{"id":2227,"date":"2016-05-15T00:07:25","date_gmt":"2016-05-15T00:07:25","guid":{"rendered":"http:\/\/fisicaevestibular.com.br\/novo\/?page_id=2227"},"modified":"2024-09-02T14:05:24","modified_gmt":"2024-09-02T14:05:24","slug":"estudo-analitico-dos-espelhos-esfericos","status":"publish","type":"page","link":"https:\/\/fisicaevestibular.com.br\/novo\/optica\/optica-geometrica\/estudo-analitico-dos-espelhos-esfericos\/","title":{"rendered":"Estudo Anal\u00edtico das Lentes Esf\u00e9ricas"},"content":{"rendered":"

Estudo anal\u00edtico dos espelhos esf\u00e9ricos<\/b><\/span><\/span><\/span><\/p>\n

\u00a0<\/span>Como voc\u00ea j\u00e1 viu no cap\u00edtulo anterior, <\/b><\/span><\/span><\/span>\u00e9 poss\u00edvel determinar graficamente <\/b><\/span><\/span><\/span>as <\/b><\/span><\/span><\/span>caracter\u00edsticas da imagem conjugada por um dado espelho esf\u00e9rico<\/b><\/span><\/span><\/span>.<\/span><\/span><\/span><\/p>\n

Mas, <\/b><\/span><\/span><\/span>existe tamb\u00e9m uma rela\u00e7\u00e3o matem\u00e1tica entre a posi\u00e7\u00e3o (localiza\u00e7\u00e3o) do objeto<\/b><\/span><\/span><\/span>\u00a0P,<\/b><\/span><\/span><\/span>\u00a0da imagem<\/b><\/span><\/span><\/span>\u00a0P\u2019<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>e a <\/b><\/span><\/span><\/span>dist\u00e2ncia focal<\/b><\/span><\/span><\/span>\u00a0f\u00a0<\/b><\/span><\/span><\/span>para os espelhos esf\u00e9ricos, <\/b><\/span><\/span><\/span>denominada <\/b><\/span><\/span><\/span>equa\u00e7\u00e3o dos pontos conjugados, num <\/b><\/span><\/span><\/span>referencial conveniente <\/b><\/span><\/span><\/span>denominado referencial de Gauss, <\/b><\/span><\/span><\/span>com as <\/b><\/span><\/span><\/span>seguintes caracter\u00edsticas<\/b><\/span><\/span><\/span>
\n\"\"<\/b><\/span><\/span> O <\/b><\/span><\/span><\/span>eixo das abscissas coincide com o eixo principal do espelho.<\/b><\/span><\/span><\/span>
\n\"\"<\/b><\/span><\/span><\/span> <\/b>O <\/b><\/span><\/span><\/span>eixo das ordenadas coincide com o espelho.<\/b><\/span><\/span><\/span>
\n\"\" A <\/b><\/span><\/span><\/span>origem coincide com o v\u00e9rtice do espelho.<\/b><\/span><\/span><\/span><\/p>\n

Equa\u00e7\u00e3o dos pontos conjugados ou equa\u00e7\u00e3o de Gauss para espelhos esf\u00e9ricos<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Equa\u00e7\u00e3o do aumento linear transversal<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Conven\u00e7\u00f5es do referencial de Gauss<\/b><\/span><\/span><\/span><\/span><\/span><\/p>\n

No <\/b><\/span><\/span><\/span>referencial de Gauss adotado no estudo <\/b><\/span><\/span><\/span>dessas equa\u00e7\u00f5es, as <\/b><\/span><\/span><\/span>posi\u00e7\u00f5es s\u00e3o transformadas em coordenadas onde:<\/b><\/span><\/span><\/span><\/p>\n

\"\" No v\u00e9rtice V <\/b><\/span><\/span><\/span>do espelho est\u00e1 <\/b><\/span><\/span><\/span>localizada a origem das abscissas e das ordenadas.<\/b><\/span><\/span><\/span><\/p>\n

