[a`\g[ab § ^8k;^d«^NieXVtu^op^8fdX(ie^d` ~u fhg iJ[8op^`X;]_XV`XVlpk;g[an'ZlXV`\k;g[an (O)
^ai\X(¨ XhsXX;n[![aopgjkVgj^dlm[!¥¦fr[a`\g[ab ^!k;^d©^!ieX;Ltu^!opX
pX(]_X;ieg~fr[aLX;lOieXTXVopgjom[
lp^N`\X;]_ X;`\X;lpkVg[an ¢
L[ag opsm[ tp[a`\§ k;XVn[ 1spX X;` § ^NgoXVlO~uieg mk[aop[ L[ag [aopg[rl1i\X
^ai\X(¨ XT1spXT[ tm[r`\kVX;n[ opX$fdX;n^hkVgjom[aopX glpk;^d`\tu^d`\[aom[ ^!oXV`\g~fr[aom[ op[fdXVnj^Fk;g~¨
om[aopX[a lpcaspn[a` ω~ l § ^l1sp n[!om[q)XV``e[ 0[ ^Lk;^dlOie`\a`\gj^LB[¦efr§ [a`g[ab § ^!opX ~u X; £ ¢ X
XV ¢ XV`g[a©gjcdsm[ag
WFsp iegji\spglop^r¨ X µ O¢ lm[X(®Ftp`X e§ ^ d¢ lp^aiJ[rlpop^r¨ XN1spXa −OP→ − −OC→ =
X1spX
=
^dieX;z¨ Xr
d~u
dt
=
O
d~u
dt
+ 2~ω ∧ ~u + ω
~ ∧ (~ω ∧ ~r) ,
C
d2
d2
dt2
d2 ~r
dt2
d2 ~r
dt2
O
m.
XXVlOi § ^p
m.
d~u
dt
d~u
dt
O
dt2
d~
u
dt O
+ 2m.~ω ∧ ~u + m.~ω ∧ ω
~ ∧ ~r = P~ + F~ + T~ + ~τ
µ ¢
= −2m.~ω ∧ ~u − m.~ω ∧ ω
~ ∧ ~r + P~ + F~ + T~ + ~τ ,
µ a¢
C
C
1 spX 5[+X;1sm[ab[r^]_slpom[aXVlOiJ[rnptp`\^hkVs`e[aom[¦
aX k;`\g~iJ[$XV `\X;n[ab § ^T[a^T`X;]_XV`XVlpk;g[an
^d`g V^al1ie[an ¬ ¢ op^ ^Fk;X[al^
rlp^riJ[alpo^r¨ X'hsX ¢ X ¬ ¢ e§ ^@^5LX ^5`\X;]_X;`\X;lpkVg[an
]_^d`b[NopX ^d`g^dnjg
?[tp`gXVgj`e[9tm[a`kVX;n[Nop^n[aop^Nopgj`\XVg~ie^NopX µ a¢
)hsX[acaX
´
lm[L[ [zsplpgjiea`\g[ m XXV©i\^Fop^ ^ ^ahYX;i\^ 1spX X^6faXV,k;^d fdX;n^hkVgjom[aopX ~u
hsp[an1spX;`@lp[ spt¯X;`]_°jkVgXr
p^dsNlp^glOieX;`\gj^d`0om[!q)XV`\`\[
opXplpgom[!lm[!]_^d`[
]´ ^d`\bV[kVX;lOie`\°jt¯X(iJ[
[ X;cdsplpop[tm[a`kVXVn[op^¦n[aop^opg`XVg~ie^oX µ d¢
h1spXT[acdX
xµ XVO ¢
_
ie^hom[ L[ \ [ lm[ st¯X;`]_°k;gXCom[q)XV``e[
B1spXV` XNL^fr[a kV^d df XVnj^Fk;gom[ropX ~u ^ds
l § ^p
X kV`gjiJ[!k;^dL^!]_^d`b[!kVX;lOie`\°~]_spcO[!lm[¦]_^d`[![ap[ag~®F^p
F~CC = −2m.~ω ∧ ~u
@ ^ ± 1sm[ao^d`XVn[i\XVog`\X;b § ^Ngjcdsm[anx
L[ opX X;lOiego^^dtu^ ie^8¥Com[9]_^d`\bV[Nxµ odX ¢
[rie`\[ab § ^zcd`\[fFg~iJ[ak;g^dlp[an
M ~r
,
F~ = −m.K.
