{"id":2386,"date":"2016-08-02T00:37:34","date_gmt":"2016-08-02T00:37:34","guid":{"rendered":"http:\/\/fisicaevestibular.com.br\/novo\/?page_id=2386"},"modified":"2024-08-23T14:40:08","modified_gmt":"2024-08-23T14:40:08","slug":"cordas-vibrantes","status":"publish","type":"page","link":"https:\/\/fisicaevestibular.com.br\/novo\/ondulatoria\/acustica\/cordas-vibrantes\/","title":{"rendered":"Cordas vibrantes"},"content":{"rendered":"

Cordas vibrantes – Harm\u00f4nicos<\/b><\/span><\/span><\/span><\/p>\n

No estudo da <\/b><\/span><\/span><\/span>ac\u00fastica,<\/b><\/span><\/span><\/span> <\/b>uma<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>harm\u00f4nica<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>de uma<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>onda<\/b><\/span><\/span><\/span> sonora\u00a0<\/b><\/span><\/span><\/span>corresponde \u00e0 uma <\/b><\/span><\/span><\/span>frequ\u00eancia <\/b><\/span><\/span><\/span>espec\u00edfica de vibra\u00e7\u00e3o <\/b><\/span><\/span><\/span>que tem a <\/b><\/span><\/span><\/span>propriedade de causar o fen\u00f4meno de<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>resson\u00e2ncia. <\/b><\/span><\/span><\/span><\/p>\n

Essas <\/b><\/span><\/span><\/span>frequ\u00eancias<\/b><\/span><\/span><\/span> s\u00e3o denominadas<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>frequ\u00eancias de resson\u00e2ncia<\/b><\/i><\/span><\/span><\/span>.<\/b><\/span><\/span><\/span>\"\"<\/p>\n

Assim, o <\/b><\/span><\/span><\/span>conjunto de todos os modos de oscila\u00e7\u00e3o poss\u00edveis <\/b><\/span><\/span><\/span>\u00e9 chamado de <\/b><\/span><\/span><\/span>s\u00e9rie<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>harm\u00f4nica<\/b><\/span><\/span><\/span>.<\/b><\/span><\/span><\/span><\/p>\n

Para que voc\u00ea <\/b><\/span><\/span><\/span>estude detalhadamente os <\/b><\/span><\/span><\/span>harm\u00f4nicos<\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>deve dividi-los em duas partes,<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>cordas vibrantes<\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>tubos sonoros<\/span><\/span><\/span>.<\/b><\/span><\/span><\/span> <\/b><\/p>\n

Cordas vibrantes<\/b><\/span><\/span><\/span><\/p>\n

As <\/b><\/span><\/span><\/span>cordas vibrantes<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>correspondem \u00e0 <\/b><\/span><\/span><\/span>fios flex\u00edveis e tracionados <\/b><\/span><\/span><\/span>(tensionados) em <\/b><\/span><\/span><\/span>seus extremos,<\/b><\/span><\/span><\/span><\/p>\n

utilizados<\/b><\/span><\/span><\/span> em <\/b><\/span><\/span><\/span>instrumentos musicais <\/b><\/span><\/span><\/span>como, <\/b><\/span><\/span><\/span>viol\u00e3o, guitarra, violino, cavaquinho, banjo, etc.<\/b><\/span><\/span><\/span>\"\"<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/span><\/p>\n

Harm\u00f4nicos de uma corda vibrante<\/b><\/span><\/span><\/span><\/p>\n

Os <\/b><\/span><\/span><\/span>harm\u00f4nicos<\/b><\/span><\/span><\/span> de uma <\/b><\/span><\/span><\/span>corda vibrante<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>s\u00e3o as <\/b><\/span><\/span><\/span>v\u00e1rias poss\u00edveis freq\u00fc\u00eancias naturais das ondas estacion\u00e1rias<\/b><\/span><\/span><\/span> que <\/b><\/span><\/span><\/span>surgem em cordas tensas <\/b><\/span><\/span><\/span>(sob a\u00e7\u00e3o de for\u00e7as tensoras de intensidade T), com <\/b><\/span><\/span><\/span>massa m e comprimento L e densidade linear de massa \u00b5.<\/b><\/span><\/span><\/span><\/p>\n

\"\" \"\"<\/span><\/p>\n

\"\"<\/p>\n

Velocidade V de propaga\u00e7\u00e3o da onda na corda<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

