2010, Wagner Corradi; Rodrigo Dias Trsia Wanderson Silva de Oliveira; Srgio Luiz Arajo Vieira; Maria Carolina Nemes; Karla Balzuweit
2010, Editora UFMG
Este livro ou parte dele no pode ser reproduzido por qualquer meio sem a autorizao escrita do Editor. Fundamentos de Fsica I / Wagner Corradi ...[et al.]
- Belo Horizonte ; Editora UFMG, 2010 p. Il (Educao a Distncia) Inclui referncias. ISBN: 1. Fsica. 2. Mecnica. 3. Gravitao I. Corradi, Wagner II. Srie. CDD: CDU:
Elaborada pela DITTI Setor de Tratamento da Informao Biblioteca Universitria da UFMG
Este livro recebeu o apoio financeiro da Secretaria de Educao a Distncia do MEC.
ASSISTNCIA EDITORIAL Eliane Sousa e Eucldia Macedo EDITORAO DE TEXTO Maria do Carmo Leite Ribeiro
PREPARAO DE TEXTO Michel Gannam REVISO DE PROVAS
FORMATAO PROJETO GRFICO E CAPA Eduardo Ferreira
PRODUO GRFICA Warren Marilac
EDITORA UFMG Av. Antnio Carlos, 6627 Ala direita da Biblioteca Central
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A elaborao deste livro nasceu da necessidade de se produzir um material didtico adequado ao Ensino a Distncia (EAD) das disciplinas de Fsica Bsica na Universidade Federal de Minas Gerais (UFMG). Ele foi construdo a partir de um conjunto de textos que vm sendo utilizados e aprimorados durante vrios anos no projeto-piloto de EAD do Departamento de Fsica da UFMG.
Acreditamos que para se fazer EAD no basta disponibilizar o material na internet, em um stio muito colorido e com vrias animaes. preciso que se tenha um material impresso de boa qualidade, com uma organizao adequada, concatenao e seqncia lgica das idias, numa linguagem coerente e acessvel ao estudante. Sem isso, quase impossvel aprender fsica estudando de maneira autnoma.
H ainda a necessidade de se fornecer acesso ao material didtico independente da disponibilidade de um computador, j que nem sempre o acesso aos recursos computacionais possvel. Mesmo quando h essa disponibilidade, difcil aprender fsica na frente do computador apenas lendo os textos durante horas e clicando nos links disponveis.
A utilizao de um livro voltado para o ensino presencial requer um professor que pondere a linguagem do material, acrescente toda a sua experincia, e modere o ritmo de estudo em sala de aula. Sem essa interveno voc no teria como saber, de antemo, qual ritmo de estudos deveria seguir em cada captulo ou seo do livro. J no EAD, o livro deve suprir a falta do professor, agindo como um roteiro de estudo. Para tanto, ele deve ser dividido em aulas, que contenham uma maior sub-diviso do contedo. No fundo, uma tentativa de se colocar no papel o que o professor faria na sala de aula.
Mas, lembre-se: esse livro no deve ser a sua nica referncia bibliogrfica. O material j consagrado no ensino presencial uma fonte imprescindvel para o completo aprendizado de fsica bsica, mesmo porque, inegvel a forte influncia destes textos na estrutura e organizao desta obra.
Os tpicos aqui apresentados seguem a forma histrica. A fsica moderna introduzida ao longo do texto sempre que possvel ou conveniente. O nvel matemtico leva
em conta que o aluno j fez ou est fazendo um curso introdutrio de clculo. Durante o desenvolvimento das equaes bsicas todos os passos so mostrados, e a matemtica introduzida medida que se faz necessria.
O trabalho de elaborao, adequao e preparao dos manuscritos e figuras que deram origem a este livro de responsabilidade dos autores da presente obra. Grande parte deste esforo contou com a colaborao imprescindvel dos estudantes de Graduao e Ps-Graduao do DF/UFMG, em particular Marcelo ngelo Diniz Alessio, Alexandre Ferreira de Freitas Lages e Gustavo Henrique Reis de Arajo Lima. Um agradecimento especial para Hugo Jos da Silva Barbosa, de onze anos de idade, que desenhou a maioria das figuras do livro. Agradecemos ainda o suporte de nossos familiares, dos vrios colegas do DF/UFMG e da Editora UFMG.
Os Autores
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864001
smc /458.792.299=
( ) Kgxumau 2710000000083,0660538782,111 ==
12,000 000 0 1,007825032
22,98976967 26,98153844 39,9625912 53,9396148 62,9296011 106,905093 196,966552
m V
Vm
=
3310 m
33101
mkg
=
33101 m
kgxgua = =gua
Vm =
Vm =
11041m = xxkg
kgkgx 4,0104m 1 == g400m =
IR V
cmP 55,8=
cmD 72,2=
cmcm
DP
72,255,8
==
143382352,3=
14,372,255,8
=== cmcm
DP
( ) ( ) ( ) 9,4cm 3,1415cm x1,11cmx 2,7cm =
mm 55,1023,214,3
=
23,12)5,1()2,8( dmdmxdm =
3101x 3100,1 x
31000,1 x
221 gth =
g
pt
( )9
89,094,002,111,199,095,097,012,199,0 st p++++++++
=
st p 95,0=
t
989,095,094,095,002,195,011,195,099,095,095,095,097,095,012,195,099,095,0 ++++++++
=t
st 08,0=
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221 gth =
221 gth =
h
wattsx 61010 mx 7101 gx 910004,0 s0030,0
Gbytes30 m19,0
mg46 pm12,0 kgx 21001980
btar += brv = ( )btasenr = )exp( btar =
0
221
041
dqqF
=
anog50
kmmi 6,11 =
52
53
54
55
x
12 xxx =
56
kmkmkmxxx isfss 0,303 === .
kmkmkmxxx ipfpp 0,35,05,3 ===
kmkmkmEWOEDt 800300500 =+=+=
57
x
kmkmkmEWOEOW 200300500 ===
x t
12
12ttxx
txvm
=
=
58
mxA 5,2= st A 0,1= mxB 25= stB 5,2=
ssmm
ttxx
txv
AB
ABm 0,15,2
5,225
=
=
=
smvm /15=
sm /15
Ax At Bx Bt AB xxx =
Ax At Bx Bt
59
AB ttt = Ax Bx
AB xxx =
x t
v
AB
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txinclinao
=
=
txvinclinao m
==
60
0t t x t
0t
Ax Bx t
0t vvm
dtdx
txv
t=
=
0lim
61
ev
t
tdv te =
smm
tdv te 5
1010 +=
=
smve /0,4=
62
22 )/0,3(10)( tsmmtx +=
12 ttt =
221 )0,1)(/0,3(10 ssmmx +=
mx 131 =
222 )0,5)(/0,3(10 ssmmx +=
mx 852 =
mmmxxx 72138512 ===
12
12ttxxvm
=
sm
ssmmvm 4
720,10,5
1385=
=
smvm /18=
dtdxv =
( )[ ] ( )( )tsmtsmmdtdv 2/0,30/0,310 222 +=+=
( ) tsmv 2/6=
( )( ) smssmv /60,1/6 21 ==
63
( )( ) smssmv /300,5/6 22 ==
ttttx 600,0200,0500,0)( 23 += x
ox ot x t
ott 0ot x t
64
o
ottxx
txv
=
=
tvx =
)( oo ttvxx =
)( oo ttvxx +=
v ox
ot
65
Grfico x-t para os dois atletas
09
18273645546372
1 3 5 7 9tempo (s)
p os i
o
x t
66
txvm
=
m/s9=
= t
xvm
m/s8=
= t
xvm
btatx +=)(
tgb =
67
vdtdxb ==
ttxA )m/s9(m1)( +=
ttxB )m/s8(m3)( +=
m/s8)(m/s9)(
=
=
tvtv
B
A
vvm =
smvvm /0,9==
m/s8== vvm
m36m1m370 === xxxA
m32m3m350 === xxxB
m36s4m/s9 === tvx AA
m32s4m/s8 === tvx BB
68
),( 00 ttvxx =
m36s4m/s90 == xx m32s4m/s80 == xx
)(tx
)(tx t
69
)0,1(600,0)0,1(200,0)0,1(500,0)0,1( 231 hhhhx +=
kmhx 90,0)0,1(1 +=
)0,3(600,0)0,3(200,0)0,3(500,0)0,3( 233 hhhhx +=
kmhx 5,13)0,3(3 =
)9,0(5,1313 KmKmxxx ==
kmx 6,12=
dttdxtv )()( =
600,0)2(200,0)3(500,0)( 2 = tttv
600,0400,050,1)( 2 = tttv
v
)0,5(600,0)0,5(200,0)0,5(500,0)0,5( 235 hhhhx +=
01112 === xxxxx
70
kmhx 5,60)0,5(5 =
)0,7(600,0)0,7(200,0)0,7(500,0)0,7( 237 hhhhx +=
kmhx 9,165)0,7(7 =
kmkmkmxxx 4,1055,609,16557 ===
hkmhkm
txvm /7,522
4,105==
=
)(tx t
btatx +=)(
)(tx a b
smdtdxbv /24===
tsmmtx )/24(15)( +=
smv /24=
71
72
32)( BtAttx =
32)( tttx = 2/40,3 sm= 3/200,0 sm= .
73
74
1x 1v 1t 2x 2v 2t
12
12ttvv
tvam
=
=
0t
dtdv
tva
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=
0lim
)(xdtdv = )(vdt
da =
2
2
dtxd
dtdx
dtda =
=
75
mtsmtsmx 9,5)/0,1()/8,9( 22 +=
!"
