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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
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{TEXT 256 19 "Oscila\347\365es For\347adas" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Vamos supor que novamente temos
 um sistema massa-mola que agora \351 sujeito a uma for\347a externa F
(t)." }}{PARA 0 "" 0 "" {TEXT -1 47 "A nossa edo do oscilador ganha um
 termo a mais:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "diff(x(t),`$`(t,2)) = -omega[0]^2*x(t)+F(t)/m;" "6#/-%%
diffG6$-%\"xG6#%\"tG-%\"$G6$F*\"\"#,&*&&%&omegaG6#\"\"!F.-F(6#F*\"\"\"
!\"\"*&-%\"FG6#F*F7%\"mGF8F7" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "onde " }{XPPEDIT 18 0 "omega[0] \+
= sqrt(k/m);" "6#/&%&omegaG6#\"\"!-%%sqrtG6#*&%\"kG\"\"\"%\"mG!\"\"" }
{TEXT -1 5 " \351 a " }{TEXT 261 10 "frequ\352ncia" }{TEXT -1 1 " " }
{TEXT 260 7 "angular" }{TEXT -1 1 " " }{TEXT 262 7 "natural" }{TEXT 
-1 14 " do oscilador." }}{PARA 0 "" 0 "" {TEXT -1 77 "Para deixar a co
isa mais concreta, vamos supor ainda que a for\347a F \351 da forma" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "F(t)
=F[0]*cos(omega*t)" "6#/-%\"FG6#%\"tG*&&F%6#\"\"!\"\"\"-%$cosG6#*&%&om
egaGF,F'F,F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 29 "Vamos res
olver isso no Maple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "edo
:=diff(x(t),t,t)=-omega0^2*x(t)+F0/m*cos(omega*t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$edoG/-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F,\"\"#,&*&)%'
omega0GF0\"\"\"F)F5!\"\"*(%#F0GF5%\"mGF6-%$cosG6#*&%&omegaGF5F,F5F5F5
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dsolve(edo);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,(*&-%$sinG6#*&%'omega0G\"\"\"
F'F/F/%$_C2GF/F/*&-%$cosGF,F/%$_C1GF/F/**%#F0GF/-F36#*&%&omegaGF/F'F/F
/%\"mG!\"\",&*$)F.\"\"#F/F<*$)F:F@F/F/F<F<" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 100 "Ok, mas vamos ser ainda mais modestos e pedir a solu\347
\343o para o caso em que o sistema sai do repouso:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 31 "dsolve(\{edo,x(0)=0,D(x)(0)=0\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&**-%$cosG6#*&%'omega0G\"\"\"
F'F/F/%#F0GF/%\"mG!\"\",&*$)F.\"\"#F/F2*$)%&omegaGF6F/F/F2F/**F0F/-F+6
#*&F9F/F'F/F/F1F2F3F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "
sol:=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG,$**%
#F0G\"\"\",&-%$cosG6#*&%'omega0GF(%\"tGF(!\"\"-F+6#*&%&omegaGF(F/F(F(F
(%\"mGF0,&*$)F.\"\"#F(F0*$)F4F9F(F(F0F0" }}}{EXCHG {PARA 257 "" 1 "" 
{TEXT -1 38 "Uma imagem vale mais que mil palavras:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 60 "# dando nome aos bois\n m:=1;omega:=2;ome
ga0:=2.2;F0:=1;\nsol;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG\"\"#" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'omega0G$\"#A!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 
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\"\"#F(F1F(F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# gr
\341fico dessa solu\347\343o\ngsol:=plot(sol,t=0..100):\ndisplay(gsol)
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255 "Nesse caso as setas n\343o s\343o desenhadas (a edo \351 de segun
da ordem, as setas n\343o s\343o informa\347\343o suficiente de qualqu
er forma nesse caso) e o n\372mero default de pontos tamb\351m n\343o \+
d\341 uma figura t\343o boa assim (tente tirar o que eu pus e verifiqu
e por si mesmo)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 93 "Agora, de volta ao meu coment\341rio. Olha s\363 o que oc
orre se eu puser um seno espec\355fico junto:" }}}{EXCHG {PARA 0 "> " 
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"Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "(Procurem depois Re
de Tupi - n\343o deve haver ningu\351m velho o bastante para se lembra
r, mas just in case)" }}{PARA 0 "" 0 "" {TEXT -1 23 "Como eu acertei o
 seno?" }}{PARA 0 "" 0 "" {TEXT -1 92 "Trigonometria. D\341 pra escrev
er a diferen\347a de dois cossenos como um produto de senos, j\341 que
" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "cos(sigma+delta)-cos(sigma-delta)
 = (-2)*sin(sigma)*sin(delta);" "6#/,&-%$cosG6#,&%&sigmaG\"\"\"%&delta
GF*F*-F&6#,&F)F*F+!\"\"F/*(,$\"\"#F/F*-%$sinG6#F)F*-F46#F+F*" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Basta ent\343o usar a f\363rmul
a acima com" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sigma=(a+b)/2" "6#/%&sig
maG*&,&%\"aG\"\"\"%\"bGF(F(\"\"#!\"\"" }{TEXT -1 3 " e " }{XPPEDIT 18 
0 "delta=(a-b)/2)" "6#/%&deltaG*&,&%\"aG\"\"\"%\"bG!\"\"F(\"\"#F*" }
{TEXT -1 20 "  que temos, j\341 que " }{XPPEDIT 18 0 "a = sigma+delta;
" "6#/%\"aG,&%&sigmaG\"\"\"%&deltaGF'" }{TEXT -1 3 " e " }{XPPEDIT 18 
0 "b=sigma-delta" "6#/%\"bG,&%&sigmaG\"\"\"%&deltaG!\"\"" }{TEXT -1 1 
"," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 
"cos(a*t)-cos(b*t)=-2*sin((a+b)/2*t)*sin((a-b)/2*t)" "6#/,&-%$cosG6#*&
%\"aG\"\"\"%\"tGF*F*-F&6#*&%\"bGF*F+F*!\"\",$*(\"\"#F*-%$sinG6#*(,&F)F
*F/F*F*F3F0F+F*F*-F56#*(,&F)F*F/F0F*F3F0F+F*F*F0" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 19 "No caso em quest\343o," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 4 "sol;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "A:=2*F0/m/(omega^2-om
ega0^2);\nsigma:=.5*(omega0+omega);delta:=.5*(omega0-omega);\n# solu
\347\343o como produto de senos\nsolalt:=-A*sin(sigma*t)*sin(delta*t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&$\"+!>w/>\"!\"*\"\"\"-%$cosG6#
,$*&$\"#A!\"\"F(%\"tGF(F(F(F0*&$\"+!>w/>\"F'F(-F*6#,$*&\"\"#F(F1F(F(F(
F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$!+!Q_4Q#!\"*" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%&sigmaG$\"$5#!\"#" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&deltaG$\"#5!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%'solaltG,$*($\"+!Q_4Q#!\"*\"\"\"-%$sinG6#,$*&$\"$5#!\"#F*%\"tGF*F*F*-
F,6#,$*&$\"#5F2F*F3F*F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "# confere que ambas express\365es s\343o a solu\347\343o\nplot(sol
-solalt);" }}{PARA 13 "" 1 "" {GLPLOT2D 774 348 348 {PLOTDATA 2 "6%-%'
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*F/FA7$$\"3iv=nj+U')**F/FX7$$\"#5F*F+-%'COLOURG6&%$RGBG$\"*++++\"!\")F
+F+-%+AXESLABELSG6$%\"tGQ!6\"-%%VIEWG6$;F(F]aw%(DEFAULTG" 1 2 0 1 10 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 12 "" 1 "" {TEXT -1 79 "Ver esse fen\364meno com molas e massas \+
\351 complicado, mas \351 f\341cil ouvir o exemplo." }}{PARA 0 "" 0 "
" {TEXT -1 197 "Para isso, pegamos dois diapaz\365es com frequ\352ncia
s angulares pr\363ximas e tocamos ambos. O som que ouvimos \351 claram
ente uma sen\363ide r\341pida modulada por uma lenta. Confira em (mais
 pro final do v\355deo) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 58 "http://www.youtube.com/watch?v=722ev4GqArY&feature
=related" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
183 "A hist\363ria de senos fren\351ticos e senos chapados \351 assim \+
se as frequ\352ncias angulares s\343o pr\363ximas uma da outra (afinal
 as duas frequ\352ncias no produto s\343o a m\351dia e a semi-diferen
\347a de " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 3 " e " }
{XPPEDIT 18 0 "omega[0]" "6#&%&omegaG6#\"\"!" }{TEXT -1 2 ")." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Uma pergu
nta \363bvia \351 \"o que ocorre se as frequ\352ncias angulares natura
l e for\347ada coincidirem?\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 257 11 "Resson
\342ncia" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
29 "Agora estamos no caso em que " }{XPPEDIT 18 0 "omega=omega[0]" "6#
/%&omegaG&F$6#\"\"!" }{TEXT -1 39 ". Vamos deixar o Maple fazer as con
tas:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart;with(plots)
:with(DETools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "edo:=dif
f(x(t),t,t)=-omega^2*x(t)+F0/m*cos(omega*t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$edoG/-%%diffG6$-%\"xG6#%\"tG-%\"$G6$F,\"\"#,&*&)%&om
egaGF0\"\"\"F)F5!\"\"*(%#F0GF5%\"mGF6-%$cosG6#*&F4F5F,F5F5F5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dsolve(edo);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,(*&-%$sinG6#*&%&omegaG\"\"\"F'F/F/%