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

\"\" No <\/b><\/span><\/span><\/span>lado onde a luz incide <\/b><\/span><\/span><\/span>as <\/b><\/span><\/span><\/span>abscissas (P e P\u2019) s\u00e3o positivas e no lado oposto, negativas.<\/b><\/span><\/span><\/span><\/p>\n

\"\" As <\/b><\/span><\/span><\/span>coordenadas (i e O) <\/b><\/span><\/span><\/span>s\u00e3o <\/b><\/span><\/span><\/span>positivas acima do v\u00e9rtice do espelho e negativas abaixo.<\/b><\/span><\/span><\/span><\/p>\n

Imagem real \"\"<\/b><\/span><\/span><\/span> <\/b>P\u2019 > 0, (positivo) \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>imagem virtual, P\u2019< 0 (negativo).<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Informa\u00e7\u00f5es importantes<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

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

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

\"\"\u00a0Toda <\/b><\/span><\/span><\/span>imagem real pode ser projetada e \u00e9 invertida <\/b><\/span><\/span><\/span>(<\/b><\/span><\/span><\/span>i <\/b><\/span><\/span><\/span>e <\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/span> tem sinais opostos) \"\" <\/b><\/span><\/span><\/span>P\u2019 \u00e9 positivo e i \u00e9 negativo<\/b><\/span><\/span><\/span><\/p>\n

\"\"\u00a0Toda <\/b><\/span><\/span><\/span>imagem virtual \u00e9 direita \"\"<\/b><\/span><\/span><\/span> <\/b>P\u2019 \u00e9 negativo e i \u00e9 positivo<\/b><\/span><\/span><\/span><\/p>\n

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

\"\" Demonstra\u00e7\u00e3o da equa\u00e7\u00e3o dos pontos conjugados ou equa\u00e7\u00e3o de Gauss<\/b><\/span><\/span><\/span><\/p>\n

Para a <\/b><\/span><\/span><\/span>demonstra\u00e7\u00e3o da equa\u00e7\u00e3o de Gauss, <\/b><\/span><\/span><\/span>vamos considerar um <\/b><\/span><\/span><\/span>objeto de altura O<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>e sua correspondente <\/b><\/span><\/span><\/span>imagem de altura i\u00a0<\/b><\/span><\/span><\/span>conjugada por um <\/b><\/span><\/span><\/span>espelho esf\u00e9rico c\u00f4ncavo, <\/b><\/span><\/span><\/span>conforme a figura abaixo.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n


\n<\/b><\/span><\/span>Os <\/b><\/span><\/span><\/span>tri\u00e2ngulos ABV e A\u2019B\u2019V s\u00e3o semelhantes<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\" Algumas vezes<\/b><\/span><\/span><\/span> voc\u00ea precisa aplicar a <\/b><\/span><\/span><\/span>equa\u00e7\u00e3o dos pontos conjugados mais que uma vez <\/b><\/span><\/span><\/span>como no exemplo a seguir:<\/b><\/span><\/span><\/span><\/p>\n

Um objeto <\/b><\/span><\/span><\/span>desloca-se ao longo do eixo principal<\/b><\/span><\/span><\/span>, em <\/b><\/span><\/span><\/span>dire\u00e7\u00e3o ao v\u00e9rtice de um espelho esf\u00e9rico c\u00f4ncavo Gaussiano, com velocidade constante de 4cm\/s.<\/b><\/span><\/span><\/span> A <\/b><\/span><\/span><\/span>dist\u00e2ncia focal do espelho \u00e9 de 10cm.<\/b><\/span><\/span><\/span>\"\"<\/p>\n

Em um <\/b><\/span><\/span><\/span>certo instante, o objeto est\u00e1 a 50cm do v\u00e9rtice. <\/b><\/span><\/span><\/span><\/p>\n

Determine<\/b><\/span><\/span><\/span>, ap\u00f3s 5s, <\/b><\/span><\/span><\/span>a <\/b><\/span><\/span><\/span>dist\u00e2ncia percorrida pela imagem <\/b><\/span><\/span><\/span>do objeto. <\/b><\/span><\/span><\/span><\/p>\n