As <\/b><\/span><\/span><\/span>ondas peri\u00f3dicas <\/b><\/span><\/span><\/span>s\u00e3o originadas por <\/b><\/span><\/span><\/span>fontes que executam oscila\u00e7\u00f5es peri\u00f3dicas <\/b><\/span><\/span><\/span>(s\u00e3o repetidas em intervalos de tempos iguais). <\/b><\/span><\/span><\/span><\/p>\n

Sendo assim, podemos dizer que a <\/b><\/span><\/span><\/span>propaga\u00e7\u00e3o de uma onda peri\u00f3dica em um meio homog\u00eaneo e is\u00f3tropo \u00e9 um movimento uniforme,<\/b><\/span><\/span><\/span> com a <\/b><\/span><\/span><\/span>onda<\/b><\/span><\/span><\/span> se propagando com <\/b><\/span><\/span><\/span>velocidade constante V, tal que V = \u0394S\/\u0394t.<\/b><\/span><\/span><\/span><\/p>\n

Equa\u00e7\u00e3o fundamental da ondulat\u00f3ria<\/b><\/span><\/span><\/span><\/p>\n

Observe na figura<\/b><\/span><\/span><\/span> as <\/b><\/span><\/span><\/span>caracter\u00edsticas<\/b><\/span><\/span><\/span> de uma <\/b><\/span><\/span><\/span>onda transversal <\/b><\/span><\/span><\/span>se propagando numa <\/b><\/span><\/span><\/span>corda<\/b><\/span><\/span><\/span>, com com<\/b><\/span><\/span><\/span> velocidade V:<\/b><\/span><\/span><\/span>\"\"<\/p>\n

\"\" Comprimento de onda (\u03bb) \"\" dist\u00e2ncia <\/b><\/span><\/span><\/span>entre duas <\/b><\/span><\/span><\/span>cristas <\/b><\/span><\/span><\/span>ou dois <\/b><\/span><\/span><\/span>vales consecutivos.<\/b><\/span><\/span><\/span><\/p>\n

\"\" \u00a0Per\u00edodo (T)<\/b><\/span><\/span><\/span><\/em> \"\"<\/b><\/span><\/span><\/span><\/em>\u00a0<\/b><\/span><\/span><\/span>que \u00e9 o <\/b><\/span><\/span><\/span>intervalo de tempo <\/b><\/span><\/span><\/span>para que <\/b><\/span><\/span><\/span>cada ponto homog\u00eaneo da onda <\/b><\/span><\/span><\/span>execute uma oscila\u00e7\u00e3o completa, ou <\/b><\/span><\/span><\/span>percorra um comprimento de onda \u03bb.<\/b><\/span><\/span><\/span><\/p>\n

\"\" Frequ\u00eancia (f) \"\" <\/b><\/span><\/span><\/span>n\u00famero de <\/b><\/span><\/span><\/span>oscila\u00e7\u00f5es completas que cada ponto homog\u00eaneo da onda executa na unidade de tempo.<\/b><\/span><\/span><\/span><\/p>\n

Observe nas <\/b><\/span><\/span><\/span>informa\u00e7\u00f5es acima <\/b><\/span><\/span><\/span>que, num <\/b><\/span><\/span><\/span>intervalo de tempo \u0394t que corresponde a um per\u00edodo T <\/b><\/span><\/span><\/span>(\u0394t = T), a <\/b><\/span><\/span><\/span>onda percorre uma dist\u00e2ncia \u0394S que corresponde a um comprimento de onda \u03bb <\/b><\/span><\/span><\/span>(\u0394S = \u03bb).<\/b><\/span><\/span><\/span><\/p>\n

V = \u0394S\/\u0394t \"\"<\/b><\/span><\/span><\/span> V = \u03bb\/T \"\" T = 1\/f \"\" V = \u03bb\/(1\/f) \"\" <\/b><\/span><\/span><\/span>V = \u03bb.f (equa\u00e7\u00e3o fundamental da ondulat\u00f3ria)<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

F\u00f3rmula de Taylor para cordas vibrantes<\/span><\/span><\/span><\/p>\n

Pode-se <\/b><\/span><\/span><\/span>provar que<\/b><\/span><\/span><\/span>, para a <\/b><\/span><\/span><\/span>propaga\u00e7\u00e3o de um pulso transversal ou de uma onda peri\u00f3dica transversal numa corda, <\/b><\/span><\/span><\/span>a <\/b><\/span><\/span><\/span>velocidade (V) <\/b><\/span><\/span><\/span>com que uma <\/b><\/span><\/span><\/span>onda peri\u00f3dica se propaga<\/b><\/span><\/span><\/span> depende da <\/b><\/span><\/span><\/span>densidade linear de massa (<\/b><\/span><\/span><\/span>\u00b5<\/b><\/span><\/span><\/span><\/em>) da corda <\/b><\/span><\/span><\/span>e da <\/b><\/span><\/span><\/span>intensidade da for\u00e7a tensora (<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/span><\/em>) a que ela est\u00e1 sujeita.<\/b><\/span><\/span><\/span><\/p>\n