$_C2GF/F/*&-%$cosGF,F/%$_C1GF/F/*&#F/\"\"#F/**%#F0GF/,&F2F/*(F*F/F.F/F
'F/F/F/%\"mG!\"\"F.!\"#F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Ag
ora j\341 vemos, que h\341 um termo novo que introduz um comportamento
 distinto: " }{XPPEDIT 18 0 "t*sin(omega*t)" "6#*&%\"tG\"\"\"-%$sinG6#
*&%&omegaGF%F$F%F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 48 "O q
ue ocorre se novamente come\347armos do repouso?" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 31 "dsolve(\{edo,x(0)=0,D(x)(0)=0\});" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG,&*&#\"\"\"\"\"#F+**%#F0GF+%\"
mG!\"\"%&omegaG!\"#-%$cosG6#*&F1F+F'F+F+F+F0*&#F+F,F+**F.F+,&F3F+*(-%$
sinGF5F+F1F+F'F+F+F+F/F0F1F2F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "sol:=simplify(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$solG,$*&#\"\"\"\"\"#F(*,%#F0GF(-%$sinG6#*&%&omegaGF(%\"tGF(F(
F0!\"\"F1F(%\"mGF2F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "\311 aq
ui que a porca torce o rabo! Novamento temos uma sen\363ide modulada p
or uma fun\347\343o A(t), mas que " }{TEXT 258 18 "cresce lineramente
" }{TEXT -1 8 " com t!!" }}{PARA 0 "" 0 "" {TEXT -1 11 "Um gr\341fico:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "# dando nome aos bois\n
 m:=1;omega:=2;F0:=1;\nA:=1/2*F0/omega/m;\nsol;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omeg
aG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F0G\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"AG#\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&#\"\"\"\"\"%F&*&-%$sinG6#,$*&\"\"#F&%\"tGF&F&F&F/F&F&F&" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "# gr\341fico da solu\347
\343o\ngsol:=plot(sol,t=0..100):\n# gr\341fico da amplitude A(t)\ngA:=
plot([A*t,-A*t],t=0..100,color=blue):\ndisplay(gA,gsol);" }}{PARA 13 "
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WG6$;F(FczF]`[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 
0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
10 "Quando em " }{TEXT 259 31 "Tintim e as J\363ias da Castafiore" }
{TEXT -1 242 " a cantora l\355rica quebra um copo com a voz (quebra, n
\343o quebra?), \351 justamente isso que vemos. Seu canto tem a mesma \+
frequ\352ncia que a vibra\347\343o natural do cristal, que partindo do
 (quase) repouso vibra com amplitude cada vez maior at\351 quebrar." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "Mas em \+
geral n\343o vemos efeitos t\343o extremos de resson\342ncia. Basicame
nte porque efeitos de resist\352ncia do meio tamb\351m exercem influ
\352ncia." }}{PARA 0 "" 0 "" {TEXT -1 78 "De fato, nesse caso a equa
\347\343o fica, lembrando o que j\341 vimos em aulas passadas" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "restart;with(plots):with(DET
ools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "edo:=diff(x(t),t,t)=-omeg
a^2*x(t)-alpha/m*diff(x(t),t)+F0/m*cos(omega*t);" }{TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$edoG/-%%diffG6$-%\"xG6#%\"tG-%\"$G6
$F,\"\"#,(*&)%&omegaGF0\"\"\"F)F5!\"\"*(%&alphaGF5%\"mGF6-F'6$F)F,F5F6
*(%#F0GF5F9F6-%$cosG6#*&F4F5F,F5F5F5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "dsolve(edo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"
xG6#%\"tG,(*&-%$expG6#,$**\"\"#!\"\"%\"mGF0,&%&alphaGF0*$,&*$)F3F/\"\"
\"F8*(\"\"%F8)%&omegaGF/F8)F1F/F8F0#F8F/F8F8F'F8F8F8%$_C2GF8F8*&-F+6#,
$**F/F0,&F3F8F4F8F8F1F0F'F8F0F8%$_C1GF8F8**%#F0GF8-%$sinG6#*&F<F8F'F8F
8F<F0F3F0F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "# atribuindo valores, com resist
\352ncia pequena\n# i.e.: alpha< 2*omega*m\nm:=1;omega:=2;F0:=1;alpha:
=(2*omega*m)*.1;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaG\"\"#" }}{PARA 11 "" 1 "" 
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\"\"&!\"\"F'\"\"\"F0F1-%$sinG6#,$**\"\"$F1F/F0\"#6#F1\"\"#F'F1F1F1%$_C
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45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
118 "A amplitude cresce, mas n\343o de forma linear. \311 f\341cil ver
 que a solu\347\343o, para tempo grande tem m\363dulo limitado por 5/4
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