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

Situa\u00e7\u00e3o inicial (t = 0)<\/b><\/span><\/span><\/span>\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>P = 50cm\u00a0 \"\"\u00a0 f = 10cm\u00a0 \"\"\u00a0 1\/f = 1\/P + 1\/P\u2019\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 1\/10 = 1\/50 + 1\/P\u2019\u00a0 \"\"\u00a0 1\/10 \u2013 1\/50 = 1\/P\u2019\u00a0 \"\"\u00a0 4\/50 = 1\/P\u2019\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>P\u2019 = 12,5cm (posi\u00e7\u00e3o inicial da imagem).<\/b><\/span><\/span><\/span>\"\"<\/p>\n

Situa\u00e7\u00e3o final (t = 5s)\u00a0\u00a0\"\"<\/b><\/span><\/span><\/span>\u00a0 V=\u0394S\/\u0394t\u00a0<\/b><\/span><\/span><\/span> \"\"\u00a0 4= \u0394S\/5\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>\u0394S=20cm\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 como ele se <\/b><\/span><\/span><\/span>aproxima do espelho <\/b><\/span><\/span><\/span>sua <\/b><\/span><\/span><\/span>nova dist\u00e2ncia do v\u00e9rtice <\/b><\/span><\/span><\/span>do espelho ser\u00e1\u00a0 \"\"\u00a0 P = 50 \u2013 20\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 P = 30cm\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>f = 10cm\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 1\/10 = 1\/30 + 1\/P\u2019\u00a0\"\"\u00a0 <\/b><\/span><\/span><\/span>P\u2019= 15cm<\/b><\/span><\/span><\/span> <\/b>(nova posi\u00e7\u00e3o da imagem).<\/b><\/span><\/span><\/span><\/p>\n

A <\/b><\/span><\/span><\/span>imagem percorreu <\/b><\/span><\/span><\/span>\u0394S = 15 \u2013 12,5\u00a0 \"\" \u00a0<\/b><\/span><\/span><\/span>\u0394S = 2,5cm.\u00a0\u00a0<\/b><\/span><\/span><\/span><\/p>\n

\"\" Exerc\u00edcio interessante:<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

A <\/b><\/span><\/span><\/span>imagem do Sol \u00e9 formada em um espelho esf\u00e9rico c\u00f4ncavo<\/b><\/span><\/span><\/span>, de <\/b><\/span><\/span><\/span>dist\u00e2ncia focal igual a 1 metro. Considerando<\/b><\/span><\/span><\/span> a <\/b><\/span><\/span><\/span>dist\u00e2ncia do Sol \u00e0 Terra 250 vezes maior que o di\u00e2metro do Sol<\/b><\/span><\/span><\/span>, <\/b><\/span><\/span><\/span>determine o di\u00e2metro da imagem formada. <\/b><\/span><\/span><\/span><\/p>\n

Sendo a <\/b><\/span><\/span><\/span>dist\u00e2ncia do Sol \u00e0 Terra muito grande<\/b><\/span><\/span><\/span>, <\/b><\/span><\/span><\/span>os raios de luz emitidos por ele chegam \u00e0 Terra <\/b><\/span><\/span><\/span>como<\/b><\/span><\/span><\/span>\"\"<\/p>\n

feixes de raios paralelos <\/b><\/span><\/span><\/span>e a <\/b><\/span><\/span><\/span>imagem P\u2019 do Sol se forma no foco\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>P\u2019 = f = 1m \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>P=250d\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>O = d\u00a0\u00a0\"\"<\/b><\/span><\/span><\/span>\u00a0\u00a0i\/O = – P\u2019\/P\u00a0 \"\" \u00a0i\/d = – 1\/250d \"\" <\/b><\/span><\/span><\/span>i = <\/b><\/span><\/span><\/span>-1\/250 <\/b><\/span><\/span><\/span>= 4.10<\/b><\/span><\/span><\/span>-3<\/b><\/span><\/span><\/sup><\/span>m\u00a0(i < 0, imagem do Sol invertida e de di\u00e2metro <\/b><\/span><\/span><\/span><\/p>\n