#"
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ox ot
ov x t v
76
o
o
ttvv
tva
=
=
)( oo ttavv +=
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ta at ( )0vv t ta 0v
ta ( )0vv
x 0ttt =
77
xttvttttarea =+= )())((21
0000
( ) ( )20000 21 ttattvxx +=
200 21 attvxx ++=
)(tx a 0x ov ot
v x ot
avvt 0=
200
00 21
+
+= a
vvaavvvxx
( )02
02 2 xxavv +=
)(xv a x 0x 0v
200
00 21
+
+= a
vvaavvvxx
( )
+
= 2
00
0 vvvavvx
( )
+= 22
000
vvvvvxa
( ) ( )00 21 vvvvxa +=
( )20221 vvxa =
20
22 vvxa =
( )02
02 2 xxavv +=
78
sssmsm
tvam 010
/5,12/7,16
=
=
2/42,0 smam =
a
2/42,0 smaa m ==
v
)( oo ttavv +=
)5(/42,0/5,12 2 ssmsmv +=
smv /6,14=
smv /6,19=
79
80
!"
tsmsmdtdxtv )/0,1(2/8,9)( 2==
tsmsm )/0,1(2/8,90 2=
2/0,2/8,9. smsmt =
st 9,4=
mssmssmsx 9,5)9,4)(/0,1()9,4)(/8,9()9,4( 22 +=
msx 9,29)9,4( =
)(vdtda = 2
2
dtxda =
2/0,2 sma =
#"
tsmsmtv )/0,2(/8,9)( 2=
81
smsvssmsmsv
smsvssmsmsv
/2,8)0,9()0,9)(/0,2(/8,9)0,9(
/2,2)0,6()0,6)(/0,2(/8,9)0,6(
2
2
=
=
=
=
$"
mtsmtsmtx 9,5)/0,1()/8,9()( 22 +=
mssmssmsx 9,5)10)(/0,1()10)(/8,9()10( 22 +=
msx 9,3)10( =
tsmsmtv )/0,2(/8,9)( 2=
)10)(/0,2(/8,9)10( 2 ssmsmsv =
smsv /2,10)10( =
00 =v 00 =x
( )02
02 2 xxavv +=
( )0202 += xav
axv
=22
82
mx 6,19= smv /6,19=
( )( )m
smxva 6,192
/6,192
22==
2/8,9 sma =
200 21 attvxx ++=
00 =x 00 =v
221 atx =
( )( )22 5,2/8,921 ssmx =
mx 6,30=
0v 1v 2v
83
2)( tttv += 2/50,2 sm= 3/200,0 sm=
84
85
g
2T
T
RGMg =
2)( hRGMgT
T
+=
22 /8,9 smR
GMgT
T ==
86
g
g g
g
87
y g y y x a g
200 21 gttvyy +=
gtvv = 0
00 =t
200 21 gttvyy +=
gtvv = 0
0y 0v g y v
kgm 5,0= smv /0,200 =
88
0=v
s2,2)m/s8,9()m/s20(2
0 === gvt
( ) ( )20 2,2212,2)2,2( sgsvsy =
( ) ( )22 2,2)/8,9(212,2)/20()2,2( ssmssmsy =
m2,22)s2,2( =y
00 =y
20 2
1 gttvy =
22 )m/s8,9(21)m/s20()( ttty =
gtvtv = 0)(
ttv )m/s8,9()m/s20()( 2=
g
m1,15)s0,1( =y m/s2,10)s0,1( =v
m4,20)s0,2( =y m/s4,0)s0,2( =v
89
m9,15)s0,3( =y m/s4,9)s0,3( =v
m6,1)s0,4( =y m/s2,19)s0,4( =v
00 =y smv /0,200 = y my 1,151 = m/s2,101 =v
kgm 721 = kgm 562 =
90
200 21 gttvyy =
20 21 gttvH +=
st 8,2=
22 )8,2)(/8,9(21)8,2)(/5,1( smssmH +=
mH 43=
g g
g
y
20 21 gttvy =
st 8,2=
22 )8,2)(/8,9(21)8,2)(/5,1( smssmH =
mH 43=
mH 43=
91
gtvv += 0
)8,2)(/8,9(/5,1 2 ssmsmv +=
smv /27=
kmm 3101 = ( )hs 360011 =
=
hkmv
360011027
3
hkmhkmxv /99/6,327 ==
st 5,1=
22 )5,1)(/8,9(21)5,1)(/5,1()5,1( ssmssmsy +=
my 13=
mh 30=
22 )5,1)(/8,9(21)5,1)(/5,1()5,1( ssmssmsy =
my 13=
92
gtvv += 0
( )( )ssmsmv 5,1/8,9/5,1 2+=
smv /2,16= .
00 =t y 00 =y
20 21 gttvy =
ht 0=v
gtvtv = 0)(
hgtv = 00
20
/8,9/1,9smsm
gvth ==
sth 93,0=
20 21 hh gttvh =
( )( ) ( )( )22 93,0/8,92193,0/1,9 ssmssmh =
mh 2,4=
93
00 =t 00 =y 00 y 00 t
00 t 00 y
ht
star 9,1=
g
94
95
!"
#"
!"
#"
96
!"
#"
CBtAtx ++= 23 ( )3/558,0 smA = ( )2/101,0 smB = mC 23,9=
!"
#"
$"
97
!"
#"
$"
%"
98
!"
#"
$"
CtBtAtx += 23 3/01,4 smA = 2/5,13 smB = smC /2,12=
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#"
99
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#"
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#"
100
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sxdias 51059,23
sxkm
txvm 51059,2
2528=
=
smvm /75,9=
0=mv
pv
sxxkmxv p 51059,22
25282=
smvp /75,9=
101
txvm
=
tvx = )18,1)(/0,99( hhKmx =
kmx 116=
vxt =
hkmkmt /3,77
116=
ht 50,1=
ttxx occ 2,20152,20 +=+=
22 23,321523,32
1 ttxx omm +=+=
ocx omx
cm xx =
0102,2062,1 2 = tt
st 13= st 48,0=
102
222 )13)(/23,3(2123,32
1 ssmtxx omm ==
mx 9,272=
atvv omm +=
)13)(/23,3(0 2 ssmvm += smvm /9,41=
( )CBtAtdtd
dtdxv ++== 23
BtAttv 23)( 2 +=
)0(2)0(3)0( 2 BAv += 0)0( =v
)0,1)(101,0(2)0,1)(/558,0(3)0,1( 23 sssmsv += smsv /88,1)0,1( =
)0,2)(101,0(2)0,2)(/558,0(3)0,2( 23 sssmsv += =)0,2( sv
mBAx 23,9)0(2)0(3)0( 2 ++= mx 23,9)0( =
103
msBsAsx 23,9)1,2(2)1,2(3)1,2( 2 ++= msx 8,14)1,2( =
mx 61,5=
txvm
=
smvm 1,2
61,5=
smvm /67,2=
vM
2/80,70 smM v
+=
smMv /90,3=
smvm /67,2=
104
)( ottvx =
kmx 7,47=
txvm
=
sxmxvm )6036(
107,47 3=
smvm /1,22=
21
1
/083,0
/083,0120
/10min2
/10
smas
smssmsma
=
===
22
2
/17,0120
/20min2
/20
smassmsma
=
==
23 /10,0min
/61116300 smsma ==
=
105
24
4
/067,0min
/4
sma
sma
=
=
25
5
/17,0min
/10
sma
sma
=
=
21
1
/33,0min
/20
sma
sma
=
=
CtBtAttx += 23)(
CBtAttxdtd
+= 23)( 2
CBtAttv += 23)( 2
023 2 =+ CBtAt
106
st 63,0=
st 12,1=
BAtdtdv 26 =
BAtta 26)( =
026 = BAt
st 12,1=
CBtAttv += 23)( 2
dtdv
BAtdtdv 26 =
vxtC
=
107
hkmkmtC /91
56=
htC 62,0=
vxtS
=
hkmkmtS /76
17=
htC 22,0=
pv
hkmv p 82,0
)1756( +=
hkmv p /0,89=
vxt =
txvm
=
hkmvm 82,0
)1756( =
hkmvm /6,47=
108
gtvv o =
tsmsm )/8,9(/50 2+=
st 51,0=
sxt 51,02=
st 02,1=
2
21 gttvy o =
109
y ov g
00,17)/00,5()/90,4( 22 =+ mtsmtsm
st 42,1=
st 44,2=
( )st 02,142,1 +=
st 44,2=
txvm
=
smvm 44,2
17=
smvm /00,7=
gtvv o =
tsmsmtv )/8,9()/0,5()( 2=
)42,1)(/8,9()/0,5()42,1( 2 ssmsmv =
smsv /9,18)42,1( =
110
Py Px Oy Ox
111
VOx VPx
OPx
OPVOVP xxx +=
( ) ( ) ( )OPVOVP xdtdxdt
dxdtd
+=
OPVOVP vvv +=
hkmvVO /10= hkmvOP /40= VPv
OPVOVP vvv +=
hkmhkmvVP /40/10 +=
hkmvVP /50=
hkmvVO /10=
hkmhkmvVP /40/10 +=
hkmvVP /30=
0=VOv
112
hkmvVP /400 +=
hkmvVP /40=
OPVOVQVP vvvv ++=
OPVOVP aaa +=
VOVP aa =
smxc /1000,3 8=
113
114
APVAVP vvv +=
VPv VAv APv
hkmhKmvVP /50/10 +=
hkmvVP /60=
PAAP vv = BPv
APBPBA vvv =
hkmhkmvBA /50/40 =
hkmvBA /90=
115
hKmvAV /10=
AVAPBPBV vvvv =
hkmhkmvBV /10/90 =
hkmvBV /100=
VBv BVv
BVVB vv =
hkmvVB /100=
116
( )12 vv ( )12 tt ta
ma t
tvam
=
117
tav m=
ma t
v 1t 2t 0t
adtdv =
=2
1
2
1
t
t
v
vadtdv
=2
1
12
t
tadtvv
001 == tt 01 vv =
+=tadtvv
00
ta
x mv t
tvx m=
0t
vdtdx =
118
=2
1
2
1
t
t
x
xvdtdx
=2
1
12
t
tvdtxx
001 == tt 01 xx =
=tvdtxx
00
)(tv 0x 0t
a
+=tdtavv
00
tavv += 0
200 21 tatvxx ++=
119
00 =t mx 0,10 =
tsmsmta )/16,0()/82,0()( 32 =
st 0,3=
ta tv tx
2/82,0 smA = 3/16,0 smB = BtAta =)(
+=tadtvv
00
( ) +=t
dtBtAv0
0
22)( tBAttv =
232 )/08,0()/82,0()( tsmtsmtv =
st 0,3= 232 )0,3)(/08,0()0,3)(/82,0()0,3( ssmssmsv =
smsv /7,1)0,3( =
+=
tdttBAtxx
0
20 2
120
320
32
0 62322 tBtAxtBtAxx +=+=
3322 )/05,0()/41,0(0,1)( tsmtsmmtx +=