µ O¢
r
|~r|
XVlpop^ [kV^al ie[alOieXcd`e[fhgjie[akVgj^dlm[an opX [fdX;lpopgjkJ¬
pkV^a 1spX[¦L[ \ [ M om[
qAX;`\`e[![ie`e[rg [L[ [¦splpg~iJa`g[ m
opX+]_^a`\L[1spX+fr[anjX+tm[a`\[ie^hom[ [ k;^dn[ri\gji\spopX
om[!q)XV`\`\[L[!XE®t`\X e§ ^9[am[ag ®^
γ
P~ = m.~g = F~ − m.~ω ∧ (~ω ∧ ~r) ,
1spX ¦[]_^a`\b[ P~ `\X spnjie[alOieXom[ ^d[zfdX(ie^d`g[an4om[L]_^a`\b[opX¦[ie`e[rb § ^cd`ex[µ fhOg~¢¨
iJ[ak;g^dlm[rnkV^d©[!]_^a`\b[!k;XVlOie`°j]_spcO[!lm[ztm[a`¡ie°jkVspn[ m
^d`ie[alOie^^faX;ie^a`5]_^a`\b[z`\X spn~iJ[alOi\X[iesm[alop^Llp[L[ \ [!splpgjiea`\g[ m kV^an^hk[aom[
XV (P )
oX k;`\g~ie^zt¯^a`0LX;opgb § ^¦opX ~u ]_XVg~iJ[¦lp^!`X;]_X;`\XVlkVg[rn ^d`g ;^dlOiJ[an ¬ ¢
pg ie^ r
lp^ ^Fk;X[al^
mtu^FoX XV`5X(®Ftp`X ^lp[z]_^d`[F
F~cf = −m.~ω ∧ (~ω ∧ ~r)
2
eso
eso
¢
^dlopXa
\
i
XV`L^@[]_^a`\b[`\X spn~iJ[alOi\X+[ri\sm[alpo^X; m
F`X;]_X;`\gop[[r^`X;]_XV`XVlpk;g[an § ^ZlpXV`¨
1
kVg[rn ^d`g ;^dlOiJ[an ¬ ¢
m1spXTXV1spg~fr[anX;l1i\X[a^z`\X(]_XV`\X;lpkVg[anop^ X;l1i\`\^om[zqAX;`\`e[ ¢
ieXV`L^ [¦]_^d`\bV[!opX;fhgo^L[r^ca`e[aopgjXVlOieXopXtp`X e§ ^ ~
2
P
e
i
V
X
`
L
^
[
¦
_
]
d
^
\
`
V
b
!
[
¯
t
X
^
~
3
ieXV`L^ [¦]_^d`\bV[!cdXV`\[aopP^d`\[z oXL[a` T~
4
ieXV`L^ [¦]_^d`\bV[!opg gtp[rieg~fa[ ~τ
5
ieXV`L^ [¦]_^d`\bV[!opX ^d`\gj^dng F~
6
m.
d~u
dt
= P~ + P~eso + T~ + ~τ + F~CC
h
o
o
o
eso
o
o
o
CC
¨ ´ X;hsp[ab § ^ ¢ [ ± 1sm[ab § ^ ³ splpom[rLX;l1ie[an4om[ gl agkV[!op^ £$k;X[alp^
6
¢
tm[a`e[CspL[Ctm[a`¡ie°k;spn[9oXL[ [Csplpg~iJa`g[ m sptu^dlpop^¨ X ω~ 6= ± n[ oX k;`\g~iJ[
XV`XVn[rb § ^N[a^N`X;]_XV`XVlpk;g[an< § ^NZlpX;`\kVg[an<om[Cq)XV``e[ @^d`\g V^dlOiJ[rn ¬ ¢\¢
?gjlpopgjk[alpo^
hsXC[[ab § ^op[ ^dL[CfdX;i\^d`\g[an
)om[ ]_^d`\bV[ i\`e[ri\gjfr[ opXL[a` opX(fhgom[ [a^Wh^dn
4sp[LX^dsie`^ tpn[rlpX;ie[
uop[z]_^d`b[zopXcd`e[aogXVlOi\XoXtp`X ^p
uom[z]_^d`\bV[opX[ri\`\gji\^p
om[z]_^a`\b[!tuX ^LXom[¦]_^d` b¢ [!opX ^d`\gj^dng
[akVX;nXV`\[z[zL[ [!spe§ lpg~iJa`g[ m
¨ ´ ]_^d`\bV[Lcd`\[aopgX;lOieXopXtp`\X \e§ ^ P~ [`\X tu^dl faXVn*tuXVn[t`\^dtm[rcO[ab § ^Lom[
d
µ
¢
kV^d``\XVlOi\X oX sptuXV`]_°jkVgjX8caXV`e[rom[ t¯X;n^fdX;lOie^p5tuXVn[tp`^dtm[acd[ab § ^om[ ^alpom[ cdX(¨
`e[aom[ tuXVnj^9fdX;l1i\^p
)hsX[ri\glpcaXV ie^hom[ [ tp`\[ag[ op[ gnj¬m[ XzkV^al1i\glpX;lOieX *tuXVn[
tp`\^dtp[acO[ab § ^opX5q splm[ag opX;kV^d``\XVlOi\X oX
[rm[an^ ' ° LgjkV^ FtuXVn[$tp`\^dtp[acO[abV[a^om[
^dlpom[opXL[a` lp[ a`X[ kV^ ieX;g`\[ oXie^hop^ ^ ^Fk;X[alp^ Xr
¯X;l1i\`\X^dsie`\[ L[ag
t¯X;n[tp`^dtm[acd[ab § ^om[ rcdsm[ [rhsXVkVgjom[ op[ a`X[ ie`\^atpgkV[ag XV.opg`XVb § ^!¥ r`\X[