A <\/b><\/span><\/span><\/span>rela\u00e7\u00e3o entre essas grandezas <\/b><\/span><\/span><\/span>foi <\/b><\/span><\/span><\/span>provada matematicamente pelo brit\u00e2nico Brook Taylor <\/b><\/span><\/span><\/span>(1685-1731), onde voc\u00ea pode <\/b><\/span><\/span><\/span>determinar a velocidade de propaga\u00e7\u00e3o de uma onda numa corda utilizando a equa\u00e7\u00e3o conhecida como F\u00f3rmula de Taylor, expressa a seguir:<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\u00a0<\/span><\/p>\n

Modos de vibra\u00e7\u00e3o (harm\u00f4nicos) em cordas vibrantes<\/b><\/span><\/span><\/span><\/p>\n

Considere uma <\/b><\/span><\/span><\/span>corda de comprimento L fixa em seus extremos.<\/b><\/span><\/span><\/span><\/p>\n

Produzindo-se uma <\/b><\/span><\/span><\/span>perturba\u00e7\u00e3o em qualquer ponto entre os extremos fixos<\/b><\/span><\/span><\/span>, esta perturba\u00e7\u00e3o <\/b><\/span><\/span><\/span>propaga-se at\u00e9 cada uma das extremidades, refletem-se e retornam em sentido contr\u00e1rio,<\/b><\/span><\/span><\/span> formando<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

ondas estacion\u00e1rias com n\u00f3s <\/b><\/span><\/span><\/span>(pontos que n\u00e3o vibram) <\/b><\/span><\/span><\/span>e ventres <\/b><\/span><\/span><\/span>(dist\u00e2ncia entre dois n\u00f3s, que chamamos de <\/b><\/span><\/span><\/span>fuso, <\/b><\/span><\/span><\/span>onde todos os pontos est\u00e3o em movimento vibrat\u00f3rio).<\/b><\/span><\/span><\/span><\/p>\n

As <\/b><\/span><\/span><\/span>figuras abaixo <\/b><\/span><\/span><\/span>mostram os <\/b><\/span><\/span><\/span>diversos modos de vibra\u00e7\u00e3o numa mesma corda <\/b><\/span><\/span><\/span>(mesmo meio, mesma velocidade).<\/b><\/span><\/span><\/span><\/p>\n

A <\/b><\/span><\/span><\/span>onda estacion\u00e1ria de freq\u00fc\u00eancia mais baixa \u00e9 chamada freq\u00fc\u00eancia fundamental. <\/b><\/span><\/span><\/span>Ela corresponde a <\/b><\/span><\/span><\/span>uma onda estacionaria com um \u00fanico ventre, o harm\u00f4nico fundamental ou primeiro harm\u00f4nico. <\/b><\/span><\/span><\/span><\/p>\n

As <\/b><\/span><\/span><\/span>demais freq\u00fc\u00eancias naturais s\u00e3o chamadas sobretons ou harm\u00f4nicos superiores,<\/b><\/span><\/span><\/span> pois as <\/b><\/span><\/span><\/span>freq\u00fc\u00eancias subsequentes s\u00e3o m\u00faltiplos inteiros da freq\u00fc\u00eancia fundamental.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

En\u00e9simo harm\u00f4nico<\/b><\/span><\/span><\/span>\u00a0\"\" <\/b><\/span><\/span><\/span>(\u201cn + 1\u201d n\u00f3s e\u00a0<\/b><\/span><\/span><\/span>n fusos<\/b><\/span><\/span><\/span>)<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Lembrando que <\/b><\/span><\/span><\/span>f<\/b><\/span><\/span><\/span>1 <\/b><\/span><\/span><\/sub><\/span>= V\/2L\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0\u00a0<\/b><\/span><\/span><\/span>f<\/b><\/span><\/span><\/span>n<\/b><\/span><\/span><\/sub><\/span> = nf<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span><\/p>\n