st 0,3= 3322 )0,3)(/05,0()0,3)(/41,0(0,1)0,3( ssmssmmsx +=
msx 3,2)0,3( =
0 5 10 15 20 25-3,5
-3,0
-2,5
-2,0
-1,5
-1,0
-0,5
0,0
0,5
1,0 Grfico a-t
a ce l e
r a
o ( m
/ s2)
tempo (s)
0 5 10 15 20 25-50
-40
-30
-20
-10
0Grfico v-t
v el o c
i d ad e
( m/ s )
tempo (s)
0 5 10 15 20 25-500
-400
-300
-200
-100
0
Grfico x-t
p os i
o ( m
)
tempo (s)
)(tv
0=dtdv
adtdv
=
BtABtAta == 0)(
121
ssmsm
BAt 1,5/16,0
/82,03
2===
st 1,5= 232 )/08,0()/82,0()( tsmtsmtv =
)(tv st 1,5= 232 )1,5)(/08,0()1,5)(/82,0()( ssmssmtv =
smvMAX /1,2=
hKmmtc /18
4000=
Atta =)(
3/92,0 smA =
122
smmtc /5
4000=
stc 800=
hKmvri /)1118( +=
hKmvri /29=
rit
hKmmtri /29
2000=
smmtri /05,8
2000=
st ri 248=
hKnvrv /)1118( =
hKmvrv /7=
hKmmtri /7
2000=
hmmtri /94,1
2000=
stri 1031=
rvri ttt +=
st 1279=
123
+=2
1)(12
t
tdttavv
( )+=2
11
t
tdtAtvv
( ) 23
2)/92,0(/0,6)0,3(
0,3
0,2
tsmsmsvs
s
+=
smsmsv /3,8)23(46,0/0,6)0,3( 22 =+=
+=2
1)(12
t
tdttvxx
++=
2 2
121 2t
t odtAtvxx
31 6
2
1
2
1tAtvxx
t
t
t
to ++=
])2()4[(692,0)0,20,4)(/0,6(0,8 33 sssssmmx ++=
mmmx 6,28]864[692,0)0,12(0,8 =++=
124
125
2)( ttta = 4/100,0 sm= 3/00,1 sm= 0=x 0=t
BAtta =)( 3/00,3 smA = 2/00,2 smB = 0=t st 00,2=
0=t st 00,2=
( ) CtBtAttx += 23 3/40,2 smA = 2/00,8 smB = smC /20,7=
2/755,0 sm
126
( ) ( )tsmsmtv 2/5,2/0,5 =
( ) ( ) mtsmtsmxA 00,1/60,3/80,1 22 ++= ( ) ( ) ( )tsmtsmtsmxB /6,3/80,1/0700,0 2233 ++=
127
128
129
130
C A B
131
C mC 5=
C
A B
C
222 BAC +=
mC 5=
CBA
=+
C
374343
=
=
==
arctgABtg
V xV
yV
yx VVV +=
132
V xV
yV
V xV
yV xV yV
xV yV
VVy=sen
VVx=cos
senVVy = cosVVx =
22yx VVV +=
i j
V
jViVV yx += 133
xV V i
yV V j
kVjViVV zyx ++=
zV V k
r
r i j k
r
kzjyixr ++=
ix jy kz x y z r
r
222 zyxrr ++==
1r
2r
r
134
r
12 rrr
=
r
22 )/5,0()0,1( tscmcmx =
tscmy )/0,3(= 33)/30,0()0,5( tscmcmz =
kzjyixr ++=
ktscmcmjtscmitscmcmtr ])/25,0(0,5[])/0,3[()])/50,0(0,1[)( 3322 ++= r
kcmjicmsrkssmcmjsscmisscmcmsr
)0,5()0()0,1()0,0(])0,0)(/3,0(0,5[)]0,0)(/0,3[(])0,0)(/50,0(0,1[)0,0( 3322
++=
++=
135
kcmjcmicmsrkssmcmjsscmisscmcmsr
)6,2()0,6()0,1()0,2(])0,2)(/3,0(0,5[)]0,2)(/0,3[(])0,2)(/50,0(0,1[)0,2( 3322
++=
++=
02 rrr
=
kcmjcmicmrkcmcmjcmcmicmcmr
)4,2()0,6()0,2()0,56,2()0,00,6()0,10,1(
+=
+=
v
kcjbiav ++=
v
( ) ( ) ( )kncjnbinavn ++=
v 0=a mmb 7= mmc 6= v
136
kmmjmmimmv )6()7()0( ++=
v
kmmjmmvkmmjmmimmv
)12()14(2)6(2)7(2)0(22
+=
++=
v v2 v
v
kmmjmmimmv )6()7()0(11 =
v v v v
12 rrr
=
)( 12 rrr+=
137
1r
2r
mv r t
t
tr
ttrrvm
=
=
12
12
r t1
t
r 2r 1r
2r 1r
138
t
dtrd
trv
t
=
=
0lim
v r
t r
0t v r
kzjyixr ++=
dtrdv =
kdtdzjdt
dyidtdxv ++=
kvjvivv zyx ++=
dtdxvx = dt
dyvy = dtdzvz =
xv yv zv v v
139
222zyx vvvvv ++==
0r 1r r 0r 1r
jyixr +=
0r
1r
jcmicmr )1,1()3,2(0 +=
jcmicmr )1,4()6,4(1 +=
140
01 rrr
=
jcmicmrjcmcmicmcmr
)0,3()3,2()1,11,4()3,26,4(
+=
+=
01 rrr
+=
tr
ttrrvm
=
=
12
12
jscmiscmvs
jcmicmv
m
m
)/0,3()/3,2(0,1
)0,3()3,2( 1
+=
+=
i j
( ) ( )itAsentr =
mmA 05.0= srad /60=
0=t 2=t
141
r `r
222` yxr +=
r `r
22 `rzr +=
222 zyxr ++=
imr )4,10(=
trvm
=
simv m 0,4)4,10(
=
ismv m )/6,2(=
142
td
tR
tdv tp ==
smvp 0,4
)2,5(=
smvp /1,4=
0=t ( ) 0)0(0 ==
iAsenr
2=t ( ) ( ) iAiAsenr 22 ==
( ) ( )immr 05,02 =
( ) ( )02
= rrr
immr )05,0(=
( ) ( )[ ]itAsendtdtrdt
dtv )( =
=
( )itAtv cos)( =
trvm
=
( )
2
05,0 immvm =
( )ismxvm /109,1 3=
143
x y ( )2,1;3,0 st 1,2= x y ( )2,5;9,1 st 3,4= x y
st 0,1= st 0,1= st 0,8=
smvx /15,0= smvy /23,0=
x y st 0,8=
st 0,8=
( ) ( ) jCtiBtAr 2 ++= mA 0,2= 2/63,0 smB = smC /1,3=
st 0,1= st 2,2=
v t
st 0,1= st 2,2=
144
Cttx =)( 2)( BtAty =
mA 0,3= 2/12,0 smB = smC /4,2=
145
1v 2v
v
ma
v v
146
tv
ttvvam
=
=
12
12
ma
v ma
a ma
t
dtvd
tva
t=
=
0lim
kvjvivv zyx ++=
dtvda =
kdtdvjdt
dvidt
dva zyx ++=
kajaiaa zyx ++=
xa ya za a
147
2
2
dtrda =
kzjyixr ++=
kdtzdj
dtydi
dtxda 2
2
2
2
2
2++=
v
ta
ca
a ta
ca
ta v ca
ta ca
148
ktsmjtsmimr )/590,0()/970,0()160,0( 4433 +=
xa ya za
kCtjBtiAr 43 += mA 160,0= 3/970,0 smB = 4/590,0 smC = dt
rdv
=
kCtjBtv 43 32 =
ktsmjtsmv )/36,2()/91,2( 3423 =
kssmjssmsv )0,1)(/36,2()0,1)(/91,2()0,1( 34231 =
ksmjsmsv )/36,2()/91,2()0,1(1 =
kssmjssmsv )0,2)(/36,2()0,2)(/91,2()0,2( 34232 =
ksmjsmsv )/9,18()/6,11()0,2(2 =
ksmsmjsmsmvvv )]/36,2()/9,18([)/91,2/6,11(12 +==
ksmjsmv )/5,16()/69,8( =
ssksmjsm
tvam 0,10,2
)/5,16()/69,8(
=
=
ksmjsmam )/5,16()/69,8( 22 =
149
kCtjBtdtvda 126 2==
ktsmjtsma )/08,7()/82,5( 243 =
kajaiaa zyx ++= 0=xa
tsmay )/82,5( 3= 24 )/08,7( tsmaz =
( ) ( ) ( ) ( )kCtjBtiAttr 432 += 2/5,2 smmA = 3/0,3 smmB = 4/0,1 smmC =
150
( ) dtrdtv
=
( ) ( ) ( ) ( )kCtjBtiAtdtrdtv 432 32 +==
( ) ( ) ( ) ( ) 34232 /0,4/0,9/0,5 tsmmktsmmjtsmmitv +=
( ) ( ) ( ) ( )kCtjBtiAdtvdta 1262 2+==
( ) ( ) ( ) ( ) 2432 /12/18/0,5 tsmmktsmmjsmmita +=
( ) mmsr 00 =
( ) ( ) ( ) ( )kCjBiAsr 16840,2 +=
( ) ( ) ( ) ( )kmmjmmimmsr 1624100,2 +=
( ) ( )srsrr 00,2
=
( ) ( ) ( )kmmjmmimmr 162410 +=
151
trvm
=
( ) ( ) ( )ss
kmmjmmimmvm 00,2162410
+
=
( ) ( ) ( )kmmjmmimmvm 0,8120,5 +=
min5,01 =t smvx /0,9= smvy /18= min1,72 =t smvx /0,3= smvy /22=
1t 2t
min5,51 =t smvx /7,9= smvy /21= min5,51 =t min172 =t
2/98,0 sma = Ox
xv yv min172 =t
min172 =t
1t 2t
2)( BtAtx =
Ctty =)(
KmA 0,9= 2/0091,0 sKmB = smC /445=
01 =t min12 =t
152
v a t
st 10=
smvx /0,5= smvy /0,4= smvx /0,4= smvy /0,5=
i j
153
ca
ma
v
ta
ta
154
ca
v
Rvac
2=
v ca ca
R2 v T
TRv 2=
2 T
T
2
=
srad /
s31 .31 sT =
f
Tf1
=
155
f 1s Hz
m
12 = 12 ttt =
tttm
=
=
12
12
st 0,11 = rad34,0201 == st 0,22 = rad87,0501 ==
sradssradrad
ttm /53,00,10,234,087,0
12
12 =
=
=
m t
dtd
tt
=
=
0lim
2 T
T
2
=
1t 1t 1 2t 2
156
v
Rv =
22
2 2 RTv
=
RTaRTRa cc2
22 221
=
=
RT
ac 224
=
RTac22
=
Rac 2=
157
mR 10= smv /5,4=
ca
222
/0,210)/5,4( smmsm
Rvac ===
2/0,2 smac =
2/0,2 smac =
)10)(/0,2()10(222
2 msmm
RaR
vRT
c
===
sT 14=
mx 61038,6
ca g
158
BA RR 3=
cmR 0,151 = cmR 00,82 =
1 2 1 2
159
0v
0v
tvxx x00 +=
xx vv 0=
0=xa
160
g 2/8,9 smg =
200 )(21 tgtvyy y +=
gtvv yy = 0
gax =
cos00 vv x = senvv y 00 =
tvxx )cos( 00 +=
cos0vvx =
200 )(21)( tgtsenvyy +=
gtsenvvy = 0
000 == yx
xvxt0
=
00 =y