t¯^dn[a`\X
d^alpopX#tuXV`\oXV i\XVt¯X;`e[ri\sp`e[F
aie^d`lm[a!¨ X L[ag opXVl [ X5[r]_splom[a8
aom[alop^
^d`\gjcdXV«¥8kVgj`\kVsn[ab § ^NopXn[a`\cO[CX k[rn[NkV^dlp¬XVkVgjom[k;^dL^Nk;g`\k;spn[rb § ^NieX;`\^d¬m[anjglm[
op^ ^Fk;X[al^
¨ ]_^d`\bV[ `\X spn~iJ[+om[ ^dL[
fdX(ie^d`g[an1om[@]_^d`\bV[+opX#[ri\`e[ab ^+ca`e[fhgjiJ[rkVg^alm[an
Xom[CO¢ ]_^d`\´ bV[k;XVlOieP`~°j]_spcO[8 [ri\sm[alpop^ XV m 9´ ]_^d`\bV[k;XVlOie`°j]_spcO[ L§ [ag glOieX;l [8lm[
a`\XV[ 1sm[ie^d`g[anAX ¦l1spn[9l^ ^an^ ^aL^ie[an
[L]_^d`b[ LX;lp^ gjl1i\XVl [
lp^ ± h± sp[aop^d`$X6®gj[zlp^ ^dnj^ i\ XV `\`X ie`\X ± \ X¦]_XVl dP~XVl^9tp`\^hops ^C[akJ¬m[6¨
iJ[aXVlOie^!op[qAX;`\`\[¦lm[ a`X[¦tu^dn[r`\X X$sp [am[asn[aXVlOie^om[ r`\X[ ± hsp[rie^d`g[ag
opX;fhgjop^
¨ ]_^d`\bV[+cdX;`e[aop^a`e[
oX0[r` ~
a1spX0tp`^Fos 0^$opX nj^FkV[aLX;lOie^+oX
´
m
T
¢
[ [r`
l § ^ Tsp[]_^d`\bV[¦cd`e[fhgjie[akVgj^dlm[anukV^d^¦cdX;`e[anjLX;lOieX X spt FX '´ ]_^d`\bV[
`X snjiJ[
om[+opg~]_XV`\X;lpb[$oX0opsm[ ]_^d`\bV[ cd`\[fFg~iJ[ak;g^dlp[ag F~
dopX;fhgjom[T[+k[rom[$[ ie`\^ToX
T~
t¯X;` X
OspL[T[riesm[rlpop^ ^ap`\X5[TL[ \ [ m k;^dn^hk[aop[+lp^Ttu^dlOie^ (P ) op^T`X;]_X;`\XVlkVg[rn
^d`g V^al1ie[an ¬ ¢ X+^dsFie`e[[riesm[rlpop^ ^dp`XT[L[ \ [ m kV^an^hk[aom[lm[^d`gcdX;op^`X;]_XE¨
`\XVlkVg[rn*
¥a^9ZlpX;`\kVg[an ¢ lp^ XVlOie`^Com[q)XV`\`\[ $´ ]_^d`\bV[ T~ ieXV gjlm[an)lpXVcd[rieg~fd^p
^ds tu^ g~iegjfa^p
*oXVtuXVlpopX;lpop^opXL^dlpopXLX iJ[L[ [ m lp^`\X;]_X;`\X;lpkVg[anop[ XVlOi\`\^
om[9qAX;`\`\[ ¢
¯t¯^a`iJ[rl1i\^Clp^9tu^dlOie^ ¢
X!op^Cn[ao^XV 1spX¦^N[ ie`\^C1spXX(®FXV`kVX
[L[ab § ^zcd`e[fhgjie[akVgj^dlm[an¯X iJzXV©`XVn[rb § ^¥zqAX;`\`e[F
m^dsNlp^Ln[aop^!^dtu^ ie^p
mX;lp1sm[alOie^L[
]_^d`\bV[opXL[rie`\[ab § ^cd`\[fFg~iJ[ak;g^dlp[an F~
*XV«1sm[rn1spXV`op^ k[ ^
)ieX; XVtp`X gjlm[an
lpXVcO[iegjfa^p
mopX[ri\`e[ab § ^p
^1spX[ opg iegjlpcdspXTk;n[a`\[aLX;lOieX
X(fdX XV`¦lp^riJ[aop^hsXC[]_^d`b[opX[rie`\[ab ^cd`e[fhgjie[akVgj^dlm[an'X;lOie`\X9[[ [om[
qAX;`\`e[X¦[z [ \ [om[?sm[zopX(fdXV`g[
uopX[akV^a`\op§^LkV^d [XV1sm[ab § ^ xµ O¢
^hk[ g^d\ lp[a`5[
kV^dnjg e§ ^op^ [ ie`\^ \ ^¦l § ^[akV^al1i\XVk;Xa
ptu^dg fhg ie^ opX £ ¢
1op^dg [ ie`\^ 1spX$l § ^
kV^dnjgopX; !tu^d`\1spXz[r¦¯^ ie`\[al n[aom[a XV ie^d`lp^CopX!sp XVlOie`^opXª[ \ [ ^r¨
¦sp
p^as8l § ^zt¯X;`ieX;lpkVX; [a^ g ieX;[!cd`\[fFg~iJ[ak;g^dlp[an¯X;©kV^al gjopXV`\[ab § ^ @^aieXE¨ X
eso
eso
h sX[?sm[8
moX]²[ie^p
ml § ^cag`e[!X; ie^d`lp^op^Lk;XVlOie`^Lop[q)XV``e[ ¢
mkV^d^zcdXV`\[an~¨
