{"id":25605,"date":"2022-07-12T00:00:33","date_gmt":"2022-07-12T00:00:33","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25605"},"modified":"2024-05-21T09:42:26","modified_gmt":"2024-05-21T09:42:26","slug":"as-transformacoes-de-galileu-e-suas-limitacoes","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/as-transformacoes-de-galileu-e-suas-limitacoes\/","title":{"rendered":"As Transforma\u00e7\u00f5es de Galileu e suas limita\u00e7\u00f5es"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>As Transforma\u00e7\u00f5es de Galileu e suas limita\u00e7\u00f5es<\/h1>\n<p class=\"eq\"><em><strong>Resumo:<\/strong><br \/>\nO princ\u00edpio da relatividade afirma que as observa\u00e7\u00f5es dependem do referencial inercial, mas de tal forma que as leis f\u00edsicas se mant\u00eam. Uma primeira e intuitiva abordagem a esse princ\u00edpio vem das Transforma\u00e7\u00f5es de Galileu, que modelam a forma como as observa\u00e7\u00f5es mudam entre referenciais inerciais na mec\u00e2nica cl\u00e1ssica. Nesta aula, estudaremos tais transforma\u00e7\u00f5es e suas propriedades, e tamb\u00e9m veremos como falham ao ser aplicadas ao fen\u00f4meno da propaga\u00e7\u00e3o de ondas.<\/br><\/em><\/p>\n<p><strong>OBJETIVOS DE APRENDIZAGEM<\/strong><br \/>\nAo concluir esta aula, o estudante ser\u00e1 capaz de:\n<\/p>\n<p><\/center><\/p>\n<ol>\n<li><strong>Reconhecer<\/strong> os conceitos fundamentais das Transforma\u00e7\u00f5es de Galileu, incluindo sua formula\u00e7\u00e3o b\u00e1sica e princ\u00edpios subjacentes.<\/li>\n<li><strong>Analisar<\/strong> a geometria galileana do espa\u00e7o e do tempo e sua separa\u00e7\u00e3o no contexto da mec\u00e2nica cl\u00e1ssica.<\/li>\n<li><strong>Avaliar<\/strong> as limita\u00e7\u00f5es das Transforma\u00e7\u00f5es de Galileu quando aplicadas a fen\u00f4menos como a propaga\u00e7\u00e3o de ondas e sua relev\u00e2ncia no avan\u00e7o para a teoria da relatividade especial.<\/li>\n<\/ol>\n<p><center><\/p>\n<p class=\"indx\"><strong>\u00cdNDICE<\/strong><br \/>\n<a href=\"#1\"><strong>Formula\u00e7\u00e3o das transforma\u00e7\u00f5es de Galileu<\/strong><\/a><br \/>\n<a href=\"#2\">A transforma\u00e7\u00e3o inversa<\/a><br \/>\n<a href=\"#3\">O tempo absoluto e a soma das velocidades<\/a><br \/>\n<a href=\"#4\">Geometria Galileana do espa\u00e7o e do tempo<\/a><br \/>\n<a href=\"#5\"><strong>A relatividade de Galileu e as leis f\u00edsicas<\/strong><\/a><br \/>\n<a href=\"#6\">Aplicada \u00e0 din\u00e2mica de Newton<\/a><br \/>\n<a href=\"#7\">Aplicada \u00e0 propaga\u00e7\u00e3o de uma onda<\/a><br \/>\n<a href=\"#8\">Qual o efeito da transforma\u00e7\u00e3o de Galileu sobre a propaga\u00e7\u00e3o das ondas?<\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ku-9nbTSaJg?si=1dmuCtjzBPUy14v_\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><br \/>\n<\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Formula\u00e7\u00e3o das transforma\u00e7\u00f5es de Galileu<\/h2>\n<p style=\"text-align:justify;\">A f\u00edsica de Newton se baseia no princ\u00edpio da relatividade modelado atrav\u00e9s das transforma\u00e7\u00f5es de Galileu, onde se estabelece o tempo como uma coordenada universal para todos os observadores inerciais; ou seja: <span class=\"katex-eq\" data-katex-display=\"false\">t=t^\\prime<\/span>. Sob esta afirma\u00e7\u00e3o, vimos que a transforma\u00e7\u00e3o linear que relaciona as observa\u00e7\u00f5es de dois referenciais inerciais <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> revistos