Um anel circular de raio R, como mostrado na figura 29, gira em torno do eixo vertical passando por um de seus di\303\242metros a uma velocidade angular constante de
\342\204\246. Uma conta est\303\241 confinada a se mover neste anel.
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^yiYweY_hvwpWqNVdTavZ>[Pv\\QFxNHsCIsAyrA_elvayvcOnaxna@a]FocaXtvHnpObNPrDF`nQaeogKNgRqgqxxpqxlHufatNadRnjJPbF@tZOt;psHqs<_Zqohpohqwhiwgiwf`ofLGoNhfPGvPwipAiuWie^\\D>qvQwdQkK>etvusHqs^eXo_QH^;Aau`pn`ptPqdoboqyxqieWgeFlB?hAp_CYwJ?nOp_[nyKoec^in>mVowIyvmywiywj>mVXtb_`xnax>mE?RWC@_DL=SAQyvayvMYtMWiWxrcHkCH;GyfwwDaVlsvayvrWE]GBPeEUugQIuNWC@Cew?WkmDXkDAqTg?VK=CoEcOOrDCHv=ykIBQYUkyVNiD]idDer<KhFIEjgDdoi`ObyexZ=r;WSyAVv]YVOYdyW@kSScfSgh?Ih`UhCUdeQfOSHekB]?fEmU_?h<]C_kdTOSkcr;=D>;XQGVRktdeHd]EnOtTWh:Mb]EhR[yccdNGsd_CZ;T\\CwqCVEuTPmTZaVEgr\\MGOOWqWuVUvlOdSuDocFZeF<kCDivlSxA[Unoh`UcZMTRIFrUDLQX^?Dy?hkosEeclgRvuDgoyuIwboyQUGrOxmEirMgi]rH[wLsYOSsQGTyQDpsYKmsP_If?lUqrttOYTweuJfxmHyXKuX<tQ:@xqHosIX>ySaQwu<pjvoynrQYuDHZwyd^xe;dJKTREF>?d<QC[[bJO:_DP=C[[fjORBCF^BkyIjie?kgUMUisT^GD:;B:;RLEdMCde?DR?6>\"\{\}-%+ANNOTATIONG6'-%)BOUNDS_XG6#$"#5!""-%)BOUNDS_YG6#$""!F*-%-BOUNDS_WIDTHG6#$"%gAF*-%.BOUNDS_HEIGHTG6#$"%![#F*-%)CHILDRENG6"-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6(-I#miGF$6#Q!F'-F#6&F+/%'familyGQ0Times~New~RomanF'/%+executableGQ&falseF'/%,mathvariantGQ'normalF'F+F1F4F7Determine a frequ\303\252ncia natural da vibra\303\247\303\243o livre do cord\303\243o em torno da posi\303\247\303\243o de equil\303\255brio est\303\241vel. Em seu c\303\241lculo, use m = 1 kg, R = 0.25 m, \342\204\246 = 10 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJHJhZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUYvNiVRInNGJ0YyRjVGMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRjUvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkUvJSliZXZlbGxlZEdGPy8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYnRj0vRjZRJ25vcm1hbEYn, g = 9.81 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 adequado de coordenadas:
O \303\242ngulo \316\270 pode ser escolhido para representar a posi\303\247\303\243o do cord\303\243o.restart:params:=[M = 1, R = 0.25, Omega = 10, g = 9.81]:T := 1/2 * M * (R*diff(theta(t),t))^2 + 1/2 * M * (R * sin(theta(t))*Omega)^2V := M * g * R * ( 1 - cos(theta(t)))L:=T - V;GovEq:=-VariationalCalculus:-EulerLagrange(L, t, theta(t))[1] = 0Eql:=subs({theta(t) = theta__0, diff(theta(t),t,t) = 0},GovEq)factor(Eql)Ent\303\243o tamb\303\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 or 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Vale a pena notar que, embora a primeira solu\303\247\303\243o esteja sempre presente, a exist\303\252ncia do segundo ponto de equil\303\255brio na verdade depende dos valores de R, g e \316\251. Para o nosso caso, o valor desta quantidade \303\251 mostrado abaixo.