LX;l1i\X X st hXa
<L[ ie`\[al n[aom[NXV«ie^a`\lp^op^ X;l1i\`\^opXª[ [ ^dsp«1spX
]_^d`\L[
k; ^d [@qAX;`\`\[ £ LX ^+tu^FoX X;`<opg~ie^+X; `XVn[ab § ^+[a^$k;^dlYsplOie^TWF^dn~¨ qAX;`\`\[
XLk;^d«L^hopg mk[ab FX t¯X;`iegjlpXVlOi\X iJ[a¦ VRXV«`\X;n[ab § ^[a^ ^dsi\`\^ tpn[rlpX;ie[ op^
g ieX;[ ^an[a`
¨ ]_^a`\b[opg gtp[rieg~fa[
FkV^d^![]_^d`b[oX ^d`\gj^dng !X(®Fg ieX+hsp[alpop^z¬p^dsh¨
fdXV`Td¢ ^fh´ gXVlOie^ ~u om [9L[ \ [ ~τm qA[anAk;^d^9lm[ ^Fk;gXVop[aopX ^hopXV `lm[
^alpopXlp[aom[
LX@^Lfhg^fdXXVlO ieX@^ l § ^¬pX¦^asfd^6X;`fhgjspLLX;[Tl1i\p^sptu`\^h^FkVo`\X [akVgXV[`$
1kV[[aops g [r opgtp^9[atub § ^d^`+l1§sm^[anjXE1®spg XV `@i\X ]_^d X
`\bVl [ § ^¬
^ds
ufd^dX;s `
\[ opX]_`\gjkVb ^ X ie[a`e t`\X XVlO ieX £ ]_X;lp^dXVlp^oX9opg P~gjtm[aT~b ^X
±
1
p
s
N
X
[
_
]
a
^
\
`
b
XVlOi\^§ op^
P^h~kVgX;om[aopX@ieX;H[ancds[[alm§ [rn^d~τcdg[$ lm[]_^a`\L[ og ]_^a `\X@k;^d1spX+^L^fhg
^Fk;X[alp^ Xzom[ ^hkVgX;om[aopX kV^dlpos VX;
4^dstuXV`\ oXV«XVlXV`\cag[ £ £$kVX rlp^NkV^dlpos
XVlpX;`\cdg[z lp[!]_^a`\TL [¦opX ^d`\opX; lm[aom[zom[¦ies`\pspn Vlpk;g[!¬p^d^dc ;lpX[F
´ X;lpXV`cdg[!tm[a`\[![
[al1si\XVlpb § ^¦om[ ^hkVgjXVom[aoX ie[a¦ VopX ^d`opXVlp[aom[ ¢
p^dstuXV`opX+X;lpXV`cdg[F
[ri\`e[f
op^¦]_XVlaLX;lp^!opXTkVg [anj¬m[aXVlOie^¯X; k;^dLtu^d`¡iJ[aop^
[ 0 ^hkVgjXVom[ropX
[rie`\[f om[
`\XVn[ab hX ogtpnj^diegkV[ ¢
F^asNi\sp`\spnX;l1i\^ +[ # ^hkVgX;om[aopX
[ri\`e[f op[zcaspXV``e[¦ie^r¨
iJ[an ¢ £ `XVkV^alp¬pXVk;gXVlOie^om[$`X[anjgom[ropX opX \ X ]²[rie^ ie^d`lm[T[T[alm[rn^dcdg[@k;^dl ie`\s°om[
opXkVXV`¡iJ[!]_^d`\L[
hfdXV`^ \ °XVn
[zog \ gtm[rb § ^zom[!XVlXV`\cag[!om[ L[a` t¯X;n^z[ri\`\gji\^ e§ ^cdX;`e[aom[ kV^dt¯^alpXVlOieX
opXzt¯X;hsXVlp^8]_splpop^8t¯^a`TkV^dt¯^alpXVlOieX tuXV`g hopgkV[ [ ^hkVg[aom[ ¥ d`pgjie[ op^WF^dn
X!om[C?sm[
gjlpopgkV[alpop^hsX!¬mCsp k;XV`¡ie^Ncd`ersopX¦opX(ieXV`Lgjlpg L^Llp^Ntp`\^hkVX \ ^8oX
cdXV`\[ab § ^+opX [ k;^dt¯^dlXVlOieX
O[a^
glOf op^+ka^ ie^aie[an1op^+^fhgXVlOie^5iesp`pspnjXVlOie^
L[LkV^al X;hs Vlpk;g[9uXV opX;i\XV`\glp[aom[N
om[opg gtp[ab[a^Lom[X;lpXV`cdg[lp^9k[ ^9om[
³ ^a`\b[ qA`\[rieg~fa[ opXª[r`
fdX;om[C[ ie`\^alp^dg[X ¦^N[rspLX;lOie^CopX¦]_^d`[Lcd`e[¨
opsm[anx
mt¯X;`\L[alpXVlOi\XXg``\X;faXV` °jfaXVn)om[zopg i alpk;g[!om[L4sp[X;,`\XVn[ab § ^L¥qAX;`\`\[
t¯^d`
`e[ FX opXLkV^dl XV`¡fr[ab § ^op^^dXVlOie^8[alcdspn[r`o^ g i\XVL[8q)XV``e[N¨+?sm[
)opX;kV^d`¨
`\XVlOi\Xom[!opgjLgjl1spgb § ^¦op^L^aLX;l1i\^z[alpcdsn[a`0op[q)XV``e[
¨ ´ ]_^d`\bV[opX ^a`\g^ang F~