na aula sobre <a href=\"http:\/\/toposuranos.com\/material\/es\/el-principio-de-relatividad-especial\/?fbclid=IwAR2_lpF1hyUlbSzFQ5B5OQpPu1Vhc_20zTGu8D4pxKPsSvzZRvLSzPdwQXU\" rel=\"noopener\" target=\"_blank\">o princ\u00edpio da relatividade<\/a> tem a forma de uma transforma\u00e7\u00e3o linear:<\/p>\n<p><a name=\"eq1\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nt^\\prime &amp;= At + Bx,\\\\ x^\\prime &amp;= Dt + Ex,\\\\ y^\\prime &amp;= y, \\\\ z^\\prime &amp;=z,\n\n\\end{array}<\/span> <strong>[1]<\/strong><\/p>\n<p style=\"text-align:justify;\">assume a seguinte forma quando os referenciais inerciais <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> est\u00e3o em configura\u00e7\u00e3o padr\u00e3o e <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> se move com velocidade <span class=\"katex-eq\" data-katex-display=\"false\">v_{ss^\\prime_x}\\hat{x}<\/span> em rela\u00e7\u00e3o a <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png\" alt=\"transforma\u00e7\u00f5es de coordenadas\" width=\"1374\" height=\"741\" class=\"aligncenter size-full wp-image-25502 lazyload\" \/><noscript><img decoding=\"async\" src=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png\" alt=\"transforma\u00e7\u00f5es de coordenadas\" width=\"1374\" height=\"741\" class=\"aligncenter size-full wp-image-25502 lazyload\" srcset=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio.png 1374w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-300x162.png 300w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-1024x552.png 1024w, http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/coordenadas-cambio-768x414.png 768w\" sizes=\"(max-width: 1374px) 100vw, 1374px\" \/><\/noscript><\/center><\/p>\n<p><a name=\"eq2\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rlr}\n\n{}t^\\prime &amp;= t  \\\\ x^\\prime &amp;= x - v_{ss^\\prime_x}t \\\\ y^\\prime &amp;= y \\\\ z^\\prime &amp;= z\n\n\\end{array}\n\n<\/span> <strong>[2]<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>A transforma\u00e7\u00e3o inversa<\/h3>\n<p style=\"text-align:justify;\">A partir de uma esp\u00e9cie de simetria alg\u00e9brica, podemos escrever a transforma\u00e7\u00e3o inversa:<\/p>\n<p><a name=\"eq3\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\nt &amp;= t^\\prime \\\\ x &amp;= x^\\prime + v_{ss^\\prime_x}t \\\\ y &amp;= y^\\prime \\\\ z &amp;= z^\\prime \\end{array}\n\n<\/span> <strong>[3]<\/strong><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>O tempo absoluto e a soma das velocidades<\/h3>\n<p style=\"text-align:justify;\">Da primeira equa\u00e7\u00e3o das transforma\u00e7\u00f5es de Galileu (qualquer uma das duas, <a href=\"#eq2\">[2]<\/a> ou <a href=\"#eq3\">[3]<\/a>) tem-se que a coordenada temporal de um evento n\u00e3o depende do referencial a partir do qual se observa, enquanto que a segunda permite obter o que usualmente se entende como \u00abo senso comum\u00bb associado \u00e0 soma de velocidades. Se uma part\u00edcula se move com velocidade constante <span class=\"katex-eq\" data-katex-display=\"false\">v_{ss^\\prime_x}<\/span> sobre o eixo <span class=\"katex-eq\" data-katex-display=\"false\">\\hat{x}<\/span> de <span class=\"katex-eq\" data-katex-display=\"false\">S,<\/span> ent\u00e3o sua velocidade em <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span> \u00e9 determinada por<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle v^\\prime_x = \\frac{dx^\\prime}{dt^\\prime} = \\frac{dx^\\prime}{dt} = \\frac{d}{dt}\\left(x - v_{ss^\\prime_x} t \\right) = v_x - v_{ss^\\prime_x}<\/span>\n<p style=\"text-align:justify;\">Derivando nesta \u00faltima express\u00e3o, mostra-se que a acelera\u00e7\u00e3o de uma part\u00edcula qualquer \u00e9 a mesma em <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> e em <span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span>, ou seja: <span class=\"katex-eq\" data-katex-display=\"false\">dv^\\prime_x\/dt^\\prime = dv_x\/dt<\/span>.