22
00
0
2 xvgxv
vyxx
y =
222
0 cos2.tan x
vgxy
=
cbxaxxyy ++== 2)(
0x xv0
161
0y yv0
xv
smv /7,270 =
R
162
030= 085=
R
xv0
tvxtvxx x )cos( 0000 +=+=
tvR )cos( 0 =
gtsenvvy = 0
0=yv
gvth
sen0=
tvR )cos( 0 =
htvR )cos(2 0 =
gvvR )sen)(cos(2 00 =
gvR 2sen
20=
cossen22sen =
163
12sen = 045=R
2
220
/80,9)/7,27(smsm
gvR ==
mR 3,78=
2
2
/80,9)52,0*2()/7,27(
smradsensmR =
mR 8,67=
2
2
/80,9)48,1*2()/7,27(
smradsensmR =
mR 6,13=
2
00 21)( gttsenvyy += 2
0 21)sen( hh gttvh =
2
000
sen21
sen)sen(
= g
vggvvh
gvh 2
sen 220 =
045=R
)/80,9(2)78,0()/7,27(
2
22
smradsensmh =
mh 3,21=
164
smv /0,140 =
165
sh 36001
sxh 41064,824
sxTT 41064,8=
TfT1
=
HzxfT 5102,1 =
sTCD 5,0=
HzfCD 0,2=
T
TT
T fT
22 ==
( )HzxradT 5102,12 =
sradxT /103,7 5=
CDCD f 2=
sradxCD /103,1=
sradxsradx
T
CD
/103,7/103,1
5=
TCD x 5108,1=
166
Ra Tc 2=
( )( )mxsradxac 65 1038,6/103,7 = 22 /104,3 smxac =
ca g
2
22
/8,9/104,3
smsmx
gac =
3105,3 = xgac
TBA
2
==
sradsA /5,38,12
==
sradBA /5,3==
0=mv
pv
tRvp =2
167
sm
tRv ApA 8,1
)45,0(22 =
=
smv pA /6,1=
sm
tRv BpB 8,1
)90,0(22 =
=
smv pB /2,3=
1=B
A
A
AA T
Rv 2=
B
BB T
Rv 2=
BA = BA TT =
A
B
B
A
B
A
TT
RR
vv
22
=
B
A
B
A
RR
vv
=
33 ==B
B
B
ARR
vv
BA vv 3=
168
Rvp =
AcA Ra 2=
BcB Ra 2=
Rvp =
pBpA vv =
22
11
2
1
RR
vvp
p
=
1
2
2
1RR
=
cmcm
0,1500,8
2
1 =
21 533,0 = 12 88,1 =
169
12 88,1 =
( )rpm6588,12 =
rpm1222 =
sradrotao
602
min1
=
sradsrpm /105,0301 ==
srad /8,122 =
srad /81,61 =
)/80,9(2)86,0()/0,14(
2 22222
0sm
radsensmgsenvh ==
mh 70,5=
smradsensm
gsenvR /80,9
)71,1()/0,14(2 220 ==
mR 8,19=
ssmradsensm
gsenvt 08,1/80,9
)86,0()/0,14(2
0 ===
slitrosslitrosE /1,2316,2
50==
170
g5,2 g0,6
20,600m/s
Kmx 81050,1
g
32
32
( ) ( ) jsmismv /04,7/1,150 +=
171
172
VOr
OPv
VPr
OPr
VOr
VPr
OPr
VOOPVP rrr
+=
( ) ( ) ( )VOOPVP rdtdrdt
drdtd
+=
OPVOVP vvv
+=
173
( ) ( ) ( )VOOPVP vdtdvdt
dvdtd
+=
OPVOVP aaa
+=
( ) jhkmvb /0,30= ( )ihkmvc /0,15=
cmbcbm rrr
+=
bmr
bcr
174
cmr
cmbcbm vvv
+=
bmv
bcv
cmv
( ) ( ) jhKmihKmvbm /0,30/0,15 +=
ismvc )/0,5(=
175
jsmismv )/0,3()/0,2(2 +=
12 vv
=
smsmsmv /6,3)/0,3()/0,2(|| 222 =+=
smv /6,31 = v1
ismv )/6,3(1 =
2121 vvv
+=
jsmismsmjsmismismv )/0,3()/0,2/6,3()/0,3()/0,2()/6,3(21 ++=++=
jsmismv )/0,3()/6,1(21 +=
( )ihKmv A /56=
176
( )ihKmv B /65=
( ) jsmv moeda /0,1=
( ) jsmv moeda /00,1=
( ) ( ) jhKmihKmv B /0,20/6,34 +=
( )ihKmv A /56=
177
2qtax =
ctbay =
2/023,0 smq = 2/08,0 smb = 3/011,0 smc =
( ) ( ) jsmismv o /50,1/10,1 +=
( ) ( ) ( ) jtsmitsmtv /0,16/50,2 2 +=
0=t
hgvo 20= h
178
ov h o
0=h
smvo /0,2= mh 1,1=
)cos()( tRtx = )()( tRsenty =
R2
179
180
181
182
smxc /103 8= cv 1,0<
183
184
185
186
F
Aa Ba
B
A
A
B
aa
mm
=
F
F F
B
A
B
A
aa
aa
=
B
AB a
am =
m
187
m
1m 2m m 1m 2m
188
1F
2F
RF
ou R
21F FFFR R
+===
nR FFFF
+++== ...F 21
189
22yx FFF +=
FsenF
FF
y
x
=
= cos
Figura 11.3 Decomposio de foras
s.
NF 751 = NF 1002 = NF 503 =
190
1F 2F 3F
N.38230)N50(N,32230cos)N50(
N,71135)N100(N,71135cos)N100(
N,3830)N75(N,6530cos)N75(
3
3
2
2
1
1
==
==
==
==
==
==
senFF
senFF
senFF
y
x
y
x
y
x
FR =
N.71)N38(N71N38N,38)N32()N71(N65
321
321
=++=++==
=++=++==
yyyyy
xxxxx
FFFFRFFFFR
xR yR FR =
.N81)N71()N38( 2222 =+=+= yx RRR
191
07,1)87,1arctan(N38N71arctanarctantan ==
===x
y
x
y
RR
RR
11818062ou 62 =+== 118=
1F 2F
192
R
193
.
194
m F
a
mFa
k
195
amkF =
k k
k k
amF =
2/11 smkgN =
F
amF =
0=F
0=a amF
=
196
pF
P N
2m/s24,1kg45N56
===
==
mFa
maFFp
x
xpx
197
f
.m/s4,1)m0m3,1(2)m/s9,1()m/s0(
)(2
),(2
222
0
20
2
02
02
=
=
=
+=
xxvva
xxavv
f
.N5,0)m/s4,1)(kg35,0( 2 ==== mafFx
198
199
gmP =
g m
2/8,9 smg = g
mgP = 2/8,9 smg =
8,91176 = m kgm 120=
NmgP 19668,9*120 ===
)/( 2smg
200
kgf Nkgf 8,91 =
2/78,9 smg =
201
am 1
Fm
aFm
amF
= 0yF
= 0xF
0= fFT
TFf = Nf 200=
f
= amF
mafF =
202
fmaF += NF 700200500 =+=
maFR = 7005,1 =RF NFR 1050=
mgP = 78,9700=P NP 6860=
PTFR = T
68601050 = T
NT 7910=
203
204
205
m
g
gmPFTL
==
LTF
206
LMF
MLF
MLF
LMF
MLF
LMF
TLF
LTF
207
gmFTL
= MLF
P
bcF
cbF
cmF
mcF
208
= 0blocoF
0=+ cbFP
cbcb FPFP ==+ 0
= 0cordaF
0=+ mcbc FF
mcbcmcbc FFFF ==+ 0
cmF
a
mcF
cmF
cbF
209
bcF
cmmc FF
=
cbbc FF
= cbF
bcF
amF bcb
=
bm
amFF cbcmc
=
cm
mcF
bcF
210
bcmc FF
>
211
212
= 0F
0=+ PFN y PFN y =+
NsenFFy 25300 == NmgP 1968,920 ===
NN 17125196 ==
213
maF = kgmTERRA 2410 x 5,97= a= )10 x 5,7(980 24 2-22 /10 x 1,64 sma =
214
215
216
217
218
kgm 0,21 = kgm 0,32 =
219
NF 6=
2m
a f
1N 2N
mg
N F
T
resam
resF
resam
resam
220
1m 2m
yresyres amgmNF ,111,
=+= xresxres amTF ,1,
==
yresyres amgmNF ,222,
=+= xresxres amTFF ,2,
=+=
1m 011 = gmN amT 1= 2m 022 = gmN amTF 2=
T
21 mmFa+
=
T
21
11 mm
FmamT+
==
2/2,1 sma = NT 4,2=
kgm 0,11 = kgm 0,22 =
1m NF 0,3=
f
221
1m amfF 1= 011 = gmN
2m amf 2= 022 = gmN
21
22 mm
Fmamf+
==
Nf 0,2=
2
2/0,12
0,2 smmfa ===
F 2m
222
T
'T
TT
='
T
'T
T P
T
2 2PT
=
223
1m 2m
224
1P 2P
1m 2m
T
2m 1m 2m 1m
1m 111 amPT =
2m 222 amTP =
1a 2a a T a T
225
ammPP )( 2112 +=
gmmmm
mmPPa
21
12
21
12+
=
+
=
a T
gmmmmT
21
122+
=
2m
1m amPT 11 =
2m amPT 22 =
2m 1m
226
1m amPT 11 = 2m amTP 22 =
gmmmm
mmPPa
21
12
21
12+
=
+
=
227
xP
yP
yP
N
mgsenPsenPx == coscos mgPPy ==
xx maP = gsenax =
228
yy maPN = cosmgN =
NsenmgPx 245300 == NmgPy 42430cos 0 ==
020cosFFx = 020senFFy =
yy PFN =+ yy FPN =
0= xx PF NPF xx 245==
229
NFPN yy 33589424 ===
0v
v N P
cosPN =
maPsen =
gsena =
a
gsenvx 2
20=
230
gvxsenH 2
20==
12 3mm =
2m
1m 0cos11 = PN amsenPT 11 =
2m amTP 22 =+
T
gmmsenmma
21
12+
=
a
231
gmmsenmmT
21
21 )1(+
+=
090=
k
kxF =
232
N 80 = 0,1 x 800== kxF
233
NN
FPN
PNFF
E
E
6780147
0
==
=
=+=
234
1m amf 1= 2m amFf 2=
21
1mmFmf+
=
Nf 0,1= 21
/0,110,1 smm
fa ===
235
N 3,20 = 0,04m x N/m 80== kxF
236
F
237
F
ef
max,ef max,efT = cf max,ef
max,ef
max,ef
max,ef
cf
238
NFat =
atF
N
239
atF
N
cinticoesttico >
240
aF
aF
,
1m 12 3mm =
241
12 3mm =
1m 2m
2m amPT 22 =
1m 0cos11 = PN amsenPfT 11 =
cos1 gmf =
gmmsenmma
21
12 )cos(+
+=
242
1m
243
1m
2m
2m amPT 22 =
1m 0cos11 = PN amsenPfT 11 =
cos1 gmf =
gmmmsenma
21
21 )cos(+
=
012 amm
244
tgPPsenPsenN
PsenPFPPN
xat
y
=
=
=
==
