Mecânica Geral II - Dinâmica e Cinemática
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Mecânica Geral II - Dinâmica e Cinemática


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z
x
y
Figura 3.5: Modelo de um oscilador harmo\u2c6nico tridimensional.
O oscilador bi-dimensional
Para o caso de movimento num plano, a equac¸a\u2dco diferencial e´ equivalente a`s duas equac¸o\u2dces
componentes
mx¨ = \u2212kx
my¨ = \u2212ky
Elas esta\u2dco separadas, e podemos, imediatamente, escrever as soluc¸o\u2dces na forma
x = A cos(\u3c9t+ \u3b1) y = B cos(\u3c9t+ \u3b2) (3.23)
onde
\u3c9 =
\u221a
k
m
As constantes A, B, \u3b1 e \u3b2 ficam determinadas a partir das condic¸o\u2dces iniciais, em qualquer
caso.
Encontramos a equac¸a\u2dco da trajeto´ria eliminando o tempo t nestas duas equac¸o\u2dces. Para
conseguirmos isto, escreveremos a segunda equac¸a\u2dco na forma
y = B cos(\u3c9t+ \u3b1 + \u2206)
88 CAPI´TULO 3. MOVIMENTO GERAL DE UMA PARTI´CULA
onde \u2206 = \u3b2 \u2212 \u3b1. Desenvolvendo, temos
y = B[cos(\u3c9t+ \u3b1) cos \u2206\u2212 sen(\u3c9t+ \u3b1) sen \u2206]
Usando a primeira das Equac¸o\u2dces (3.23), obtemos
y
B
=
x
A
cos \u2206\u2212
\u221a
1\u2212 x
2
A2
sen \u2206 (3.24)
que e´ uma equac¸a\u2dco quadra´tica em x e y. A equac¸a\u2dco quadra´tica geral
ax2 + bxy + cy2 + dx+ ey = f
pode representar uma el´\u131pse, uma para´bola, ou uma hipe´rbole, dependendo do discriminante
b2 \u2212 4ac
ser negativo, zero, ou positivo, respectivamente. No presente caso o discriminante e´\u2212(2 sen \u2206/AB)2.
A Figura 3.6 mostra que a trajeto´ria e´ el´\u131ptica pois o discriminante da equac¸a\u2dco e´ negativo.
No caso particular em que a diferenc¸a de fase \u2206 seja igual a pi/2 a equac¸a\u2dco da trajeto´ria
se reduzira´ a
x2
A2
+
y2
B2
= 1
que e´ a equac¸a\u2dco de uma el´\u131pse cujos eixos coincidem com os eixos coordenados. Por outro
lado, se a diferenc¸a de fase for 0 ou pi, enta\u2dco a equac¸a\u2dco da trajeto´ria se reduzira´ a uma linha
reta dada por
y = ±B
A
x
O sinal positivo e´ va´lido quando \u2206 = 0, e o negativo para \u2206 = pi. E´ poss´\u131vel mostrar que
no caso geral o eixo da trajeto´ria el´\u131ptica formara´ um a\u2c6ngulo \u3c6 com o eixo x definido por
tg 2\u3c6 =
2AB cos \u2206
A2 \u2212B2 (3.25)
Como exerc´\u131cio, deixamos para o leitor a deduc¸a\u2dco da expressa\u2dco acima.
O Oscilador Harmo\u2c6nico Tri-dimensional
Para o caso de movimento tridimensional a equac¸a\u2dco do movimento e´ equivalente a`s tre\u2c6s
equac¸o\u2dces
mx¨ = \u2212kx my¨ = \u2212ky mz¨ = \u2212kz
que esta\u2dco separadas. As soluc¸o\u2dces sa\u2dco da forma (3.23), isto e´
x = A1 sen\u3c9t+B1 cos\u3c9t
y = A2 sen\u3c9t+B2 cos\u3c9t (3.26)
z = A3 sen\u3c9t+B3 cos\u3c9t
3.10. O OSCILADOR HARMO\u2c6NICO EM DUAS E TRE\u2c6S DIMENSO\u2dcES 89
y
x
A
O
-B
B
-A \u3d5
Figura 3.6: Trajeto´ria el´\u131ptica do movimento de um oscilador harmo\u2c6nico bi-dimensional.