Generalizando:<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Observe que existe uma <\/b><\/span><\/span><\/span>rela\u00e7\u00e3o simples entre o comprimento L da corda <\/b><\/span><\/span><\/span>e o <\/b><\/span><\/span><\/span>comprimento de onda \u03bb <\/b><\/span><\/span><\/span>da <\/b><\/span><\/span><\/span>onda estacion\u00e1ria de frequ\u00eancia f<\/b><\/span><\/span><\/span>n<\/b><\/span><\/span><\/sub><\/span> <\/b>que nela se estabelece. <\/b><\/span><\/span><\/span><\/p>\n

Generalizando, para o <\/b><\/span><\/span><\/span>en\u00e9simo harm\u00f4nico \"\"<\/b><\/span><\/span><\/span> <\/b>n = 1, 2, 3, 4, 5, … <\/b><\/span><\/span><\/span><\/p>\n

O <\/b><\/span><\/span><\/span>n\u00famero inteiro n corresponde ao n\u00famero do harm\u00f4nico<\/b><\/span><\/span><\/span>: <\/b><\/span><\/span><\/span>n = 1<\/b><\/span><\/span><\/span>, para o <\/b><\/span><\/span><\/span>harm\u00f4nico fundamental; n = 2,<\/b><\/span><\/span><\/span> para o <\/b><\/span><\/span><\/span>segundo harm\u00f4nico; n = 3, <\/b><\/span><\/span><\/span>para o <\/b><\/span><\/span><\/span>terceiro harm\u00f4nico; <\/b><\/span><\/span><\/span>e assim por diante.<\/b><\/span><\/span><\/span><\/p>\n

Da <\/b><\/span><\/span><\/span>equa\u00e7\u00e3o de Taylor, para o en\u00e9simo harm\u00f4nico,<\/b><\/span><\/span><\/span> teremos \"\" <\/b><\/span><\/span><\/span>V =\"\"<\/b><\/span><\/span><\/span>,<\/b><\/span><\/span><\/span> <\/b>que, <\/b><\/span><\/span><\/span>substitu\u00edda em f<\/b><\/span><\/span><\/span>n <\/b><\/span><\/span><\/sub><\/span>= n\"\"<\/b><\/span><\/span><\/span>,<\/b><\/span><\/span><\/span> nos <\/b><\/span><\/span><\/span>fornece<\/b><\/span><\/span><\/span>:\u00a0\u00a0\u00a0 <\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Observe na <\/b><\/span><\/span><\/span>express\u00e3o acima que temos tr\u00eas vari\u00e1veis<\/b><\/span><\/span><\/span>, <\/b><\/span><\/span><\/span>comprimento da corda\u00a0<\/b><\/span><\/span><\/span>L<\/b><\/span><\/span><\/span>,<\/b><\/span><\/span><\/span> <\/b>densidade linear de <\/b><\/span><\/span><\/span><\/p>\n

massa<\/b><\/span><\/span><\/span> (corda mais grossa ou mais fina)\u00a0<\/b><\/span><\/span><\/span>\u00b5<\/b><\/span><\/span><\/span>\u00a0e <\/b><\/span><\/span><\/span>for\u00e7a de tra\u00e7\u00e3o\u00a0<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/span>.<\/b><\/span><\/span><\/span><\/p>\n

As <\/b><\/span><\/span><\/span>cordas de um viol\u00e3o, <\/b><\/span><\/span><\/span>por exemplo, s\u00e3o <\/b><\/span><\/span><\/span>dedilhadas com o polegar, indicador, m\u00e9dio e anular da m\u00e3o direita <\/b><\/span><\/span><\/span>e, para <\/b><\/span><\/span><\/span>variar o comprimento da corda L, <\/b><\/span><\/span><\/span>o m\u00fasico coloca os <\/b><\/span><\/span><\/span>dedos da m\u00e3o esquerda fazendo press\u00e3o no espa\u00e7o entre os trastes, produzindo assim as diversas notas musicais.<\/b><\/span><\/span><\/span><\/p>\n

Para <\/b><\/span><\/span><\/span>variar a densidade linear\u00a0\u00b5, <\/b><\/span><\/span><\/span>o <\/b><\/span><\/span><\/span>m\u00fasico muda de uma corda para a outra <\/b><\/span><\/span><\/span>e, para <\/b><\/span><\/span><\/span>afinar o instrumento ele varia a for\u00e7a de tra\u00e7\u00e3o girando as cravelhas ou tarraxas ( roscas para essa finalidade).<\/b><\/span><\/span><\/span><\/p>\n