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Geometria Galileana do espa\u00e7o e do tempo<\/h3>\n<p style=\"text-align:justify;\">Se considerarmos dois eventos <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">B<\/span> que t\u00eam coordenadas <span class=\"katex-eq\" data-katex-display=\"false\">(t_A,x_A,y_A,z_A)<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">(t_B,x_B,y_B,z_B),<\/span> respectivamente. \u00c9 f\u00e1cil ver que as quantidades <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta t = t_B - t_A<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r^2 = \\Delta x^2 + \\Delta y^2 + \\Delta z^2<\/span> s\u00e3o separadamente invariantes sob as transforma\u00e7\u00f5es de Galileu, isso nos leva a considerar o espa\u00e7o e o tempo como entidades separadas. Por outro lado, <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r^2<\/span> sugere que isso \u00e9 uma propriedade geom\u00e9trica do pr\u00f3prio espa\u00e7o. Reconhecemos <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta r^2<\/span> como o quadrado da dist\u00e2ncia entre os eventos no espa\u00e7o euclidiano. Isso define a geometria do espa\u00e7o e do tempo no contexto da mec\u00e2nica de Newton.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>A Relatividade de Galileu e as Leis F\u00edsicas<\/h2>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Aplicada \u00e0 Din\u00e2mica de Newton<\/h3>\n<p style=\"text-align:justify;\">No trecho anterior, vimos que, no contexto da f\u00edsica newtoniana, dois referenciais inerciais distintos sempre observar\u00e3o as mesmas acelera\u00e7\u00f5es. Isto, juntamente com a segunda lei de Newton, implica que todos os referenciais inerciais observar\u00e3o sempre a mesma din\u00e2mica. Ou seja:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F_x = m\\frac{dv_x}{dt}= m\\frac{dv^\\prime_x}{dt^\\prime} = F^\\prime_x.<\/span> <\/p>\n<p style=\"text-align:justify;\">Esta \u00faltima express\u00e3o nos diz que a <strong>f\u00edsica n\u00e3o muda ao realizar transforma\u00e7\u00f5es de Galileu,<\/strong> o que \u00e9 equivalente a dizer que: a f\u00edsica \u00e9 a mesma para todos os observadores inerciais.<\/p>\n<p><a name=\"7\"><\/a><\/p>\n<h3>Aplicada \u00e0 Propaga\u00e7\u00e3o de Ondas<\/h3>\n<p style=\"text-align:justify;\">Embora essa persist\u00eancia da f\u00edsica diante das mudan\u00e7as de observadores inerciais seja algo esperado, primeiro porque \u00e9 o que observamos ao nos mover, e segundo porque \u00e9 o que foi obtido atrav\u00e9s dos c\u00e1lculos anteriores, a verdade \u00e9 que nem sempre se cumpre dessa maneira. O caso mais not\u00e1vel de um fen\u00f4meno que n\u00e3o se preserva sob transforma\u00e7\u00f5es de Galileu \u00e9 o caso da propaga\u00e7\u00e3o de ondas; em geral, a equa\u00e7\u00e3o que modela a propaga\u00e7\u00e3o de uma onda <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> no espa\u00e7o e no tempo \u00e9 da forma<\/p>\n<p><a name=\"eq4\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\nabla^2 \\psi = \\frac{1}{v_0^2}\\frac{\\partial^2 \\psi}{\\partial t^2}<\/span> <strong>[4]<\/strong><\/p>\n<p style=\"text-align:justify;\">onde <span class=\"katex-eq\" data-katex-display=\"false\">v_0<\/span> \u00e9 a velocidade de propaga\u00e7\u00e3o da onda.