==
cos
cos
245
bvF = b
.bmgvt =
2CvF = C
.Cmgvt =
246
247
N
P
N 784 = m/s2) kg)(9,80 (80,0== gmP
N
NgamN
mamgNF
1024)38.9(80)( =+=+=
==
248
RVac
2=
V R
249
T
2
=
Tf1
=
f 2=
250
R2 R
RV =
Rac 2=
RVac
2=
smV
mV
sm
/7,29882
1502
28,9*6,0
==
=
251
2 rad .min/9,11/198,0150
7,29 radsrad
RV
===
=
252
253
NPFc +=
PNFc =
NFc =
254
255
256
257
258
)(rF
259
d F
F d
dFW =
F d
dF
cosdF
F
A B BA
cosABBA = i j k
260
090cos1110cos11====
====
kjkijikkjjii
kAjAiAA zyx ++= kBjBiBB zyx ++=
zzyyxx
zyxzyx
BABABABAkBjBiBkAjAiABA
++=
++++= )()(
F
cosFdW =
261
cosF F
d
F d
cos W
cos W
o90=
262
TW
nT WWWWW +++== ... 21
F
F NF 3000= Nf 1400= TW
263
P 0=PW
N 0=NW
J2,88ouN.m6,88236)m32)(2,23cos()N3000( kWF ==
fW J 44,8-ou J -44800)m32)(1)(N1400( kWf ==
43,4kJ )J8,44(J2,8800 =+++=+++= kWWWWW fFNPT
=+=+= N1036,1)N1400()2,23cos()N3000()(cos 3xfFFx
43,4kJ J1034,4)m32)(N1036,1( 43 === xxWT
264
2
20
1)(
cv
mvm
=
0m m c 0m
265
cv =
20 cUmm +=
Q
2cQm =
E
2cEm =
2cmE = XX
0)90cos( =
0== NP WW J70,1ouN.m1696)m19)(20cos()N95( kWF ==
266
1cos = fW J399)m19)(1)(N21( ==fW
1,30kJou J1297)J399(J70,100 =+++=+++= kWWWWW fFNPT
=+=+= N3,68)N21()20cos()N95()(cos fFFx
1,30kJ J1297)m19)(N3,68( ===TW
267
maFr = FdWT = adVV if 222 +=
2222if
TmVmVW =
22mV
cE K
TW
268
icfcT EcEEW ==
ciE cfE
TW
TW
TW
F
xF
xF
269
==2
1
2
1
x
x xx
xdxFxdFW
xxxF 25)( 3 = mx 0= mx 1=
F 0x x F x
kxF =
F
=2
1
x
x xdxFW
x kxdx0
221 kx
270
))((21 xkx 22
1 kx
x
271
=2
1
x
xdxdt
dvmW vdtdx
=
2221
22
2
1
mvmvmvdvWv
v== icfcT EcEEW ==
JxxdxxxW 25,0145
22
45)25(
241
03 ==
==
272
273
tWPmed
=
dtdWP =
sJW /11 =
274
vFP =
FvP =
NmgF 7840==
JFdW 51035,2 ==
WxxP 35
1031,11801035,2
==
275
276
22 21
21= ABAB vmvmW
m
sdFWB
AAB
=
F
sd
F F
sd
sd sd
m
h
sd
sd
gmF = sd
x ax bx
hgmxxgmdxgmdxcosgmdscosFsdFW abbx
axbx
axbx
ax
B
AAB =)(=====
h
gmF = sd
sd sd x dxds =
hgmxxgmdxgmdxcosgmsdFW baax
bxax
bx
A
BBA=)(==)(0==
BAAB WW =
0== BAABAA WWW +
L h
x dxsd |=|
dxcosgmsdFW LBAAB
0==
Lhsencoscos /==)(=
hgmdxLhgmW LAB == 0
1C 2C
1C 2C
gmf =
agmdsgmdscosfW aaAD === 00
)(===00
cbgmdygmdxgmWWW cbBDABAD ++
cba +< ADW
1C 2C
1C 2C )()(= 21 CWCWWAB + xy x 0,dx dxh,
0=2==)( 01 dxcosgmWCWdx
AD
2C
hgmdxcosgmWCW hDB ,===)( 02
hgmWWW ABDBAD =+ =
U UWAB =
)(= ABB
AUUsdF
F
sdFUU BAAB
=
F
0=AU )()( AUBU BU
sdFBU BA
=)(
0=)(AU
F
F
sdFsdFBU AB
B
A
==)(
F
0=U
h
hgmxxgmdxcosgmdscosFsdFAUBU abbx
axbx
ax
B
A=)(====)()(
L
0=)(CU 0=ax
LgmdxgmdxcosgmsdFCUAUAULL
ax
cx ==)(0==)()()(
00
)()( AUBU
x xkF
=
ax bx
)(21=|2
1===)()( 222 abbxaxbx
axbx
axxxkxkdxxksdFAUBU
20 2
1===)()(=)( bbxB
OxkdxxksdFOUBUBU
m
0x
kgmxdquegmxk =:= 00
x
x
ge UUU +=
xgmxkdxgmdxxkU xx 200 21==
1m 2m 1m 2m
2212221
12 /.106.67== KgmNGrmmGF
r 2m 1m r 2m 1m
2m 21F
1m r 2m 1m
2m 1m r 1m
rmmG
rmmGdrrmmGdrrmmGrU
rrr 21
21221221 =1=1==)(
F
sdFdUUU BA
B
AAB
==
dsFdscosFsdF s=)(=
sF F sd
dsdUFs =
ExUmv =+ )(21 2
)]([2 xUEmdtdxv ==
dtmxUEdx 2
)]([=
0xx = 0tt =
=tt
xx dtmxUE
dx00
2)]([
x t
m k Ox
x
221)( kxxU =
0=t 00 =v mxx =
2220 21
21
21
mm kxkxmvE +=
=
tt
xx
m
dtmxxkdx
m 0
2
][222
=
tt
xx
mdtm
kxx
dxm 022
tmk
xxarcsen
x
xm m=
tmkarcsenx
xarcsenm
=
)1(
tmk
xxarcsenm
=
2
=
= tm
kxtmksenxx mm cos2
E )(xU
E
v
)(xUE v
)( 0xUE = 0x 1E 1x 2x 1x 2x UE =1
1x 2x UE 1 0x 0x 0x 2x 2x
1E 3E 3x + 4E +
dxdUF =
U x
1x
1x 2x 1x 2x
0x
0EE > 0x
5x 55 UE = 0=v 5x
5x
4x
4xx >
Ox
4xx < Ox
hgmxxgmdscosFsdFBU abbx
ax
B
A=)(===)(
)(BU )()( AUBU 0=)(AU
)(21=|2
1===)( 222 abbxaxbx
ax
B
AxxkxkdxxksdFBU
)(AU )()( AUBU 0=)(AU
)(21=|2
1===)( 222 baaxbxax
bx
A
BxxkxkdxxksdFAU
x
senxgmxkhgmxkUgUU e + 22 21=2
1==
x senxh =
( ) 3xxU = 4/30,1 mJ=
mx 90,0=
NFx 5=
)()(= AEBEW ccAB
)]()([= AUBUWAB
ABW
)]()([=)()( AUBUAEBE cc
)()(=)()( AUAEBUBE cc ++
E
)(21=)(= 2 xUvmxUEE c ++
1F 1F nF
in
ic UEE +
1==
6.0=m 30=
3101.12= k 5.5=d
O H O h O
0x
O
HgmcorpoUmolaUEc =)(0=)(0= 111
HgmHgmcorpoUmolaUEE c =00=)()(= 1111 ++++
hgmcorpoUxkmolaUEc =)(21=)(0= 12012
hgmxkhgmxkcorpoUmolaUEE c +++ 20202222 21=2
10=)()(=
hgmxkHgm 2021=
D
enxDHeenxh s)(=s= 00
h H
enxgmxkenxDgm s21=s)( 0200
sengm
xkD 2=20
0.06=D
m
Rh 5=
rvmN2
=
RH 5= )(PEc
)(5=)( RgmPU
221=)( vmQEc
RgmQU =)(
RgmvmRgm ++ 221=50
gmRvmN 8==2
N gm
gmgmR 65=18= 22 +
,13,.7=81==
arctgN
gmarctg
N
)()(=)()( AEBEdisWconsW ccABAB +
)()(=)()]()([ AEBEdisWAUBU ccAB +
)]()([)]()([=)()()()(=)( AUAEBUBEAUBUAEBEdisW ccccAB +++
)()(=)( AEBEdisWAB
E
20,0= D 055,0=0x
cosgmfa =
D
DcosgmW =
HgmHgmmolaUgravUKEi =00=)()(= ++++
20210=)()(= xkhgmmolaUgravUKE f +++
HgmxkhgmDcosgm + 2021=
2021)(= xkHhgmDcosgm ++
senDHh + =
2021= xksenDgmDcosgm +
D
2021=)( xkDcossengm
)(12=20
tgsengmxkD
055,0=0x 6=m 3101.12= k 30= 20,0=
cmD 0.09= D
)(1= tgDD
enDgmhHgmcorpoU s=)(=)(1 +
0=)(1 molaU
0=)(2 corpoU
[ ] 200020
0
0
02 2
1=21===)( xkxkdxxksdFmolaU xxx
2021=s xkenDgm
sengm
xkD 2=20
gmN +
Ox
RvmgmN2
=+
v
gmN =
Rvmgm2
=2
Rgmvm 2=2
v
)(221= 2 RgmvmE +
hgmE +0= h
221)(2= vmRgmhgm +
2vm
Rgmhgm 3=
Rh 3=
2410983,5 =tM 81049598,1 =r
30
im ),,( iiii zyxr
zyx ,,
),,( CMCMCM ZYXR
iN
iii
N
iCM mMrmMR 1=1=
=1=
iiN
iCMii
N
iCMii
N
iCM zmMZymMYxmMX 1=1=1=
1=1=1=
M M3 2=D
Ox M3 M
DXX MM =0=3
M mD 2= M3
CMX
.21
42
4)2(=3
)(03= mmMM
MmM
MMDMMXCM ==
+
+
mXCM 50,0= m50,0 M3
2410972,5= TM 300,81 310378,6= TR 510564,3= d
510067,4= D
121096= a 96 5,.104=
1=Hm 2710660,1 16=16= HO mm 18=M h
hYXYXYX OOHHHH =/2=0==0=0= ''
2=18/2)(16101= ++CMX
hhhYCM 98=18
16=18160101= ++
HOH HH O HH rad824,1=5,.104
pmradsenpmsena 76=)912,(096=/2)(=/2)( pmradcospmcosah 59=)912,(096=/2)(= 76=CMX 52=CMY
x y
dm dV r dVrdm )(= R
dVdmMdmrMR VVVCM ==1=
),,( zyx ),,( zyxdV
dmzMZdmyMYdmxMX VCMVCMVCM 1=1=1=
),( yx
)(x
L
LM = dm x
2=21==1=
2
0LL
LM
MdxxMdmxMXL
LCM
R
M a
z
1/222 )(= zar dz dzzadm )(= 22 z
aaadzzazMZa
z 83=4
3=)(=4
322
0=
0,3=cM 0,24=R 0,30=H 0,1=eM
Oz O 0,15=/2= HZc
z 0,9=0/8,240,3= z O 0,39=0,90,30== 0 ++ zHZe
cmMMZMZMZ
ce
cceeCM 0,21=0,10,3
0,150,30,390,1==+
+
+
+
Oz 0=CMX 0=CMY
0,30=R 50,0 M 0,5=r 0,15
x
0=dX 15=bX 2= RM 2= rm
cmrRr
rRrRXCM 42,0=25900
2515=15=150= 222
22
22
LM LT MM 300,81= x