Variando<\/b><\/span><\/span><\/span> dessa maneira <\/b><\/span><\/span><\/span>essas tr\u00eas grandezas o m\u00fasico obt\u00e9m as <\/b><\/span><\/span><\/span>v\u00e1rias notas musicais (harm\u00f4nios, freq\u00fc\u00eancias).<\/b><\/span><\/span><\/span><\/p>\n

\u00a0<\/span><\/p>\n

O que voc\u00ea deve saber, informa\u00e7\u00f5es e dicas<\/b><\/span><\/span><\/span><\/p>\n

\"\" A<\/b><\/span><\/span><\/span> <\/b>frequ\u00eancia de uma onda \u00e9 sempre a mesma frequ\u00eancia da fonte produtora da onda. <\/b><\/span><\/span><\/span><\/p>\n

Dessa forma, <\/b><\/span><\/span><\/span>independentemente do meio em que a onda se propaga, <\/b><\/span><\/span><\/span>sua <\/b><\/span><\/span><\/span>frequ\u00eancia n\u00e3o vai ser modificada.<\/b><\/span><\/span><\/span> <\/b><\/p>\n

\"\" A <\/b><\/span><\/span><\/span>velocidade de propaga\u00e7\u00e3o de uma onda mec\u00e2nica em um determinado meio s\u00f3 depende das propriedades el\u00e1sticas do meio.<\/b><\/span><\/span><\/span><\/p>\n

\"\" Procure entender <\/b><\/span><\/span><\/span>e memorizar as <\/b><\/span><\/span><\/span>informa\u00e7\u00f5es e as express\u00f5es matem\u00e1ticas fornecidas acima.<\/b><\/span><\/span><\/span><\/p>\n

\u00a0<\/span><\/p>\n

\"\" Analise<\/b><\/span><\/span><\/span> atentamente <\/b><\/span><\/span><\/span>as resolu\u00e7\u00f5es desses exerc\u00edcios interessantes:<\/b><\/span><\/span><\/span><\/p>\n

\"\" 01- (UNICAMP-SP) <\/b><\/span><\/span><\/span>Para a <\/b><\/span><\/span><\/span>afina\u00e7\u00e3o de um piano <\/b><\/span><\/span><\/span>usa-se um <\/b><\/span><\/span><\/span>diapas\u00e3o com frequ\u00eancia fundamental igual a 440Hz, que \u00e9 a frequ\u00eancia\u00a0 da nota L\u00e1.<\/b><\/span><\/span><\/span><\/p>\n

a)<\/b><\/span><\/span><\/span> <\/b>A <\/b><\/span><\/span><\/span>nota L\u00e1 <\/b><\/span><\/span><\/span>de certo <\/b><\/span><\/span><\/span>piano est\u00e1 desafinada <\/b><\/span><\/span><\/span>e <\/b><\/span><\/span><\/span>o seu <\/b><\/span><\/span><\/span>harm\u00f4nico fundamental<\/b><\/span><\/span><\/span> est\u00e1 representado na curva tracejada do gr\u00e1fico.<\/b><\/span><\/span><\/span>\"\"<\/p>\n

a)<\/b><\/span><\/span><\/span> Obtenha<\/b><\/span><\/span><\/span> a <\/b><\/span><\/span><\/span>freq\u00fc\u00eancia da nota L\u00e1 desafinada.<\/b><\/span><\/span><\/span><\/p>\n

b)<\/b><\/span><\/span><\/span> <\/b>O <\/b><\/span><\/span><\/span>comprimento dessa corda do piano \u00e9 igual a 1m <\/b><\/span><\/span><\/span>e sua <\/b><\/span><\/span><\/span>densidade linear \u00e9 igual a 5,0.10<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span>g\/cm. Calcule<\/b><\/span><\/span><\/span> o\u00a0 <\/b><\/span><\/span><\/span>aumento de tens\u00e3o<\/b><\/span><\/span><\/span> na corda <\/b><\/span><\/span><\/span>necess\u00e1rio para que a nota L\u00e1 seja afinada.<\/b><\/span><\/span><\/span><\/p>\n