Determinam-se as seis constantes de integrac¸a\u2dco a partir da posic¸a\u2dco e velocidade iniciais da
part´\u131cula. As Equac¸o\u2dces (3.26) escritas na forma vetorial ficam
~r = ~A sen\u3c9t+ ~B cos\u3c9t
onde as componentes de ~A sa\u2dco A1, A2 e A3 e da mesma forma para ~B. Fica claro que o
movimento se faz inteiramente em um u´nico plano que e´ o plano comum aos dois vetores
constantes ~A e ~B, e que a trajeto´ria da part´\u131cula nesse plano e´ uma el´\u131pse, como no caso
bidimensional. A ana´lise relativa a` forma da trajeto´ria el´\u131ptica feita no caso bidimensional
tambe´m se aplica no caso tridimensional.
Oscilador na\u2dco Isotro´pico
Na discussa\u2dco acima consideramos o oscilador tridimensional isotro´pico, onde a forc¸a res-
tauradora era independente da direc¸a\u2dco de deslocamento. Se a forc¸a restauradora depender
da direc¸a\u2dco de deslocamento, temos o caso do oscilador na\u2dco-isotro´pico. Escolhendo adequa-
damente os eixos do sistema de coordenadas, as equac¸o\u2dces diferenciais do movimento para o
oscilador na\u2dco isotro´pico sa\u2dco
mx¨ = \u2212k1x
my¨ = \u2212k2y (3.27)
mz¨ = \u2212k3z
Aqui temos um caso de tre\u2c6s freque\u2c6ncias de oscilac¸a\u2dco diferentes:
\u3c91 =
\u221a
k1
m
\u3c92 =
\u221a
k2
m
\u3c93 =
\u221a
k3
m
e as soluc¸o\u2dces sa\u2dco
x = A cos(\u3c91t+ \u3b1)
90 CAPI´TULO 3. MOVIMENTO GERAL DE UMA PARTI´CULA
y = B cos(\u3c92t+ \u3b2) (3.28)
z = C cos(\u3c93t+ \u3b3)
Tambe´m nesse caso, as seis constantes de integrac¸a\u2dco das equac¸o\u2dces acima ficara\u2dco determi-
nadas a partir das condic¸o\u2dces iniciais. A oscilac¸a\u2dco resultante da part´\u131cula fica inteiramente
contida dentro de uma caixa retangular (cujos lados sa\u2dco 2A, 2B e 2C) centrada na origem.
Se as freque\u2c6ncias \u3c91, \u3c92 e \u3c93 forem redut´\u131veis a uma medida comum, isto e´, se
\u3c91
n1
=
\u3c92
n2
=
\u3c93
n3
(3.29)
onde n1, n2 e n3 sa\u2dco inteiros, a trajeto´ria, chamada de Figura de Lissajous, se fechara´, porque
depois de um tempo 2pin1/\u3c91 = 2pin2/\u3c92 = 2pin3/\u3c93 a part´\u131cula retornara´ a` sua posic¸a\u2dco
inicial e o movimento se repetira´. (Na Equac¸a\u2dco (3.29) cancelamos os fatores inteiros comuns).
Por outro lado, se os \u3c9\u2019s na\u2dco forem redut´\u131veis a um fator comum, a trajeto´ria na\u2dco sera´
fechada. Neste caso dizemos que a trajeto´ria pode encher completamente a caixa retangular
mencionada acima, pelo menos no sentido de que se esperarmos um tempo suficientemente
longo, a part´\u131cula chegara´ arbitrariamente perto de qualquer ponto dado.
Em muitos casos a forc¸a restauradora l´\u131quida exercida em um dado a´tomo numa substa\u2c6ncia
cristalina so´lida e´ aproximadamente linear nos deslocamentos. As freque\u2c6ncias de oscilac¸a\u2dco
resultantes normalmente ficam na regia\u2dco infravermelho do espectro: 1012 a 1014 vibrac¸o\u2dces
por segundo.
3.11 Movimento de Part´\u131culas Carregadas em Campos
Ele´tricos e Magne´ticos
Quando uma part´\u131cula eletricamente carregada estiver
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