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\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd\ufffd 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 restauradora era independente da direc¸a\u2dco de deslocamento. Se a forc¸a restauradora de- pender da direc¸a\u2dco de deslocamento, temos o caso do oscilador na\u2dco-isotro´pico. Escolhendo adequadamente 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 na vizinhanc¸a de outras part´\u131culas carregadas, ela sofrera´ a ac¸a\u2dco de uma forc¸a. Dizemos