Resolu\u00e7\u00e3o:<\/b><\/span><\/span><\/span><\/p>\n

a)<\/b><\/span><\/span><\/span> Observe no <\/b><\/span><\/span><\/span>gr\u00e1fico <\/b><\/span><\/span><\/span>que representa o <\/b><\/span><\/span><\/span>harm\u00f4nico fundamental <\/b><\/span><\/span><\/span>da <\/b><\/span><\/span><\/span>nota L\u00e1 desafinda que o per\u00edodo <\/b><\/span><\/span><\/span>(tempo decorrido entre duas cristas ou vales consecutivos) <\/b><\/span><\/span><\/span>vale<\/b><\/span><\/span><\/span> <\/b>T = 2,5.10<\/b><\/span><\/span><\/span>-3<\/b><\/span><\/span><\/sup><\/span>s\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>f = 1\/T\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 f = 1\/2,5.10<\/b><\/span><\/span><\/span>-3<\/b><\/span><\/span><\/sup><\/span>\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>f = 400Hz (frequ\u00eancia do harm\u00f4nico funfamental da nota L\u00e1 desafinada).<\/b><\/span><\/span><\/span><\/p>\n

b)<\/b><\/span><\/span><\/span> <\/b>Tra\u00e7\u00e3o<\/b><\/span><\/span><\/span> (tens\u00e3o) <\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span> da nota L\u00e1 desafinada <\/b><\/span><\/span><\/span>de <\/b><\/span><\/span><\/span>frequ\u00eancia f=400Hz, densidade linear \u00b5 = 5,0.10<\/b><\/span><\/span><\/span>-3<\/b><\/span><\/span><\/sup><\/span>kg\/m.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Tra\u00e7\u00e3o<\/b><\/span><\/span><\/span> (tens\u00e3o) <\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span> da nota L\u00e1 afinada (frequ\u00eancia do diapas\u00e3o) f = 440H z, densidade linear \u00b5 = 5,0.<\/b><\/span><\/span><\/span><\/p>\n

10<\/b><\/span><\/span><\/span>-3<\/b><\/span><\/span><\/sup><\/span>kg\/m.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

O <\/b><\/span><\/span><\/span>aumento pedido de tens\u00e3o na corda <\/b><\/span><\/span><\/span>necess\u00e1rio para que a <\/b><\/span><\/span><\/span>nota l\u00e1 seja afinada <\/b><\/span><\/span><\/span>ser\u00e1\u00a0<\/b><\/span><\/span><\/span>\u0394T =T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0\u2013 T<\/b><\/span><\/span><\/span>1 <\/b><\/span><\/span><\/sub><\/span>= <\/b><\/span><\/span><\/span>3.872 \u2013 3.200 <\/b><\/span><\/span><\/span>= <\/b><\/span><\/span><\/span>672N.<\/b><\/span><\/span><\/span><\/p>\n

\"\" 02- (UFMG-MG)<\/b><\/span><\/span><\/span> Bruna afina a corda mi de seu violino,<\/b><\/span><\/span><\/span> para que <\/b><\/span><\/span><\/span>ela vibre com uma freq\u00fc\u00eancia m\u00ednima de 680 Hz.<\/b><\/span><\/span><\/span><\/p>\n

A <\/b><\/span><\/span><\/span>parte vibrante das cordas do violino de Bruna mede 35 cm de comprimento,<\/b><\/span><\/span><\/span> como mostrado nesta figura:<\/b><\/span><\/span><\/span>\"\"<\/p>\n

Considerando <\/b><\/span><\/span><\/span>essas informa\u00e7\u00f5es,<\/b><\/span><\/span><\/span><\/p>\n

a) Calcule<\/b><\/span><\/span><\/span> a <\/b><\/span><\/span><\/span>velocidade de propaga\u00e7\u00e3o de uma onda na corda mi desse violino.<\/b><\/span><\/span><\/span><\/p>\n

b)<\/b><\/span><\/span><\/span> <\/b>Considere que a <\/b><\/span><\/span><\/span>corda mi esteja vibrando com uma freq\u00fc\u00eancia de 680 Hz. <\/b><\/span><\/span><\/span><\/p>\n

Determine o <\/b><\/span><\/span><\/span>comprimento de onda, no ar<\/b><\/span><\/span><\/span>, da <\/b><\/span><\/span><\/span>onda sonora produzida por essa corda.<\/b><\/span><\/span><\/span><\/p>\n