300,821=)(0= Z
MMZ
MMZMMX
L
TLT
LTCM =
++
+
Z 510564,3= = dZ 310330,4 =CMX 510067,4= = DZ 310942,4 =CMX
0=0==/2==/2= OOHHHH YXhYXhYX
0=18016/21/21= ++ CMX
hhhYCM 91=18
016)(1)(1= ++
h 0=CMX 7= CMY
h(8/9) h(1/9)
dRdL = ),( R )/(= RM
[ ] 0===)(=1= 000
senRdcosRdRcosRRdLxMX LCM
[ ] RcosRdsenRdRsenRRdLLyMYCM 2===)(=1= 000
),,( r ),,( rdm
dddrsenrdVrdm 2==),,( cossenrx = sensenry = cosrz =
332= aM
dcosddrsenrMdddrsenrcossenrMX
aaCM
2
023/2
0022
0
/2
00=)(=
[ ] 0== 202/20
30
sendsendrrMX
aCM
dsenddrsenrMdddrsenrsensenrMY
aaCM
/2
023/2
0022
0
/2
00=)(=
[ ] 0== /202/2
03/2
00
cosdsendrrMYa
CM
ddrdcossenrMdddrsenrcosrMZ
aaCM
2
03/2
0022
0
/2
00=)(=
CMZ
aaaa
Mdrrsen
MZR
CM 83=4(2/3)=42
12=22=
4
3
43
0
/2
0
2
y Y
X Y z
Z
c
cm 2=
)(2= 2 cm rM
cc rRM )(= 22
42,0= cx
0,15=mx
Ox
]2)[(2)(== 222
222
rrRxrxrR
MMxMxMX
c
mccc
mc
mmccCM +
+
+
+
0,30=R 0,5=r
2
2
2
2
22
222
1025,910825,3=0,251000,9
0,150,25242),0(0),251000,(9=2)(=
+
+
+
+
rRxrxrRX mcCM
41,0= +CMX
N im t
iiN
iCM rmRM
1=
=
dtrdmrmdt
dRMdtd i
i
N
iii
N
iCM
1=1==)(=)(
iiN
iCM vmVM
1=
=
CMV
iv
iiN
iCM amAM
1=
=
CMA
ia
i
CMiN
iAMFF
==1=
i j ijF j
j i jiF i
jiij FF
=
CME AMF
=
EF
0,3=1m 0,4=12m 0,8=3m
,0,6=1 iNF ,0,12=2 iNF .0,16=3 jNF
1m )]0,2(),0,2[( 2m )]0,3(,)0,1[( 3m )]0,1(,)0,4[( cmA
cmX cmY
mkgmkgmkgmkgX cm 0,2=)0,80,40,3(0,40,80,10,4)0,2(0,3=
++
++
mkgmkgmkgmkgYcm 13,0=)0,80,40,3(0,10,8)0,3(0,4)0,(20,3=
++
++
NFNNNF yx 0,16=0,60,60,12= =
44,69=667,21,17= 22 ox
yyx F
FtgNFFF ===+
F Ox
cmA
2
321/1,140,0=0,15
0,160,150,6== smjijkg
NikgN
mmmFAcm ++++
./2,1)1,1()4,0(= 2222,2, smAAA ycmxcmcm =+=+
0=CMA
x
mkgkgmkgmkg
mMLmLMX i 60,2=60140
0,4600,2140=/2)(=+
+
+
+
mkgkgmkgmkg
mMmLMX f 40,1=60140
0,0600,2140=0/2)(=+
+
+
+
mmmXXX if 20,1=60,240,1==
mXx 20,1==
1m ,2m
2m 1m
2m 1m
CMX
520=1m 0=1x 5=2x 480=2m 1m
CMXmmxmm )(=0 21221 ++ CMX 1m 0,5=2x
cmcmcmxmmmXCM 4,2=1000
2400=5480520480== 2
212
++
4,25,2 1,0 1m
ECM FAmm =)( 21 + 1m 2m TF 2=
TgmgmAmm CM 2)(=)( 2121 ++
T
amTgm 11 = + amTgm 22 =+
a a T
gmmmmTgmm
mma21
21
21
21 2==++
T2 CMA
gmmmmACM
+
221
21)(
41=
gACM 0.0016=
1m 2m
ggmmmma 04,0==21
21+
a
CMA a
TgmgmFE 2)( 21 += TgmgmAmm CM 2)(=)( 2121 ++
gmmmmgmmF
21
2121
4)(=+
+
ggmmmm
mmFACM 0016,0=)(
41=)(= 22121
21
+
+
( ) ( ) jsmismv /0,2/0,31 += ( ) ( ) jsmismv /0,3/0,52 +=
ism )/0,5(
iiN
ii
N
ivmpP
1=1===
CMVMP
=
CMAMdtPd
=
CME AMF
=
CME AMdtPdF
==
100=m 0,40 =v o30=
3,01 =t 0,22 =t 12 tt
jgtvisenvjvivv yx )cos( 00 +=+=
senmvmvp xx 0== gtcosmvmvp yy == 0
3,01 == tt
smkgsensmkgp ox /.20,030/0,4100,0 == smkgssmsmkgp oy /.05,03,0/8,930cos/0,4(100,0 2 ==
jijpipp yx 05,020,01 +=+=
0414)250,0(=)20,005,0(== =
arctgarctgpparctgx
y
0,22 == tt smkgsensmkgp ox /.20,030/0,4100,0 ==
smkgssmsmkgp oy /.61,10,2/8,930cos/0,4(100,0 2 ==
jijpipp yx 61,120,02 =+=
07277)05,8(=)20,061,1(== =
arctgarctgpparctgx
y
12 ttt =
jijippp 56,10)61,105,0()20,020,0(12 =+==
constantePdtPdFE =0=0=
22102,1 2310,6.
2610,5.
depoisantes PP
=
e x y
0=ne ppp
++
0= cospp n 0=senpp ne
smkgsmkgppp en /.1036,1=/.)10(6.4)10(1.2== 2222322222 ++
242=898),(1== arctgpparctg e
vmp = )/(2=)/(2=/2= 2222 mpmvmvmK
.1059,1=108,52)/.()1036,(1= 1926
2222Jkg
smkgK
50=m 1000=v
180=F
vmpb =
VMpm =
vMmVVMvmpp mb ++ =0==
V
n F
vmnF =
16,3= sn 220=606,3= N
M 0v m
relv
v relvv
mMvvmvMvv relcm +
+ )(== 0
mMvmvvv rel+
== 0
m M
h
u V
v
Vuv
+=
jsenuiVcosuv )()(=
0=)( VcosumMV +
uMmcosmV+
=
22 21
21= vmVmmgh +
v
cosVuuVv 2= 222 +
u V v V
]))[((2= 2
22
cosmMmmm
cosmhgV++
m v dt dm u dmm vdv +
t v tt + mm vv + m u
tvmumvvmm
dtPdF
tE
++==
]))([(lim0
tvmumvmvmvmvm
dtdPF
tE
++==
0lim
vm m v
dtdmvudt
vdmFE ][
+=
dtdm
dtdm
dtdm
dtdmvudt
vdmFE ][
=
][ vu m mm m mm ][ vuvrel
=
dtdmvdt
vdmF RELE
=
1F
NsmskgskgsmdtdmvF REL 4241 1028,1/.1028,1/70/183 ====
NskgsmdtdmvF REL 42 1056,3/9,72/488 ===
NFFF 4412 1028,210)12856,3( ===
MWWsmNvFP 17,41017,4/1831028,2 64 ====
W745,5 = HP 1
HPHPWWP 3
61060,5/5,745
1017,4=
=
dtdmvFdt
vdm RELE
=
][ vuvREL
=
dtdmvdt
vdM REL
=
MdMvvd REL
=
relv
=M
MREL
v
v MdMvvd
00
0M 0v
= MMvvv REL 00 ln
00 =v
0,310000/300000 ==MM
( ) hkmhkmv /64020,3ln/5500 ==
0=dtvd
v vvrel
=
dtdMvF
=0
dtdMvF
=
dtdM
N
x x
dtdpsgL
MN +=
gxLMvL
Mdtvdx
LM
dtvdM
dtdp 2)( 2 ====
N
xgLMgxL
MxgLMN 32 =+=
1010628,1 111039,9
dt
dtFpd
=
if ttt =
dtFpp ftit
if
=
I
dtFI ftit
=
pppI if
==
2121 = pppp++
x
cospcosppp 2121 = ++
senpsenp 21=0
1p 1p
2p
2p
QEEEE cccc +++ 2121 =
i
ici m
pE 2=2 i Q
11/pp 12/pp 12/pp 12/mm Q
0=Q 0Q
0Q
1m 2m 1u 2u
1v 2v
22112211 = vmvmumum
++
0=Q
cosvmcosvmumum 22112211 = ++
senvmsenvm 2211=0
222211
222
211 = vmvmumum ++
30
0=2u 30= 1u 21 = mm cosvcosvu 211 = + senvsenv 21=0 22
21
21 = vvu +
1v 2v cosvcosvu 211 =
221121
21 =2 vcosvuvu +
cosvuv 1121 2=2
smcosuv /260== 11
smsenucosuvuv /150==1== 12121212
60=0.866=2=1 senvvsen
1u 2u
1m 2m 1v 2v
22112211 = vmvmumum
++
222211
222
211 = vmvmumum ++
x 1m
22112211 = vmvmumum ++
222211
222
211 = vmvmumum ++
x
)(=)( 222111 uvmvum
)(=)( 2222221211 uvmvum
2211 = uvvu ++
1221 = vvuu
1m 2m 2m 1m
2v
2112 = uvuv + 2v
221
21
21
211
2= ummmumm
mmv
+
+
+
1v
221
121
21
12
2= ummmmumm
mv
+
+
+
1m
2m
21 = mm 1221 == uvuv
2m 1m
12 >> mm 0211 == vuv
21 >> mm 1211 2vvuv ==
1m 2m
21
21
21
21
21 1=== u
vuvu
KKKFi
fi
0=2u
121
211 = umm
mmv
+
2
2121
2
2121
)(4=1= mm
mmmmmmF
+
+
12 206= mm 02,0=F 12 12= mm 28,0=F 12 = mm 1=F
vmmumum )(= 212211 ++
M m
vmMum )(= +
y
ygMmvMm )(=)(21 2 ++
u
ygmMmu 2= +
m M y
50,0=
smkgvmpi /.210=0,370== +
smkgvmp f /.210==
smkgvmppp if /.420=2==
24 /.102,4=0.01420== smkgt
ppF if
324 1042=/.102,4== smkgFF
cosuv 11 = senuv 12 =
sensenusencosu 11 =
sencos = /2= /2= +
31 100,2= m 21 1000,1= u 8,27= 22 1000,5= m 22 1020,1= u 3,33= O Ox 0>1u 0
0,56=V 8,9=g 50,0= 7,24=X
),,,( zyxOS ),,,,( zyxOS S
S S
S S
z r
,r
x
x
12= ttdt
12=
dtd
=
mh5623
2
2== dtd
dtd
2
3=t 23 25= tt +
dt
dtd tt
00 =
dtd tt
00 =
23 34= tba a b
)(= 00 tt +
00 =t
t +0=
200 21= tt ++
t
)(2= 0202 +
0,3= 120= 0,4=t 1 t
21 21= tt +
sradtt /0,24=0,42
0),(40,31202=22=
22
1
1