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h4>Qual o efeito da transforma\u00e7\u00e3o de Galileu sobre a propaga\u00e7\u00e3o das ondas?<\/h4>\n<p style=\"text-align:justify;\">Para isso, existe uma resposta curta e uma longa. A resposta curta \u00e9 que \u00abmesmo observando o mesmo fen\u00f4meno, observadores inerciais diferentes ver\u00e3o &#8216;uma f\u00edsica&#8217; diferente\u00bb. A resposta longa consiste em ver como muda a equa\u00e7\u00e3o de propaga\u00e7\u00e3o da onda quando se aplica a transforma\u00e7\u00e3o de Galileu; para fazer isso, primeiro tomamos a equa\u00e7\u00e3o <a href=\"#eq3\">[4]<\/a> e a expandimos sobre cada uma de suas coordenadas obtendo:<\/p>\n<p><a name=\"eq5\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial ^2 \\psi}{\\partial x^2} + \\frac{\\partial ^2 \\psi}{\\partial y^2} + \\frac{\\partial ^2 \\psi}{\\partial z^2} = \\frac{1}{v_0^2} \\frac{\\partial ^2 \\psi}{\\partial t^2}.<\/span> <strong>[5]<\/strong><\/p>\n<p>Com esta equa\u00e7\u00e3o em m\u00e3os, agora devemos usar as equa\u00e7\u00f5es de <a href=\"#eq3\">[3]<\/a>, para reexpressar as derivadas no outro referencial inercial.<\/p>\n<h5>Transforma\u00e7\u00e3o das Primeiras Derivadas<\/h5>\n<p style=\"text-align:justify;\">Seguindo as express\u00f5es de <a href=\"#eq3\">[3]<\/a> e derivando cada vari\u00e1vel em rela\u00e7\u00e3o \u00e0s vari\u00e1veis primadas, obtemos:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial x^\\prime}{\\partial x} = \\frac{\\partial y^\\prime}{\\partial y}= \\frac{\\partial z^\\prime}{\\partial z}= \\frac{\\partial t^\\prime}{\\partial t}= 1<\/span>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial x^\\prime}{\\partial t} = - v_{x_0}<\/span>\n<p style=\"text-align:justify;\">Enquanto todas as outras se anulam:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial t^\\prime}{\\partial x} = \\frac{\\partial t^\\prime}{\\partial y} = \\frac{\\partial t^\\prime}{\\partial z} = \\frac{\\partial x^\\prime}{\\partial y} = \\frac{\\partial x^\\prime}{\\partial z} = \\frac{\\partial y^\\prime}{\\partial x} = \\frac{\\partial y^\\prime}{\\partial z} = \\frac{\\partial y^\\prime}{\\partial t} = \\frac{\\partial z^\\prime}{\\partial x} = \\frac{\\partial z^\\prime}{\\partial y} = \\frac{\\partial z^\\prime}{\\partial t} = 0\n\n<\/span>\n<p style=\"text-align:justify;\">Com isso em m\u00e3os, agora podemos calcular as derivadas de <span class=\"katex-eq\" data-katex-display=\"false\">\\psi<\/span> atrav\u00e9s da regra da cadeia:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{\\partial \\psi}{\\partial x} = \\frac{\\partial \\psi}{\\partial x^\\prime} \\underbrace{\\frac{\\partial x^\\prime}{\\partial x}}_{=1} + \\frac{\\partial \\psi}{\\partial y^\\prime} \\underbrace{\\frac{\\partial y^\\prime}{\\partial x}}_{=0} +\n\n\\frac{\\partial \\psi}{\\partial z^\\prime} \\underbrace{\\frac{\\partial z^\\prime}{\\partial x}}_{=0} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\underbrace{\\frac{\\partial t^\\prime}{\\partial x}}_{=0} =  \\frac{\\partial \\psi}{\\partial x^\\prime}.