Velocidade do som no ar = 340 m\/s<\/b><\/span><\/span><\/span><\/p>\n

Resolu\u00e7\u00e3o:<\/b><\/span><\/span><\/span><\/p>\n

a) <\/b><\/span><\/span><\/span>Pelo enunciado a <\/b><\/span><\/span><\/span>frequ\u00eancia m\u00ednima de vibra\u00e7\u00e3o <\/b><\/span><\/span><\/span>que ele fornece \u00e9 <\/b><\/span><\/span><\/span>a frequ\u00eancia fundamental f<\/b><\/span><\/span><\/span>1 <\/b><\/span><\/span><\/sub><\/span>= 680 H<\/b><\/span><\/span><\/span>z<\/b><\/span><\/span><\/sub><\/span>\u00a0\"\" comprimento da corda L = 35cm<\/b><\/span><\/span><\/span> \"\" <\/b><\/span><\/span><\/span>comprimento de onda <\/b><\/span><\/span><\/span>do <\/b><\/span><\/span><\/span>primeiro harm\u00f4nico \u03bb<\/b><\/span><\/span><\/span>n<\/b><\/span><\/span><\/sub><\/span> = 2L\/n <\/b><\/span><\/span><\/span> \"\" <\/b><\/span><\/span><\/span>n= <\/b><\/span><\/span><\/span>1 \"\"\u00a0\u03bb<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span> = 2.35\/1 \"\" \u00a0 <\/b><\/span><\/span><\/span>\u03bb<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span> = 70 cm<\/b><\/span><\/span><\/span><\/p>\n

V =\u03bb.f \"\"<\/b><\/span><\/span><\/span> V = 0,7.680 \"\" <\/b><\/span><\/span><\/span>V = 476 m\/s (velocidade da onda na corda mi, constante).<\/b><\/span><\/span><\/span><\/p>\n

b)<\/b><\/span><\/span><\/span> Dados: <\/b><\/span><\/span><\/span>frequ\u00eancia de vibra\u00e7\u00e3o da corda mi f = 680 Hz que <\/b><\/span><\/span><\/span>\u00e9 a mesma que no ar<\/b><\/span><\/span><\/span>, <\/b><\/span><\/span><\/span>pois a<\/b><\/span><\/span><\/span> frequ\u00eancia independe do meio de propaga\u00e7\u00e3o<\/b><\/span><\/span><\/span> e <\/b><\/span><\/span><\/span>velocidade <\/b><\/span><\/span><\/span>de <\/b><\/span><\/span><\/span>propaga\u00e7\u00e3o do som no ar V = 340 m\/s. <\/b><\/span><\/span><\/span><\/p>\n

Equa\u00e7\u00e3o fundamental da ondulat\u00f3ria \"\" v = \u03bb.f<\/b><\/span><\/span><\/span> \"\" 340 =\u00a0\u03bb.680 \"\" <\/b><\/span><\/span><\/span>\u03bb\u00a0= <\/b><\/span><\/span><\/span>340\/680 <\/b><\/span><\/span><\/span>= 0,5 m =\u00a0<\/b><\/span><\/span><\/span>50 cm<\/b><\/span><\/span><\/span><\/p>\n

\"\" 03-(UFG)<\/b><\/span><\/span><\/span> <\/b>Na <\/b><\/span><\/span><\/span>experi\u00eancia de resson\u00e2ncia em cord<\/b><\/span><\/span><\/span>as representada na figura, <\/b><\/span><\/span><\/span>dois fios <\/b><\/span><\/span><\/span>de <\/b><\/span><\/span><\/span>densidades diferentes est\u00e3o tensionados,<\/b><\/span><\/span><\/span> atrav\u00e9s de <\/b><\/span><\/span><\/span>roldanas ideais, <\/b><\/span><\/span><\/span>por <\/b><\/span><\/span><\/span>um bloco que pende deles dois<\/b><\/span><\/span><\/span>. <\/b><\/span><\/span><\/span><\/p>\n

As <\/b><\/span><\/span><\/span>extremidades esquerdas de ambos est\u00e3o ligadas a uma fonte <\/b><\/span><\/span><\/span>que produz <\/b><\/span><\/span><\/span>pequenas vibra\u00e7\u00f5es <\/b><\/span><\/span><\/span>com <\/b><\/span><\/span><\/span>freq\u00fc\u00eancia conhecida. <\/b><\/span><\/span><\/span>\"\"<\/p>\n

A <\/b><\/span><\/span><\/span>dist\u00e2ncia entre a fonte e as roldanas \u00e9 L. Verifica-se que<\/b><\/span><\/span><\/span>, quando a <\/b><\/span><\/span><\/span>freq\u00fc\u00eancia da fonte atinge o<\/b><\/span><\/span><\/span> <\/b>valor f, ambos os fios entram em resson\u00e2ncia, <\/b><\/span><\/span><\/span>o <\/b><\/span><\/span><\/span>mais denso no terceiro harm\u00f4nico<\/b><\/span><\/span><\/span> e o outro, <\/b><\/span><\/span><\/span>na freq\u00fc\u00eancia fundamental.<\/b><\/span><\/span><\/span> <\/b>Dados:<\/b><\/span><\/span><\/span> <\/b>v = \"\"<\/b><\/span><\/span><\/span> (velocidade da onda na corda).<\/b><\/span><\/span><\/span><\/p>\n