Fk;^d^¦[]_^d`b[og \ gtm[iegjfr[ ~τ ¦X(®Fg ieX X$¬p^dsh¨
O
¢
fdXV`5^fFgjLX;lOie^ ~u om[![ [ m
mL[ l § ^ spL[]_^d`b[!psp`^Fk;`eri\gk[¦XT`X[an kV^d^
XVn[ '´ ]_^d`b[opX ^a`\g^ang 5]_^d`b[g`\`X[anmt`\^hops Vgjom[t¯X;n^L^fhgXVlOie^opX@`\^aie[ab § ^
om[Lq)XV``e[
kV^aL^L^ ie`\[z[ X;cdspgjl1i\XXVogb § ^X; tuXVl [aLX;lOie^ WFsptu^dlp¬m[z1spX
op^dg [fhg hX
rfFg i\^ o^$`\X(]_XV`\X;lpkVg[anhom[ ± ie`\X;n[ ³ g~®[ ZlXV`\k;g[an ¢ £ ¢
6tm[a`¡iJ[a op^
^dnj^
[a^n^alpcd^!opX st¯X;`]_°k;gXX ] V`gk[zcdX;^dt¯^rieXVlkVg[rn
l1spL[[rie^ ]_X;`e[ X;©fhg ¨
kV^ gom [aoX+1spX+l § ^^ [a``e[ ieXa
hXV og`\X;b § ^¥k;gom[aoX@oX X;n V,
g~iesm [rom[t ` ®FgL [
om[Lnjglp¬m[¦o^ 1sm[aop^a`
q)XV``\X ie`X ³ gc [zL^ ie`\[z1spX[a¦u^ Xopgj`\gcaXV8
pXV
ie`e[YX;i a`\g[ XV± HopX fhg^
op^ ^a#n^ ´ [ri [ k; gom[ropX
oX X;n V XV#tu^dg opX$[rncdspL[
¬p^d`e[
X+[q)XV`\`\[l § ^ X
L^faXV`V
hg ie^ a
X ω~
1XVnjX kJ¬pXVcd[a.¥ X;n V @^aieXE¨
X1spX[¦]_^d`\bV[ P~ T1spXTtp`^6f [fa[r`\g[rb § ^¦oXTopg`XVb § ^!op^ fdX(ie^d`X fdX;n^hkVgjom[aopX
a eso
CC
eso
o
Referencial Inercial
ω =0
x
Quito
x
x
z
z
ω
ω
x
xQuito x Belem
Belem
Terra
y
Polo Norte
Polo Norte
Polo Norte
z
x
Quito
Terra
y
Terra
y
Polo Sul
Polo Sul
Polo Sul
Fig3(b)
Fig3(a)
x
xBelem
Fig3(c)
³ jg cdsp`e[
op^ [fFg FX opsp`e[rl1i\X^ ie`e[YX;ie^ o^ dnj^ [ri XVn ;
^d^T[+q)XV``e[+i\XV ^fhgXVlOie^
opX0`\^riJ[ab § ^ ω~ 6= 0
sptu^dlpop^¨ X#1spX ^ [fhg hX
l § ^ XZY\[a,[a``e[ iJ[aop^ t¯X;n[![ri\L^ ]_XV`\[
1spX X^fFgjLX;lOiJ[¦kV^a©[!qAX;`\`\[
mg ie^ a
r[at [ X [ ¬p^a`e[ opXie`e[YX;ie^@XVnjX tm[ [a ^dp`\X[ [ l § ^+[ieXV``\g [a
~τ
lm[ ¢ kVgjom[aopX+opX $spgji\^ g~iesm[aop[lp^ ± hsp[aop^d` '´ ³ gc C^ i\`e [^!^hkV^a`\`\gjop^p
hfFg i\^
opXsp`X;]_XV`XVlpk;g[anBZlpX;`\kVg[an £ ¢ $g i\^op^`\X;]_X;`\X;lpkVg[anl § ^ZlpX;`\k;g[an ^d`g ;^dlOiJ[an
¬XVsN¢
)t^`\XfF gj^8L¥X;lO$ie^¦spX$g~ie^p^z
[^fhg [§ fh^¦g § 1^CspX+hsfdXzX;g^!faXVopgj^8^ o ^ ^dn^¦ ^d
n^^d`¡WFieXsn XT XzopopX X n ^hnkV^h^dkVsN^as ¥¥CogX `\ X;1gjiesp[XV`opom^ [N opXVs^
L^fhgXVlOie^ztm[r`e[!kJ¬pXVcd[a`\X;,[ri $spg~ie^!kV^aL^z^ i\`e[z[ ³ gc k
@^aieXE¨ X1spX¦^LL^fhgXVlOie^op^ [fFg FX
lm[ sptuXV`]_°jkVgjXX ] ;`\gkV[cdX;^dt¯^rieXVlkVg[rn
fFg i\^ op^ `X;]_X;`\XVlkVg[rn ¬ ¢ tp`X ^ ¥kVgjom[aopXNopX $spgji\^p
e§ ^opX kV`gji\^ kV^d^ X8[
qAX;`\`e[ l § ^ ieg~fdX X^fFgjLX;lOie^ oX`\^aie[ab § ^ ω~
51spXa
5fhg ie[ op^ `X;]_XV`XVlpk;g[an £ ¢
om[a[uq)X(¨ XV `X!`e[1
spX![Tgjq)lmXV[a`o`efa[XVi\`XVi\g om[r L$X;g l1 iJi\[LXTt¯opX;X `\k; XVp¢ gj
om^d[s k;^d ¬ ¢ ^!