<\/span>\n<p>E de forma an\u00e1loga se ter\u00e1 para as outras duas vari\u00e1veis espaciais:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{\\partial \\psi}{\\partial y} =  \\frac{\\partial \\psi}{\\partial y^\\prime}.<\/span>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\frac{\\partial \\psi}{\\partial z} =  \\frac{\\partial \\psi}{\\partial z^\\prime}.<\/span>\n<p style=\"text-align:justify;\">No entanto, a derivada temporal apresentar\u00e1 algumas diferen\u00e7as:<\/p>\n<p id=\"eq\">\n<span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\frac{\\partial \\psi}{\\partial t} &amp;= \\displaystyle\\frac{\\partial \\psi}{\\partial x^\\prime}\\underbrace{\\frac{\\partial x^\\prime}{\\partial t}}_{=-v_{x_0}} + \\frac{\\partial \\psi}{\\partial y^\\prime}\\underbrace{\\frac{\\partial y^\\prime}{\\partial t}}_{=0} + \\frac{\\partial \\psi}{\\partial z^\\prime}\\underbrace{\\frac{\\partial z^\\prime}{\\partial t}}_{=0} + \\frac{\\partial \\psi}{\\partial t^\\prime}\\underbrace{\\frac{\\partial t^\\prime}{\\partial t}}_{=1}\\\\ &amp;=\\displaystyle -v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime},\n\n\\end{array}\n\n<\/span>\n<h5>Transforma\u00e7\u00e3o das Segundas Derivadas<\/h5>\n<p style=\"text-align:justify;\">Para a parte espacial, poderemos continuar sem grandes dificuldades, os resultados s\u00e3o:<\/p>\n<p><a name=\"eq6\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial^2 \\psi}{\\partial x^2} = \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2}.<\/span> <strong>[6]<\/strong><\/p>\n<p><a name=\"eq7\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial^2 \\psi}{\\partial y^2} = \\frac{\\partial^2 \\psi}{\\partial {y^\\prime}^2}<\/span> <strong>[7]<\/strong><\/p>\n<p><a name=\"eq8\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\partial^2 \\psi}{\\partial z^2} = \\frac{\\partial^2 \\psi}{\\partial {z^\\prime}^2}<\/span> <strong>[8]<\/strong><\/p>\n<p style=\"text-align:justify;\">Mas a parte temporal, como j\u00e1 poder\u00edamos antecipar desde as primeiras derivadas, mostra grandes diferen\u00e7as:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle\\frac{\\partial^2 \\psi}{\\partial t^2} &amp;=\\displaystyle \\frac{\\partial}{\\partial t}\\left( -v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\right)\\\\ &amp; \\displaystyle = -v_{x_0} \\frac{\\partial }{\\partial t} \\left(\\frac{\\partial \\psi}{\\partial x^\\prime} \\right) + \\frac{\\partial }{\\partial t} \\left(\\frac{\\partial \\psi}{\\partial t^\\prime} \\right)\\\\ &amp;\\displaystyle = -v_{x_0} \\frac{\\partial }{\\partial x^\\prime} \\left(\\frac{\\partial \\psi}{\\partial t} \\right) + \\frac{\\partial }{\\partial t^\\prime} \\left(\\frac{\\partial \\psi}{\\partial t} \\right)\\\\ &amp;\\displaystyle = -v_{x_0} \\frac{\\partial }{\\partial x^\\prime} \\left(-v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\right) + \\frac{\\partial }{\\partial t^\\prime} \\left(-v_{x_0} \\frac{\\partial \\psi}{\\partial x^\\prime} + \\frac{\\partial \\psi}{\\partial t^\\prime} \\right) \\end{array}<\/span>\n<p><a name=\"eq9\"><\/a><\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle\\frac{\\partial^2 \\psi}{\\partial t^2}  = v_{x_0}^2 \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2} - 2v_{x_0}\\frac{\\partial^2 \\psi}{\\partial x^\\prime \\partial t^\\prime} + \\frac{\\partial^2 \\psi}{\\partial {t^\\prime}^2}.