Conhecendo <\/b><\/span><\/span><\/span>a <\/b><\/span><\/span><\/span>densidade linear de massa\u00a0\u00b5\u00a0do fio mais denso, determine<\/b><\/span><\/span><\/span> a <\/b><\/span><\/span><\/span>densidade linear de massa do outro fio;<\/b><\/span><\/span><\/span><\/p>\n

Resolu\u00e7\u00e3o:<\/b><\/span><\/span><\/span><\/p>\n

Cada fio <\/b><\/span><\/span><\/span>possui possui a <\/b><\/span><\/span><\/span>mesma frequ\u00eancia f de oscila\u00e7\u00e3o <\/b><\/span><\/span><\/span>que \u00e9 a da<\/b><\/span><\/span><\/span> fonte,<\/b><\/span><\/span><\/span> f<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span> = f<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span> = f \"\"<\/b><\/span><\/span><\/span> a for\u00e7a de tra\u00e7\u00e3o tamb\u00e9m \u00e9 a mesma para cada fio, pois(suportam o mesmo peso\u00a0(P\/2), T<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span> = T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span> = T.<\/b><\/span><\/span><\/span><\/p>\n

\u00a0O <\/b><\/span><\/span><\/span>comprimento L <\/b><\/span><\/span><\/span> de cada fio <\/b><\/span><\/span><\/span>\u00e9 o mesmo <\/b><\/span><\/span><\/span>(<\/b><\/span><\/span><\/span>dist\u00e2ncia<\/b><\/span><\/span><\/span> entre <\/b><\/span><\/span><\/span>fonte e roldanas iguais<\/b><\/span><\/span><\/span>).<\/b><\/span><\/span><\/span><\/p>\n

O <\/b><\/span><\/span><\/span>fio com tr\u00eas fusos \u00e9 a mais denso (\u00b5<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>), <\/b><\/span><\/span><\/span>pois como <\/b><\/span><\/span><\/span>T \u00e9 a mesma (<\/b><\/span><\/span><\/span>v = \"\"<\/b><\/span><\/span><\/span>)<\/b><\/span><\/span><\/span> <\/b>V e\u00a0\u00b5\u00a0s\u00e3o inversamente proporcionais<\/b><\/span><\/span><\/span> e <\/b><\/span><\/span><\/span>V \u00e9 menor na mais densa. <\/b><\/span><\/span><\/span><\/p>\n

Da <\/b><\/span><\/span><\/span>equa\u00e7\u00e3o V = \u03bbf, <\/b><\/span><\/span><\/span>como <\/b><\/span><\/span><\/span>f \u00e9 a mesma, menor V implica em menor\u00a0\u03bb, <\/b><\/span><\/span><\/span>que \u00e9 o da <\/b><\/span><\/span><\/span>corda com tr\u00eas fusos (\u00b5<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>).<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\u00a0\"\"<\/span><\/p>\n

 <\/p>\n

Confira os exerc\u00edcios<\/span><\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"

Cordas vibrantes – Harm\u00f4nicos No estudo da ac\u00fastica, uma\u00a0harm\u00f4nica\u00a0de uma\u00a0onda sonora\u00a0corresponde \u00e0 uma frequ\u00eancia espec\u00edfica de vibra\u00e7\u00e3o que tem a propriedade de causar o fen\u00f4meno de\u00a0resson\u00e2ncia. Essas frequ\u00eancias s\u00e3o denominadas\u00a0frequ\u00eancias de resson\u00e2ncia. Assim, o conjunto de todos os modos de oscila\u00e7\u00e3o poss\u00edveis \u00e9 chamado de s\u00e9rie\u00a0harm\u00f4nica. Para que voc\u00ea estude detalhadamente os harm\u00f4nicos\u00a0deve dividi-los em duas partes,\u00a0cordas vibrantes\u00a0e\u00a0tubos sonoros. Cordas<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":2325,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-2386","page","type-page","status-publish","hentry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2386","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/comments?post=2386"}],"version-history":[{"count":2,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2386\/revisions"}],"predecessor-version":[{"id":2395,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2386\/revisions\/2395"}],"up":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2325"}],"wp:attachment":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/media?parent=2386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}