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Isso indica que, para o nosso problema, temos tr\303\252s pontos de equil\303\255brio separados localizados emLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYzLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiVRIjBGJ0Y1RjcvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R0Y/L0krbXNlbWFudGljc0dGJFEnYXRvbWljRictSSNtb0dGJDYuUSI9RidGNUY3LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZ25GPC1GSjYuUSIsRidGNUY3Rk0vRlBGQUZRRlNGVUZXRlkvRmZuUSYwLjBlbUYnL0ZpblEsMC4zMzMzMzMzZW1GJy1GSjYuUSJ+RidGNUY3Rk1GT0ZRRlNGVUZXRllGXm8vRmluRl9vLUZKNi5RIitGJ0Y1RjdGTUZPRlFGU0ZVRldGWS9GZm5RLDAuMjIyMjIyMmVtRicvRmluRmpvLUYvNiZRJ2FyY2Nvc0YnRjJGNUY3LUkobWZlbmNlZEdGJDYlLUYjNiYtRj02JVEnMC4zOTI0RidGNUY3LyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRidGNUY3RjVGN0Ziby1GSjYwUSRhbmRGJy8lJWJvbGRHRkFGNS9GOFElYm9sZEYnLyUrZm9udHdlaWdodEdGYHFGTUZPRlFGU0ZVRldGWUZeb0Zlb0Ziby1GSjYuUSomdW1pbnVzMDtGJ0Y1RjdGTUZPRlFGU0ZVRldGWUZpb0ZbcEZccEZfcEZncEY1Rjc=Agora que identificamos os pontos de equil\303\255brio, precisamos entender quais deles s\303\243o est\303\241veis. Para isso, linearizaremos as equa\303\247\303\265es governantes em torno dos pontos de equil\303\255brio e observaremos suas pr\303\263prias frequ\303\252ncias.
A equa\303\247\303\243o governante pode ser escrita 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 where 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:params:=[M = 1, R = 0.25, Omega = 10, g = 9.81]:f:=g/R*sin(theta) - Omega^2/2*sin(2*theta):df:=diff(f,theta):Se linearizarmos a equa\303\247\303\243o governante em torno do ponto de equil\303\255brio em 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_f:=eval(f,theta = 0) + eval(df,theta = 0)*(theta - 0):NewEq:=diff(theta(t),t,t) + subs(theta = theta(t), Linear_f) = 0Para os valores atuais dos par\303\242metros, esta equa\303\247\303\243o se reduz a
subs(params, NewEq)Como podemos ver, a solu\303\247\303\243o desta equa\303\247\303\243o n\303\243o \303\251 oscilat\303\263ria, pois uma das ra\303\255zes da equa\303\247\303\243o das caracter\303\255sticas \303\251 positiva. Conclus\303\243o: Para os valores presentes, a oscila\303\247\303\243o em torno 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N\303\203O \303\251 est\303\241vel.Agora linearizando sobre o outro ponto de equil\303\255brio.
Linear_f:=subs(params, eval(f, theta = arccos(0.3924)) + eval(df, theta = arccos(0.3924)) * (theta - arccos(0.3924))):NewEq:=diff(theta(t),t,t) + subs(theta = theta(t), Linear_f) = 0Agora temos uma equa\303\247\303\243o diferencial n\303\243o homog\303\252nea, que precisamos verificar para estabilidade. A forma homog\303\252nea correspondente \303\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Esta solu\303\247\303\243o desta equa\303\247\303\243o \303\251 puramente oscilat\303\263ria, pois ambas as solu\303\247\303\265es s\303\243o n\303\272meros complexos. A frequ\303\252ncia natural pode ser obtida diretamente a partir desta equa\303\247\303\243o comoomega__n:=sqrt(84.60)*Unit(rad/s)