¯^d¨ lpopXl § ^ X XVlOi\Xz[`\^aie[ab § ^
ω
~
ω
~
^dlF]_^d`\X+[X(®Ftp`\X \\§ ^
F[fdX;n^hkVgop[aopX5iJ[alpcaXVlpk;g[anm¨ u~
hopX$kV[aom[s op^
¡
6
a
µ
¢
[fhg FX
hsXLlm[ ³ gjc [Cl § ^N^Fk;^d`\`X!t¯^ag ω~
4lm[ k;^dlpopgjb hX om[ ³ gc X
k@kV^a
1tm[ [[X(®Fg iegj`'X+[rspLX;lOiJ[kV^d[opgjLgjlhsgb ^Top^fr[anj^d`[a ^dnsFie^
om[ n[iegji\spω~opX6= 0£[asp XVlOie^ opXfaXVnj^Fk;gom[aoX Xomlm[ X § [`e[ § ^ X; `\XV n[ab § ^
ieXVtu^p
kV^aR 1spX!^ [fhg hX k;`\sV[a[ n[ri\gjiesopX ´ fdXVnj^Fk;gom[ropX ie[alpcdX;lpkVg[ag
¨ u~ \§ ^tuXV`t¯X;lpopgjkVspn[a`\X ¥ faXVn^hkVgjom[aopX ~u k;^dl iJ[alOieX X; hopspnj^ ¢ op^ [fhg hX
kVsY^ X;lOiego^ ^ dnj^WFspn~¨ hsp[aop^d`TX an^8
^d`¡ieXE¨ 1sm[aop^d`X X ^d[rRfdXE¨
ie^d`g[anjL X;l1i\X 5e§ X snjiJ[r op[±kV^dtu^ gjb § ^[ i\`e[6YX(i d`±g[ glpogk[rom[ lm [ ³ gjk k
^ai\X(¨ X$1spXTlp^ dnj^ [faXVnj^Fk;gom[aoX
iJ[rlpcdXVlkVg[rn ¨ u~ +lhsn[
h[ XV X;n V8
h^ds
Tsgjie^
<lm[a`X[ ± 1sm[ri\^d`\g[anBXV cdXV`\[an
'[8fdXVnj^Fk;gom[ropXLie[alpcdX;lpkVg[an ®Fgj[NXV
µ
T
T
T
fa[rn^d`#[r ^dnjsie^
£ opX nj^FkV[aLX;lOie^
m¥X 1spXV`om[X¥opg`XVgjie[op^ L^fhgXVlOie^ op^ [fhg hX e§ ^p
XVlOi § ^
Agjl1i\XV`tp`\X(iJ[aop^ lp^`X;]_X;`\XVlkVg[rn ^d`g V^al1ie[an ¬ ¢ opX $spgji\^p
?kV^d^Ck[rs [ao^
t¯^d`
spL[]_^d`b[LtuXV`t¯X;lpopgjkVspn[a`+¥opg`XVb § ^opXtp`\^atm[acO[ab § ^9@^d`i\X!WFsnAop^ [fhg hX
kJ¬m[aL[aom[]_^d`b[opX ^a`\g^ang
XVlp1sm[alOi\^81spXC
¯fFg i\^ op^C`X;]_XV`XVlpk;g[anAZlpXV`kVg[an*om[
ie`XVn[ ³ g~®[ £
X;nX l ^!XE®g i\XV©X$tu^d`¡iJ[alOie^!l ^!t¯^hopX; XV`#k[rs [ao^ t¯^d`
]_± ^d `\bV[[rncdspL[
³ gjc ¢ ± § £ ¢ [tm[ \ [rcdXV ^dp§ `X $spg~ie^op XC]_^d` [n[ri\ XV`e[rn
[rkVie^a^ ^ ]_X; `e[[ fhg FXVX fafh^Og [akVlp^ o ^gopn[a[opieXaXV`\
[a!XnE®FXVkVlOni\s Xa
dgjfrL[a[ XVlOi\[rX¦l1i\opXVX;lpfhogjom^[C[[aop^Cg`XVb ^6§ fh^gjL@X;^dl1`i\i\X(^9¨WFopspX!n ¢ `
O^aXViJ[¨
b § ^zom[!q)XV`\`\[!XT^ ie`e[YX;ie^ #e§ ^ztuXV`\k;^d`\`gop^ lp^zLX ^!glOieX;`fr[anj^!opX$ieX;Ltu^!1spX
opsp`e[r`e[a ^ f F^ ^dnj^C
^d`¡ieXE¨ X;n V X ^dn^NWhspn~¨ XVn V8
¯lm[ sptu^ gjb § ^CoX¦spL[
qAX;`\`e[ X;,L^fhgXVlOi\^!opX`\^aie[ab § ^zk;^dL^!lp[ ³ gjc [