0,004 m)<\/b><\/span><\/span><\/span><\/p>\n

Comparando as dist\u00e2ncias aparentes entre um espelho esf\u00e9rico convexo e um espelho plano<\/b><\/span><\/span><\/span><\/p>\n

Para <\/b><\/span><\/span><\/span>evitar acidentes de tr\u00e2nsito, <\/b><\/span><\/span><\/span>foram <\/b><\/span><\/span><\/span>instalados espelhos convexos <\/b><\/span><\/span><\/span>em <\/b><\/span><\/span><\/span>alguns cruzamentos.<\/b><\/span><\/span><\/span><\/p>\n

A experi\u00eancia <\/b><\/span><\/span><\/span>n\u00e3o foi bem sucedida <\/b><\/span><\/span><\/span>porque, como os <\/b><\/span><\/span><\/span>espelhos convexos fornecem imagens menores, perde-se completamente a no\u00e7\u00e3o de dist\u00e2ncia. <\/b><\/span><\/span><\/span>\"\"<\/p>\n

Para <\/b><\/span><\/span><\/span>perceber esse efeito<\/b><\/span><\/span><\/span>, suponha que um <\/b><\/span><\/span><\/span>objeto linear seja colocado a 30 m de um espelho convexo de 12 m de raio, perpendicularmente a seu eixo principal.<\/b><\/span><\/span><\/span><\/p>\n

a) <\/b><\/span><\/span><\/span>A que <\/b><\/span><\/span><\/span>dist\u00e2ncia do espelho convexo seria vista a imagem desse objeto?<\/b><\/span><\/span><\/span><\/p>\n

b) <\/b><\/span><\/span><\/span>Se <\/b><\/span><\/span><\/span>substitu\u00edssemos o espelho convexo por um espelho plano, a que dist\u00e2ncia deste espelho seria vista a imagem daquele objeto?<\/b><\/span><\/span><\/span><\/p>\n

a) <\/b><\/span><\/span><\/span>Objeto a P=30m\u00a0do espelho convexo de dist\u00e2ncia focal\u00a0 f = – 6m\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>1\/f = 1\/P + 1\/P\u2019\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 -1\/6 = 1\/30 + 1\/P\u2019\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>P\u2019 = – 5m (observe que para um objeto a 30m do espelho convexo, a imagem seria vista como se estivesse a 5m do mesmo, aparentando assim, ser menor).<\/b><\/span><\/span><\/span><\/p>\n

b) No <\/b><\/span><\/span><\/span>espelho plano <\/b><\/span><\/span><\/span>a imagem \u00e9 <\/b><\/span><\/span><\/span>sim\u00e9trica<\/b><\/span><\/span><\/span> (objeto e imagem s\u00e3o equidistantes do espelho) e de <\/b><\/span><\/span><\/span>mesmo tamanho do objeto. Assim voc\u00ea veria a dist\u00e2ncia real que \u00e9 de 30m. <\/b><\/span><\/span><\/span><\/p>\n

 <\/p>\n

Confira os exerc\u00edcios com resolu\u00e7\u00f5es comentadas<\/span><\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"

Estudo anal\u00edtico dos espelhos esf\u00e9ricos \u00a0Como voc\u00ea j\u00e1 viu no cap\u00edtulo anterior, \u00e9 poss\u00edvel determinar graficamente as caracter\u00edsticas da imagem conjugada por um dado espelho esf\u00e9rico. Mas, existe tamb\u00e9m uma rela\u00e7\u00e3o matem\u00e1tica entre a posi\u00e7\u00e3o (localiza\u00e7\u00e3o) do objeto\u00a0P,\u00a0da imagem\u00a0P\u2019\u00a0e a dist\u00e2ncia focal\u00a0f\u00a0para os espelhos esf\u00e9ricos, denominada equa\u00e7\u00e3o dos pontos conjugados, num referencial conveniente denominado referencial de Gauss, com as seguintes<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":2137,"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-2227","page","type-page","status-publish","hentry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2227","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=2227"}],"version-history":[{"count":3,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2227\/revisions"}],"predecessor-version":[{"id":10938,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2227\/revisions\/10938"}],"up":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2137"}],"wp:attachment":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/media?parent=2227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}