0,4=t 0=t
0=0 24.0=1
st 0,8=30,24== 01
r dt ds d
drds =
d r
dtdrdt
ds =
dtds/ v
rv =
ra =
A B
BAC
=
C senBAC = A B
a B A
B
C A
B
A
B
A B
kBjBiBBkAjAiAA zyxzyx == ++++
kji ,, Ox Oy Oz
=)()(= kBjBiBkAjAiAC zyxzyx ++++
+++ )()()(= kiBAjiBAiiBA zxyxxx
++++ )()()( kjBAjjBAijBA zyyyxy
)()()( kkBAjkBAikBA zzyzxz +++
i j k
0=(0)s==)(/2)(s==)(/2)(s==)(/2)(s=0=(0)s==)(/2)(s==)(/2)(s==/2)(s=0=0s=
enkkiienjkjjenikiienkjenjjkkenij
jjenkikkenjienii
iBAjBAiBAkBAjBAkBAC yzxzzyxyzxyx = ++
kBABAjBABAiBABAC xyyxzxxzyzzy )()()(= ++
zyx
zyx
BBBAAAkji
BAC
==
ABBA
= CABACBA
++ =)(
)()( CBACBA
CBABCACBA)()(=)(
BAC
=
v
rv =
r
dtrdrdt
drdtd
dtvda
+
=)(==
== dtdevdt
rd
)(= rra +
rrr 2)(=)(
r
rra 2=
rat
=
rvrrvr /)/(= 2222 r
rac
2=
2100,8 16=R
srad/49,3=602333,33=
r rar 2=
gaouamgm rr =:=
16== Rr
222 /1095,1=16)/49,(3=)( scmcmsradRar
21080,9= g 2 20,0=
S
S
sradsrad
msmhshrad
diarad /1029,7=86160
28,6=/6056/36002328,6=1
2= 5+
ttttdtd
dtd 415=)2(5== 223 ++
430=)4(15== 2 ++ tttdtd
dtd
3=t 147= 94= 2
0=0t 0=0
342300
=)3(4== tbtadtbtadt tt
0=0
45=)(==45
3400
tbtadttbtadt tt
sadRv /r74=0.75
2s3600m100000==
0= sad/r74=0 230=
222
02
/r14=230274=2= sad
Ra = xavv 2= 202 +
m73=5,106,771=2=
20R
vx
43 ctbtat +=
32 0,43 tta +=
F
r senF
F r
Fr
=
senFr=
5,1= 0,2=m
30=
gm =
mgsen 30= ,14.=8,90,250,05,1=
gm
r F
F PF
PFrFrFr
+ ==
im
ir iF
iiiiii amrFr
==
iii ra
= i
=i
)(= iiii rrm
])()[(= iiiiii rrrrm
+
0= ir
2= iii rm
iiN
ii
iFr
1=1=
==
21=
= iiN
irm
21=
= iiN
irmI
2
I=
M L
+
2=22=222 LMLMLMI
M
dVrrIV
2)(=
dV r
L M
x dx x LM/=
2
0
32
02
0 31=3=== LM
xLMdxxL
MdxxIL
LL
a
ddrrdS =
2=42=2===
24
23
0232
00222
00Maa
aMdrra
MddrraMdSrI aaa
a
dddrsenrdV 2= senr
dddrsenrdVrI aa 3400
2
02
00
2
0==
drrcoscosdsendrrI aa 40
0
33
04
0 32=2=
25
3 52=53
4432= aMaaMI
a
r dr drrdA 2= 2/= aM
2=2==2
302
20
aMdrraMdArI aa
r
dz 22= zar
dzzadzrdm )(== 222
( ) ( ) dzzadzzazadmrdI 222222222 2=])([21=2
1=
( )
+ dzadzzadzadzzaI
a
a
a
a
a
a
a
a4224222 22=2=
25 52=15
82= aMaI
3(4/3)= aM
OI z GI z ),,( zyxr
),,( zyxr ),,( ZYXR
z 22= yxd +
22= yxd + Rrr
+=
yYxXYXyxYyXxyx +++++++++ 22=)()(= 22222222
dVyYdVxXdVYXdVyxdVyxIO ++++++ 22)()(=)(= 222222
GI
x y
222 =)(= DMIYXMII GGO +++
2= DMII CMO + CMI D
12=23==222
2 LMLMLMDMII OG
a
z R /4R /2R z
gmrsengmrpesodotorque =/2)(=: 1
gmrsenNrnormaloreadatorque =/2)(=: 2
0=)(s=: 3 enfratritodeafordatorque a
L
22 =0= LMLMMI +
2/2
/2
32/2
/22/2
/2 121=3=== LM
xLMdxxL
MdxxIL
L
L
L
L
L
a
22 23== aMaMII cm +
z bcz III +=
c b
2(1/2)= RMIc Oz 2(1/2)= amIb m a
2
2
2
2==||4= Ra
Ra
MmeRa
Oz Oz /2= Rd
222
2222329=4
2162
1=)2(21=)()(2
1= RMRRmdamdmamIb
+++
222 327=32
921= RMRMRMIz
I=
m M R I
mg
mg
I=
IsenRT =/2)(
2(1/2)= RMI
RMT2=
m
m
amTmg =
a m T
)(2=2= agRMm
RMT
m Ra =
)(2= RgRMm
gRMm
Mm 2=21
+
Rg
mMm
+ 22=
gmMmMT
+ 2=
m T T
gmMma
+ 22=
im iT
iT T x
amgmTamgmT22
11
==
T a
gmmmma
+
21
12=
gmmmmT
+ 21
212=
M R 2(1/2) RM 1m 2m
af
P N af R
0=gMN 0=Rfa
amgmT 111 = + amgmT 222 =
0==12 amfTT ca
cm
IsenRfa =/2)(=
RaIIRfa ==
Ra = afTT =12 af
212 = RaITT
12 TT
ammgmmTT )(=)()( 211212 +
12 TT
gRImmmma
++
)/(= 22112
gMmmmma
++
/2)(= 2112
0,4=1m
4,0= 0,3=2m
0,2=3m
1T
amgmT 111 =
2T 2T 1T
amgmT 222 =
cIRTT =)( 12
Ra/= /2= 23RmIc
amTT 312 21=
12 TT
gmmmmma /2= 32112
++
2/35,0=0,10,30,40,34,00,4= sma
++
22
/2,1=30,0/35,0= sradmsm
NagmTNagmT
8,37=35),081,(90,4=)(=1,12=35,081),9(0.400,3=)(=
22
11
++
R M
amTgm 222 =
amTsengm 111 =+ a
RaIIRTT ==)( 12
ammgenmmTT )(=)s( 211221 ++
2(1/2)= RMI
aMRaRMTT 2
1=21=)( 2212
12 TT
ammgenmmaM )(=)s(21
2112 ++
a
gMmmenmma /2s=
21
12++
V
S
r
S
S
rvd =
S
VrVvv d
++ ==
S
Rvv ba ==
R S
VRVvv aa
++ ==
VRVvv bb
++ == av bv V S
VVVVRVvvAPonto aa 2==='=: +++
0==='=: VVVRVvvBPonto bb
+++
0== RVv
+
Rv =
Vv >
Vv
I=
R
x
cmamfsengm =
0=cosgmN
cma
IFr ==
cIsenfR =/2)(
cI
,=,= RaRV cm
V
2= RaIf cmc
cmcm
c amRaIsengm =2
gRmIsenac
cm )/(1= 2+
R
sengacm 32=
I
2222 23=2
1== RmRmRmRmII c ++
cI
IFr ==
P
Pr
=
senRgmsengmRPr ===
r
223= RmsenRgm
Rseng
32=
Racm =
sengacm 32=
0y
P T
cmmaTgm =
m
221= RmRT
Racm /=
cmamT 21=
gmacm 23=
cma 0y
00 3=2= ygmyav cm
0v
Rv =
mgf =
cmammg =
252= RmRgm
gacm = Rg
25=
0=t 0v
tgvvcm 0=
0=0
tRg
25=
Rv =
RtRgtgv 2
5=0
gvt0
72=
v t
00
0 75=7
2= vgvgvv
gvvgg
vvx
202
0
20
2
4912=2
(25/49)1=2=
vr rv/= rv =
sengaRMI cc 32=,2
1= 2
sengaRMI ec 75=,5
2= 2
ce aa >
2= RMI
2= RmRgm Rg
=
gacm =
Rv =
RtRgtgv =0
tRg
=
gvt2=0
v t
00
0 21=2= vg
vgvv
gv
gvv
gvvx
20
20
20
20
2
83=2
/4=2=
hkmV /40= v
hkmv /30= hkmv /50=
R v
0v
Lv0
Tv0
/78,9 sm /67,1 sm
221=)(2
1= vmvvmEc
m v
im iv
2
1=1= 21=)(2
1= iiN
iiii
N
ic vmvvmE
im
ir ir ii rv
=
222
1=
2
1= 21=)(2
1=21= Eii
N
iii
N
ic IrmvmE
EI
3133
sradsrad /49,3=0,60
2333,33=
2322 .106,5=)15,(0500,0500,0=21= mkgmkgRMI
JsradmkgIEc 22232 1041,3=)/49,(3.106,5500,0=21=
CMV
ir ii rv
= i
CMii Vvv
+=
i
]2[21
])()([21=)(2
1=21=
221=
1=2
1=2
1=
CMiCMiiN
i
CMiCMiiN
iCMii
N
iii
N
ic
VvVvm
VvVvmVvmvmE
++=
=+++
ii rv =
21=1=
221= 2
1)(21= VmvmVrmE i
N
iii
N
iCMii
N
ic
+
+
0=ii rm
0=ii vm
21=
22 =,21
21= ii
N
iccCMc rmIcomIVME +
cI
smmsradRVCM /523,0=15,0/49,3==
JJsmkgIVME cCMc 102,0=1041,3)/523,(0500,0500,0=21
21= 2222 ++
F
d dt F
r dt ds d
drds =
F sd
)()(=== drcosFdscosFsdFdW
F sd cosF
rcosF )(
ddW =
=== dtd
dtdWP
I=
dtdIdt
ddtdIdt
dW
==
12= ttt
2122
2
1
2
1
2
1 21
21====
IIdIdtdt
dWdWW tt
t
t
amTgm =
raRMITR c 22
1==
gmMma 22=+
gmMmMT 2= +
a
T
TRW ==
T
22
/6,26=15,0/80,9
0,1292,20,12=2
2== sradmsm
kgkgkg
Rg
mMm
Ra
+
+
0,2=t
radssradtt 2,53=)0,(2/6,26210=2
1= 2220 ++
T
+
RgmMmMRTW 2==
JradmsmKgKgKgKgW 4,46=2,5315,0/80,90,1292,2
0,192,2= 2 +
0E
JWEEc 4,46== 0 +
U
E
icCM UIVME ++ 22 21
21=
L
h
hgmU =
021
21=2
121 22202 ++++ cCMcCM IVmhgmIVm
senLh = RV =
222 43=4
121= CMCMCM VmVmVmsenLgm +
senLgVCM 34=
m r h
Rh 6=
RvmF2
=
v
RgmIvmRgm c ++ 22 21
21=6
2(2/5)= rmIc rv/=
222
222
107=5
121=5
221
21=5 vmvmr
vrmvmRgm
++
Rgv 750=2
gmF 750=
sradssradt /2,53=0,2/6,260== 20 ++
2202
21
21
21= IIIW
I
sradmKg
JoulesRM
WIWc
/2,53=)15,(092,24,464=(1/2)
2=2= 22
2= RmIc
RgmRmvmRgm ++ 222 21