X(fdXE¨ Xlp^aie[a`4hsXBfhg ie^ opX £ ^ fdX(ie^d`X fdXVnj^Fk;gom[ropX'op^ [fFg FX
lm[ sptuXV`¨
]_°kVgjXcaXV^dtu ^aieX;lpkVg[an
ml § ^Li\ gjfdX; `e[a sm¢ [ opgj`\X;b hX
^d`¡ieX(¨WFspn)L^h opg mk[aop [ opsp `e[rl1i\X
^ ie`\[6YX;i\^ [ri $spg~ie^phieg~fdXV`\[a©[rt¯X;lm[ [ # sm[ kV^dtu^dlpXVlOi\X
^a`ieX+¨BWFspn
tp`\^r¨
YX;iJ[rom[ lp^tpn[alp^om[ ³ gc
u[aspXVlOiJ[rom[ XV, Fopsn^p
poX;fhgop^¥]_^d`b[!fhg`i\sm[an
op^ ^aie^a`\X
1lm[X L[`\[ § ^ieX;Ltu^d`\[an kV^dH1spX$k;`\sV[a [ n[iegji\spopX opX
spL[
[aspncdtusXV`¡]_°[ kVgjX[X ] XV;`lp\g^ kV[
opX;[hs]_^dgtu`b^a[ieX;lpkVg[a
hnO1omsp[
X
q)^ XV``e[ [rl1 i ^dV`¡iJ[alplO[i\^p
rspfht¯g X; iJ`[@]_°jkVopgX X5cd£ X;^d¢ tul ^a§ ie^+X;lp¬mkV5g[a]_^dnp`\X@bV[[
op^ ^aie^a`\X
¯1spX^ ^fFgjLP~X;lOiJ[a,lpX \ [ sptuXV`¡]_°kVgjXauopX]²[ri\^L]_^dcaspX;i\X
¯t¯^ag lm[
kV^dlpogb § ^8oX ~τ 9^ ^aie^d`X [C¬ ;ngjkVXa
^ds¥Y\[ie^Cop^ [fhg hX
e§ ^Ngl Fk;sp^ ¢
hsX^ ^6fr[!X^ ieX;lp¬m[]_X;gji\^¦ie`e[rl n[aom[a`#opX X;n V©[ri $spgji\^zlm[og`\X;b § ^ ± i\X
¨ £TX i\X
+g ie[CopX ¢
?^as ¬ ¢
4tu^d`T^dsi\`\^Nn[ao^p
4[8XE®Fg i Vlpk;g[CopXzspL[9]_^d`\bV[Nfhg`¡iesm[rn
gL[acdglpa`\g[
m1spX!l § ^9spom[C[opgj`\X;b § ^Cop^ fdX(ie^d`X fdX;n^hkVgjom[aopX¦o^ [fhg hX
[
hsXtp`^6fa^FkV[^opX n^hk[aXVlOie^Ln[ri\XV`\[an ¥zopg`XVg~iJ[z^ds¥X hsXV`\op[9op^^fFgjLX;lOie^ ¢
opX X fdX(ie^d`X
F[rkVXVnjXV`\[alpop^r¨ ^ n[rieX;`e[anjLX;lOieXa
rXV sp[+[rie^ ]_X;`e[+^alpopX ~τ F
X;
fFg k;^ gjom[aopXr
lpX;kVX r`\g[F
Otm[a`e[@1spX XZY\[a XVlOieX;lpopgop[
dlX X#`\X(]_XV`\X;lpkVg[an
a[ i\`e[¨
YX;i a`\g [ op^ [fhg FX \XV lOie`X#^ ^dn^ X $spgji\^p
³ gc kaOX0 [+]_^d`\\bV [
fhg`i\sm[anhopX ^d `g^dnjg
ieXV gop^@n[r`\cO[aXVlOi\X'XVtp`\X;cO[aom[#tm[a`\[@^@XVlOi\XVlpopgjLX;lOie^+o^
^fFgjLX;lOie^5`\XV[anm
rg i\^
a
pfhg kV^ ^p
mkV^d ~τ 6=
pom[ k;^d`\`XVlOieX ^hkVX alpgjk[
+g ie[ opX ^ds ¬ [ k;^d`\`XVlOieX opX sptuXV`¡]_°kVgjX0op^ ^hkVXV[alp^
1cdX;`e[aop[ t¯X;n^
fdXVlOi\^
e§ ^8opX pX;¢ iegjom[ tu¢ XVn[9]_^d`b[CopX ^d` g^dnjg z´ ^d `\`XVlOieXzop ^ `\[ gjn
? lp^N¬pXE¨
Lg ] V`g^WFspnx
