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

Trreo Campus Pampulha 31270-901 Belo Horizonte/MG

Tel.: +55 31 3409-4650 Fax: +55 31 3409-4768 www.editora.ufmg.br editora@ufmg.br

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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

1500 3105,1 x 9000 000.000.000.300 3109x 11103x

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=

( )st p 08,095,0 =

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

ABttxx

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

t=

=

0lim

)(xdtdv = )(vdt

da =

2

2

dtxd

dtdx

dtda =

=

75

mtsmtsmx 9,5)/0,1()/8,9( 22 +=

!"

#"

$"

ox ot

ov x t v

76

o

o

ttvv

tva

=

=

)( oo ttavv +=

)(tv a ov ot atvv o +=

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=

!"

#"

99

$"

%"

&"

!"

#"

!"

#"

100

$"

%"

&"

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