<\/span> <strong>[9]<\/strong><\/p>\n<h5>Aplicando as transforma\u00e7\u00f5es de Galileu sobre a propaga\u00e7\u00e3o das ondas<\/h5>\n<p style=\"text-align:justify;\">Deste modo, \u00e9 poss\u00edvel realizar a transforma\u00e7\u00e3o de Galileu sobre a equa\u00e7\u00e3o de propaga\u00e7\u00e3o da onda substituindo as equa\u00e7\u00f5es [<a href=\"#eq6\">6<\/a>,<a href=\"#eq7\">7<\/a>,<a href=\"#eq8\">8<\/a>] e [<a href=\"#eq9\">9<\/a>] sobre [<a href=\"#eq5\">5<\/a>], resultando em:<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2} + \\frac{\\partial^2 \\psi}{\\partial {y^\\prime}^2} + \\frac{\\partial^2 \\psi}{\\partial {z^\\prime}^2} = \\frac{1}{v_0^2} \\left(\\color{red}{ v_{x_0}^2 \\frac{\\partial^2 \\psi}{\\partial {x^\\prime}^2} - 2v_{x_0}\\frac{\\partial^2 \\psi}{\\partial x^\\prime \\partial t^\\prime}} + \\frac{\\partial^2 \\psi}{\\partial {t^\\prime}^2} \\right).\n\n<\/span> <strong>[10]<\/strong><\/p>\n<p style=\"text-align:justify;\">Onde se observa que a forma da propaga\u00e7\u00e3o das ondas n\u00e3o se mant\u00e9m sob transforma\u00e7\u00f5es de Galileu devido ao surgimento dos termos adicionais marcados em vermelho. Embora isto por agora n\u00e3o tenha grandes consequ\u00eancias, em aulas posteriores veremos que isto \u00e9 justamente o ponto que \u00abrompe\u00bb, por assim dizer, com a f\u00edsica cl\u00e1ssica, dando passagem \u00e0 relatividade especial.<\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\">\n<h2>Conclus\u00f5es<\/h2>\n<p style=\"text-align:justify;\">\n        As Transforma\u00e7\u00f5es de Galileu, fundamentais na mec\u00e2nica cl\u00e1ssica, estabelecem um quadro para compreender como as observa\u00e7\u00f5es mudam entre diferentes referenciais inerciais. Atrav\u00e9s deste estudo, reconhecemos o conceito de tempo absoluto e a soma de velocidades como pilares da geometria galileana do espa\u00e7o e do tempo. No entanto, descobrimos limita\u00e7\u00f5es significativas dessas transforma\u00e7\u00f5es, especialmente na sua aplica\u00e7\u00e3o \u00e0 propaga\u00e7\u00e3o de ondas. Esta an\u00e1lise sublinha a necessidade de uma abordagem mais complexa para descrever o universo f\u00edsico, levando-nos em dire\u00e7\u00e3o \u00e0 relatividade especial e al\u00e9m da intui\u00e7\u00e3o cl\u00e1ssica. Em resumo, enquanto as Transforma\u00e7\u00f5es de Galileu proporcionam uma base s\u00f3lida em f\u00edsica cl\u00e1ssica, sua insufici\u00eancia perante certos fen\u00f4menos destaca a evolu\u00e7\u00e3o constante do nosso entendimento do universo.\n    <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>As Transforma\u00e7\u00f5es de Galileu e suas limita\u00e7\u00f5es Resumo: O princ\u00edpio da relatividade afirma que as observa\u00e7\u00f5es dependem do referencial inercial, mas de tal forma que as leis f\u00edsicas se mant\u00eam. Uma primeira e intuitiva abordagem a esse princ\u00edpio vem das Transforma\u00e7\u00f5es de Galileu, que modelam a forma como as observa\u00e7\u00f5es mudam entre referenciais inerciais na [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25587,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":40,"footnotes":""},"categories":[637,693],"tags":[],"class_list":["post-25605","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fisica-pt","category-relatividade"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>As Transforma\u00e7\u00f5es de Galileu e suas limita\u00e7\u00f5es - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Descubra as Transforma\u00e7\u00f5es de Galileu na mec\u00e2nica cl\u00e1ssica e suas intrigantes limita\u00e7\u00f5es nesta 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