opX mX(iegjom[¦¥X 1spX;`\om[!opX XVsC^fhgXVlOie^X$[ ^a`\`\X;lOieX$op^ ^dn~]_^p
lp^C¬pXVg ] V`g^L
^d`¡ieXa
¥9opg`XVg~iJ[zop^ X;sL^fhgXVlOie^
´ ^alpom[opX[a` ie[a¦ V
^a]_`XC[[ab § ^om[ F~ X9lp^¬pX;Lg ] V`g^WFspnx
Xl^¬pXVg ] V`\gj^@^d`i\X ¢
*cdg`\[XV
eso
CC
ie^d`lp^opX ^dlOi\^ ´ l mop` dLgjkV^ ^alpopX
[L[a` l1spn[ ¢
hopX mX;i\glpop^¨ X+¥X 1spXV`om[
Xzlp^N¬pXVg ] V`g^C@^d`i\Xa
)¥9opg`XVg~iJ[
opX sm[Copgj`\X;b § ^NopX!tp`^dtm[acd[ab § ^CX;cdX;`e[an
^dlpopgjb hX `\X \ ^alm[alOieX tu^FopX;R]²[ ;¢ XV`[8t`\^dtm[rcO[ab § ^Com[8^alpom[NopXz[a` !lm[a^ X;`
[r]_X;ie[aom[!t¯X;n[¦]_^d`b[!opX ^d`\gj^dng
kV^d``\X;l1i\X ! sppXV` [ [a^n^dlpca^op^ ± 1sm[aop^a`V
<fhg iJ[ o^`\X(]_XV`\X;lpkVg[an ¬ ¢
´
kV^dca`e[aopgjXVlOieX!oXt`\X e§ ^lm[Copg`XVb § ^NopXL£TX i\Xtp[a`e[ ± i\X
?1spXL[ t FX;XV
L^fhgXVlOie^
Olp^¦£$k;X[alp^ ´ ien alOiegjkV^p
Olp^ [rkV° pkV^X@l^ lpogkV^
pk[aH[at`\g g^dlp[aom[
[a^nj^dlpcd^zom[ngl¬m[X;1sm[rie^a`\g[rn
popX(fhgop^[ F~ WFX^a`\gslpom[ op^L¬pXVg ] V`\gj^9Whspn
kV`\s [a ^ 1sm[rop^d`#X ^zopXmX(iegjom[ ¥¦opgj`\X;gjiJ[op^ X;s^fhgXVlOie^tm[a`e[ ieXa
moX
iJ[an ]_^a`\L[¦±[!kV`\s [a`XV©\§lp^fr[aXVlOieX[¦ njglp¬p[XV1sm[ri\^d `\g[an
F^dlpopXr
plp^z¬pX;Lg ] V`± g^zWhspn
^aXVs]_`XVLR^fhlg^6frXV[alOi\^!XVlOtmi\[aX!`e[C[ [ab § i\^CXa
omkV[`s ]_^d[r`\lpbV[9op^opX^ds ie`\^d[¦`gfd^dXnjg
¯[!1nspgjlpXz¬m[ [¦ opop^ X mX(hiespX[a¥9op^dX ` 1spXV`om[Cop^
±
±
£=^fFgjLX;lOie^ [rtp`\g g^alm[aop^ kV^dlOi\gl1sm[a`g[ glpopX mlpgjom[aXVlOieX tm[a`e[ ieXr
l ^
¬p^dsfaX \ XV ^ kV^al1i\glpX;lO ieX om[ ´ XV`gkV[op^ WFspn
#op[ ]_`gkV[ X[ gnj¬m± [ om[ Z§ lh¨
op^dl g[F
?m[a`\`\[alpop^N[9t`\^dtm[rcO[ab § ^CopX [ kV^a`\`\X;lOieX
^ai\X(¨ Xa
?X;lOie`\X(iJ[alOi\^p
?1spX
ea§ `\XV^[ X \ 1[ sm [mie^d[a```\g[aX;nxg`e
<[ op ^ 1spfaX9XVlOtuieXV^ `L[ag~ienj°X;«XV^ ^opXVX tpgjnie¬mXN[atm[rXV`elO[i\^£$omX [ ie Xaa
cdlpsp^[ n[a
Aopt¯^X;n£$[X [rb ie§ XC^oplm^ [
^Fk;X[alp± ^ X[zcaXV`e[rb § ^! op^ cd`e[a ogXVlO i\X opXtp±`X \\§ ^z1spXL^ fhgXVlOiJ[r [ k;^d `\`XVlOieX
XV1sm[ri\^d`\g[ag # sppXV` [ XV©og`\X;b § ^zopX£$X ieXtm[a`\[ ± ieX
CC