21=6
Rv/=
22 21
21=5 vmvmRgm +
gmRvmFeRgmv 5==5=2
cm30 kg5,1
srad /6
m R V
221= ec IE
cI
t =W
h 1 = 1VVCM =
2 = 2VV = 21 < 21 VV <
m
)(=== vrmvmrprLo
r v
r v 2
dtvdrmvdt
rdmdtLd o
+=
0== vvevdtrd
Frdtvdmrdt
Ld o
==
oo
dtLd
=
o
gmP
= T
P
=
sengmPsen ==
vmpL
==
vmvsenmL =/2)(=
sengmamdtLd
==
m
z l
r v
z l
)(902 z
z
senrml 2=
cossenrml 2=
z
22== senrmsenllz
=senr
2= mlz
2= mI z
Ilz =
z l
I l
l
vmlz
=
zz Il =
vmlz
=
z
gmP =
vmLP
==
vRmvsenmLPRPsensenP =)(==)(==
Rsen =
PRgRmdtdvRmdt
dL ===
hkm /320 m30
kg3 m5 Hz5 Hz3
xl yl zl
xv yv zv m x y z
g500 cmx 1= cmy 1= jiF +=
jiv += jiv 3 =
kg2 mr 5,0= smv /5=
iN
ilL
1=
=
iN
iiN
i dtld
dtLd
1=1=
==
i i
eiN
idtLd
1=
=
ei
i
1m 2m
2l
1l z
2m 1m
z 1l
2l
21= llL
+
z
z
IL =
IdtLd =
I
)()(= 12 gmRgmR
+
RgmRgmN 12=
RvIRvmRvmIRvmRvmL cc ++++ 1212 ==
cI v
dtdv
RI
dtdvRmdt
dvRmdtdL c++ 12=
R
dtdv
RImmgmm c
++ 22112 =)(
gRImmmmadt
dvc
221
12/== ++
m R
TRsenTR =/2)(=
RvIIL cc ==
/2= 2MRIc
dtdv
RI
dtdL c=
aRI
dtdv
RITR cc ==
a
amgmT =
amgmT =
aRIamgm c2=)(
a
gmamm =)21( +
ggmmma 3
2=/2= +
M R
af
afR=
RvIIL ==
RaIIdt
dL ==
RaIfR a =
)(== agsenmfamfsengm aa
RaIRamRsengm =
a
gsenRMI
a)/(1
1= 2+
2(2/5)= RMI
gsena 75=
1m 2m 3m 1l 2l 3l v
5,1 400 0,2=
m R
M r F
edtLd
=
z
zz IL =
zL
1410
5100,7
2(2/5)= RMI
nnee II =
e n
sraddiasdiarad
Te /104,2=/40086302=2= 6
T
sradRR
II
enee
nen /106,4=104,216
100,7=== 36252
srad /20=
2.9,1 mkgI pi =
2.2,1 mkgI pf =
M m v d
V
VMvm ++ 0=0
cIdvm ++ 0=0
222 21
210=02
1cIVMvm +++
V m
VmMv =
222 = MIVVm
M c+
IMVd
Imvd
==
222222 = dV
IM
MIVVm
Mc
c+
21= dIM
mM
c+
m /12= 2LMIc
22
2
12= LdLMm+
R I M 1v
2v 1v M R I
R
IRvMRvM ++ 21 =0
Rv /= 2
RvIRvMRvM 221 = +
)/(1= 21
2 RMIvv
+
M r F
F
FdtPd
=
FrdtLd
=
r
dtFMv t00 =
0v
oI
FRdtdIo )(= +
dtFRI to + 0=
RIvM o+
=0
=0v
RIM o+
=
cmI 2222 2=== RMRMRMRMII cmo ++
22=)( RMRM +
0=2 22 RR +
R= R2= R=
F
L
mgr =
r r mg
dtLd
=
Ld
Ld
L
dt dtLd
=
dt L Ld
Ld
L L
dt
L
L
L
dtd=
L xy x
dtdL =
senLdt
senLdLd ==
senrgm= IL =
Irgm
senLsenrgm
senL ===
60=M 0,3=R 0,8=h
IL =
0
L
senrrsen
rsenRvac 2
22 )(= ==
senrmmaF c 2= =
gmN = Nfa = gmfa
rgsen 2
40,0=
fpfipi II =
piI i
pfI f
f
sradmkgmkg
II
ipfpi
f /20.2,1.9,12
2
=
=
sradf /32=
221MRIcm = 2220 2
321 MRMRMRI =+=
0=23 22 RR +
( )R17= ( )R17= + ( ) RR 6,117=
IrgM=
sradsrad /1051,1=242= 21
2222 .1062,1=10)0,(30,603=10
3= cmgcmgRMI
cmcmhr 0,6=40,83=4
3=
sradsradcmg
cmscmg /4,14=/1051,1.1062,1
0,6.1080,90,60= 22222
rgsen 2=
40,0= 22 .1080,9= scmg cmr 0,6= srad/4,14=
315,0=0,6)/4,(14.1080,940,0= 2
22
cmsradscmsen
37,18=
0
2
0q
))((2= 2 ImrmMgh ++
ghv 2= ghmMmv 2)(= + v
2MrI =
eF
cme amF =
dtLd
e
=
0=cmv 0=
0=eF
0=e
m a gmP = 1F 2F 1F 2F
0=21 gmFF +
0=202 21 FagmFa
+
2== 21gmFF
30=w 0,3=c 4,2=a 8,0 80=W
W w 1F hF1 vF1 hF1
2F
b
macb 8,1== 22
y x W 0x 0y
13,53=33,1=80,140,2==
mmtgb
atg
m5,1=
mcosmcosx 90,0=13),(535,1==0
msenmseny 20,1=13),(535,1==0
8,0 0b
mcosmb 48,0)13,53(80,00 ==
0=21 FF h 0=1 wWF v
0=002 bwxWaF
048,03090,0804,22 = mNmNmF
NF 36=2
NFF h 36== 21
NNNF v 21 1010,1=3080= +
40,0=
150=W 3,3= 0,2=d 30 100=w
hT vT F
hF vF W w
x y
0=wWTF vv + 0=hh TF
0=)30(21)30()30( WcosFsenFcos hv
hT vT hF vF
NwWTF vv 250== ++
NmFmFm hv 250866),03,(3500,0500),03,(3=866),03,(3 +
NFF hv 125577,0= +
hT
vTT
tgTT hv =
3030=
cossendtg
hhv TmmmTT 122,0=866,03,3
500),03,(30,2=
hv TNF 122,0250= hh TF =
NTF hv 125577,0= +
NFNT vh 228=179=
NNNFNT vv 22=228250=250=
NTTT vh 180== 22 +
NFFF vh 290== 22 +
M m im
iF iF
m
g gmF ii =
m im
gMgmF i ==
F
iF
im ir
( ) grmgmr iiii
==
R
( )ii rmRM =
gRM =
gMF =
F
pb
F
rb R
)(= asfordasbrioequilRFN +
)(= OaorelaemasfordastorquesdosbrioequilRbFb rp
pb rb
r
p
bb
FR =
rp bb > FR > F rp bb / F R
r
p
bbV =
rb
rp bb <
0=22222 21 gmFaFa +
+
gmalFaF =)()( 21 ++
21 = FgmF
gmalFalFgm =)()()( 22 ++
aFagm 22=
2=2= 12gmFegmF
d
0=02 bwcosdWaF
mNmcosdNmF 48,03013),(53804,22
2F d
mmNmdNF 4,2
48,03060),(080=2+
NNNwWFF vh 44=)30(8040,0=)(== 11 ++
21 = FF h
NmmNmdN 44=4,2
48,03060),(080 +
md 9,1=
msendH 5,1=800,083,1=13),(53=
100=0.022.0= m
mbb
Vrp=
0,60 0,15 0,20
P R
30,0
40 10
0,1 10,0
600
L
29= 60=
2,7 x
2,2 80 30 30
600 60.2 88
40,2 76 60 80,1
10 0,3 50,1=h
70 < 70
1m 2m r
221= r
mmGF
1110673,6= G 2 2
1m 2m 2m 1m
G 1010 2 2
1m 2m r 2m 1m
12F
1m 2m 21F
2m 1m
r 1m 2m
rrmmGF = 22112
12F
2m 1m 1221 = FF
rrmmGF = 2 2121
r
100=m 00,1
Nm
kgkgkgmNF 72222211 107,6=
)00,1(1000,11000,1/.10673,6=
G
15=a 300=1m 400=2m 150=3m 100=4m 100=5m
/2
Ox /4
/4)(cos/4)(cos/4)(cos/4)(cos= 51413121 FFFFFx ++ /4)(cos/4)(cos/4)(cos/4)(cos= 51413121 FFFFFy +
2a 1m
22= ar 2=
22 ar
)(4cos2= 543221 mmmma
mGFx ++
)(4cos2= 543221 mmmma
mGFy +
kgmkgkgmNFx 100),0100,0150,0400,0(785,015,0
300,0/.10673,62= 222211 ++
kgmkgkgmNFy 100),0100,0150,0400,(0785,015,0
300,0/.10673,62= 222211 +
1010492,3= xF 1010889,4= yF
NFFF yx 1022 10008,6== + 54,125==
xy
FF
tgarc
Ox
0,2=1m 0,5=2m 0,1=3m 0,3=a 1m 1m 3m 2m
G
G
m d
=
IT 2=
I
22= dmI
222 )2(4=
Tdm
M m M m
G
G
G
1110003),0673,(6= G 2 2
241098,5= TM 301099,1= SM 111050,1= r
Nm
kgkgkgmNF 2222224302211 105,3=
1050,11098,51099,1/.10673,6=
G
2m x 1m 1m 3m
0=)( 223
221
+
xammGx
mmG
0=)( 23
21
xam
xm
+
0=24 22 axax +
62= aax
a
mxmx 2,10=8,1= 21
2m 1m 3m 8,1=x
1m 61 =m 32 =m 103 =m 1m
4m 3m 21 =m 22 =m 53 =m 30=L
000,11
5=M 1=m M m 3
m 15
1
M r t r m R
dS )s(2 enr dr t
denrtdV s2= 2
denrtdVdM s2== 2
dM m m 1Fd
m 2Fd
m
m Px Py
Py Px cos1dF cos2dF
Py Px
coss2=cos== 222 denrtxGm
xdMmGdVFF x
V x
xrRcos cos=
cos2= 222 rRrRx +
RxrRr 2=cos222 +
denrRdxx s2=2
dxrRxden =s
xRxrR
2=cos222
dxxrR
RrtmGF rR
rR
+
1= 2
22
2
rdxxrRrR
rR4=12
22
+
2
2 )(4= RmtrGF
Mtr =4 2
2= RmMGF
Rr >
rR >
m r
2= rmMGF
R
gmP = g
2= rMGg
dg dr
32= rdrMGdg
Rh
rdr
gdg 22= =
6104,6= R h
hrdr == Rr =
2 2
21 0114,6378
2 2
0 50 10 60 20 70 30 80 40 90
m
Ox
camNRmMG =2
ca
camRmMGN 2=
N N egmN = eg
)(= 2 EquadornoaRMGg ce
2
222 4=2== T
RRTRac
T 41064,8= T 61037,6= R 0336,0=ca 2
cosRRMGge 22=
eg g
2= RMGge
M 0m M 0m
FM
FM
F
M M M
M 0m
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