gab t2v2.2011.1

Esta é uma pré-visualização de arquivo. Entre para ver o arquivo original

with(DEtools):
 
 
 edo:=diff(x(t),t,t)+gamma*diff(x(t),t)+x(t);
 
 LCgtSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUYkNiQtSSJ4RzYiNiNJInRHRitGLUYtIiIiRilGLiomSSZnYW1tYUdGJUYuRidGLkYu
 
 
 dsolve({edo,x(0)=1,D(x)(0)=0});
 
 Ly1JInhHNiI2I0kidEdGJSwmKipJJmdhbW1hRyUqcHJvdGVjdGVkRyIiIiksJiokKUYqIiIjRiwhIiIiIiVGLCNGLEYxRjItSSRleHBHNiRGK0koX3N5c2xpYkdGJTYjLCQqKEY0RixGJ0YsRipGLEYyRiwtSSRzaW5HRjc2IywkKihGNEYsRi1GLEYnRixGLEYsRiwqJkY1RiwtSSRjb3NHRjdGPkYsRiw=
 
 
 x:=rhs(%);
 
 LCYqKkkmZ2FtbWFHJSpwcm90ZWN0ZWRHIiIiKSwmKiQpRiQiIiNGJiEiIiIiJUYmI0YmRitGLC1JJGV4cEc2JEYlSShfc3lzbGliRzYiNiMsJCooRi5GJkkidEdGM0YmRiRGJkYsRiYtSSRzaW5HRjE2IywkKihGLkYmRidGJkY3RiZGJkYmRiYqJkYvRiYtSSRjb3NHRjFGOkYmRiY=
 
 
 Apenas para ilustra\303\247\303\243o, eis o gr\303\241fico de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== para uma escolha particular de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, por exemplo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRJDAuM0YnRjJGMg==:
 
 
 plot(subs(gamma=.3,x),t=0..20);
 
 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
 
 
 Do gr\303\241fico acima podemos ver que LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdGPS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUY7NiRRJDAuM0YnRj5GPkY+Rj4= vale aproximadamente LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRkMtSSNtbkdGJDYkUSQwLjZGJ0YvLUYsNi1RIi5GJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRJjAuMGVtRicvRkVGTkYv
 
 
 Para encontrar um valor de m\303\255nimo da fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRInRGJ0YvRjIvRjNRJ25vcm1hbEYnRj1GPQ== , temos que deriv\303\241-la e igualar a zero.
 
 
 derivada:=diff(x,t);
LCYqLCMiIiIiIiNGJSlJJmdhbW1hRyUqcHJvdGVjdGVkR0YmRiUpLCYqJEYnRiUhIiIiIiVGJUYkRi0tSSRleHBHNiRGKUkoX3N5c2xpYkc2IjYjLCQqKEYkRiVJInRHRjNGJUYoRiVGLUYlLUkkc2luR0YxNiMsJCooRiRGJUYqRiVGN0YlRiVGJUYtKipGJEYlRi9GJUY4RiVGKkYlRi0=
 
 
 solve(derivada=0,t);
 
 IiIh
 
 
 Opa, o Maple deu uma solu\303\247\303\243o que n\303\243o interessa; teremos que nos virar de outro modo.
 
 
 Note que essa fun\303\247\303\243o LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGL0YyRjlGOS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLRk9GTUY5 \303\251 igual a 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 uma constante (que depende de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NDc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy).
 Assim, o primeiro valor positivo de LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= para o qual LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGL0YyRjlGOUY5 se anula \303\251 quando LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjFGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRi82JFEiMkYnRjIvJS5saW5ldGhpY2tuZXNzR0YxLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRjwvJSliZXZlbGxlZEdRJmZhbHNlRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGMi8lJmZlbmNlR0ZBLyUqc2VwYXJhdG9yR0ZBLyUpc3RyZXRjaHlHRkEvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlYtRiM2Ji1JJm1zcXJ0R0YkNiMtRiM2Jy1GQzYtUSomdW1pbnVzMDtGJ0YyRkZGSEZKRkxGTkZQRlIvRlVRLDAuMjIyMjIyMmVtRicvRlhGXm8tRiM2JC1JJW1zdXBHRiQ2JS1JI21pR0YkNiVRKCZnYW1tYTtGJy8lJ2l0YWxpY0dGQUYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjItRkM2LVEiK0YnRjJGRkZIRkpGTEZORlBGUkZdb0Zfby1GLzYkUSI0RidGMkYyRkItRmZvNiVRInRGJy9Gam9RJXRydWVGJy9GM1EnaXRhbGljRidGMi1GZm82I1EhRidGMg== vale LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjA7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy , isto \303\251, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= vale:
 
 
 t0:=2*Pi/sqrt(-gamma^2+4);
 
 LCQqKCIiIyIiIkkjUGlHJSpwcm90ZWN0ZWRHRiUpLCYqJClJJmdhbW1hR0YnRiRGJSEiIiIiJUYlI0YlRiRGLUYl
 
 
 subs(t=t0,x);
 
 LCYqKkkmZ2FtbWFHJSpwcm90ZWN0ZWRHIiIiKSwmKiQpRiQiIiNGJiEiIiIiJUYmI0YmRitGLC1JJGV4cEc2JEYlSShfc3lzbGliRzYiNiMsJCooSSNQaUdGJUYmRidGLEYkRiZGLEYmLUkkc2luR0YxNiNGN0YmRiYqJkYvRiYtSSRjb3NHRjFGOkYmRiY=
 
 
 x0:=simplify(%);
 
 LCQtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKihJI1BpR0YmIiIiKSwmKiQpSSZnYW1tYUdGJiIiI0YtISIiIiIlRi0jRi1GM0Y0RjJGLUY0RjQ=
 
 
 A f\303\263rmula acima \303\251 a resposta do item (a).
 
 
 Resposta do item (b):
 
 
 plot(x0,gamma=0..2);
 
 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
Notamos que a medida que o atrito se aproxima do valor cr\303\255tico LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnJiM5NDc7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIjJGJ0YyRjI= , a dist\303\242ncia mais \303\240 esquerda atingida pela massa se aproxima de zero.
Por exemplo:
 
 
 plot(subs(gamma=1.6,x),t=0..10);
 
 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