{"id":2456,"date":"2016-09-11T01:06:04","date_gmt":"2016-09-11T01:06:04","guid":{"rendered":"http:\/\/fisicaevestibular.com.br\/novo\/?page_id=2456"},"modified":"2024-08-22T17:22:04","modified_gmt":"2024-08-22T17:22:04","slug":"sistema-de-unidades-analise-dimensional","status":"publish","type":"page","link":"https:\/\/fisicaevestibular.com.br\/novo\/introducao\/sistema-de-unidades-analise-dimensional\/","title":{"rendered":"Sistema de Unidades \u2013 An\u00e1lise Dimensional"},"content":{"rendered":"

Introdu\u00e7\u00e3o \u00e0 F\u00edsica<\/b><\/span><\/span><\/p>\n

Sistema de Unidades \u2013 An\u00e1lise Dimensional<\/b><\/span><\/span><\/span><\/p>\n

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

Grandezas F\u00edsicas<\/b><\/span><\/span><\/span><\/p>\n

Quando voc\u00ea est\u00e1 <\/b><\/span><\/span><\/span>estudando um determinado fen\u00f4meno, <\/b><\/span><\/span><\/span>voc\u00ea <\/b><\/span><\/span><\/span>analisa as vari\u00e1veis que participam dele<\/b><\/span><\/span><\/span>,<\/b><\/span><\/span><\/span> <\/b>de modo <\/b><\/span><\/span><\/span>que elas forne\u00e7am dados relevantes sobre o\u00a0 mesmo. <\/b><\/span><\/span><\/span><\/p>\n

Estas <\/b><\/span><\/span><\/span>vari\u00e1veis<\/b><\/span><\/span><\/span> se <\/b><\/span><\/span><\/span>denominam<\/b><\/span><\/span><\/span>\u00a0grandezas f\u00edsicas\u00a0<\/b><\/span><\/span><\/span>que correspondem a\u00a0<\/b><\/span><\/span><\/span>tudo que voc\u00ea pode contar, medir, pesar, enfim, enumerar.<\/b><\/span><\/span><\/span><\/p>\n

Escolhida a grandeza f\u00edsica <\/b><\/span><\/span><\/span>a ser <\/b><\/span><\/span><\/span>analisada no fen\u00f4meno <\/b><\/span><\/span><\/span>voc\u00ea <\/b><\/span><\/span><\/span>deve\u00a0<\/b><\/span><\/span><\/span>medi-la\u00a0<\/b><\/span><\/span><\/span>o que <\/b><\/span><\/span><\/span>significa compar\u00e1-la com outra de mesma natureza, <\/b><\/span><\/span><\/span>ou seja, com um\u00a0<\/b><\/span><\/span><\/span>padr\u00e3o<\/b><\/span><\/span><\/span>\u00a0denominado\u00a0<\/b><\/span><\/span><\/span>unidade.<\/b><\/span><\/span><\/span><\/p>\n

Assim<\/b><\/span><\/span><\/span>,<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>a medi\u00e7\u00e3o \u00e9 a t\u00e9cnica por meio da qual voc\u00ea atribui um n\u00famero a uma grandeza f\u00edsica e, para avali\u00e1-la, voc\u00ea deve compar\u00e1-la com outra similar tomada como padr\u00e3o, denominada unidade.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Exemplo: <\/b><\/span><\/span><\/span>O <\/b><\/span><\/span><\/span>jovem <\/b><\/span><\/span><\/span>da figura est\u00e1 <\/b><\/span><\/span><\/span>medindo sua massa (48kg) <\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/span> sua altura (1,5m) <\/b><\/span><\/span><\/span>onde <\/b><\/span><\/span><\/span>massa e altura s\u00e3o, respectivamente, as<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>grandezas f\u00edsicas, 48 e 1,5 <\/b><\/span><\/span><\/span>(n\u00fameros) e <\/b><\/span><\/span><\/span>kg e m<\/b><\/span><\/span><\/span> os <\/b><\/span><\/span><\/span>nomes das unidades empregadas.<\/b><\/span><\/span><\/span><\/p>\n

Unidades de medidas<\/b><\/span><\/span><\/span><\/p>\n

Por <\/b><\/span><\/span><\/span>muito tempo, <\/b><\/span><\/span><\/span>o mundo usou <\/b><\/span><\/span><\/span>medidas imprecisas, <\/b><\/span><\/span><\/span>como aquelas <\/b><\/span><\/span><\/span>baseadas no corpo humano:<\/b><\/span><\/span><\/span><\/p>\n

palmo, p\u00e9, polegada, bra\u00e7a, c\u00f4vado. <\/b><\/span><\/span><\/span><\/p>\n

Isso acabou gerando <\/b><\/span><\/span><\/span>muitos problemas, <\/b><\/span><\/span><\/span>principalmente no com\u00e9rcio, <\/b><\/span><\/span><\/span>devido \u00e0 falta de um padr\u00e3o para determinar quantidades de produtos.<\/b><\/span><\/span><\/span><\/p>\n

Por <\/b><\/span><\/span><\/span>esse motivo <\/b><\/span><\/span><\/span>a <\/b><\/span><\/span><\/span>Academia de Ci\u00eancia da Fran\u00e7a,<\/b><\/span><\/span><\/span> em 1789, <\/b><\/span><\/span><\/span>criou um sistema de medidas baseada <\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

numa constante padr\u00e3o e n\u00e3o arbitr\u00e1ria denominado\u00a0<\/b><\/span><\/span><\/span>Sistema M\u00e9trico Decimal<\/b><\/span><\/span><\/span>\u00a0que era formado inicialmente por tr\u00eas unidades fundamentais b\u00e1sicas: o metro, o litro e o quilograma.<\/b><\/span><\/span><\/span><\/p>\n

Sistema M\u00e9trico Decimal<\/b><\/span><\/span><\/span>\u00a0Atual<\/b><\/span><\/span><\/span><\/p>\n

Atualmente o <\/b><\/span><\/span><\/span>metro <\/b><\/span><\/span><\/span>\u00e9 definido <\/b><\/span><\/span><\/span>como <\/b><\/span><\/span><\/span>sendo o comprimento do trajeto percorrido pela luz no v\u00e1cuo, <\/b><\/span><\/span><\/span>durante um <\/b><\/span><\/span><\/span>intervalo de tempo de 1\/299.792.458 de segundo;\u00a0<\/b><\/span><\/span><\/span>o <\/b><\/span><\/span><\/span>quilograma <\/b><\/span><\/span><\/span>corresponde a <\/b><\/span><\/span><\/span>massa de um modelo\u00a0internacional de platina iridiada (feito<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

de\u00a0iridio\u00a0e\u00a0platina) <\/b><\/span><\/span><\/span>que se <\/b><\/span><\/span><\/span>encontra conservado no\u00a0Escrit\u00f3rio Internacional de Pesos e Medidas<\/b><\/span><\/span><\/span><\/p>\n

(BIPM), situado no <\/b><\/span><\/span><\/span>parque de Sant Cloud, nas proximidades de Paris (Fran\u00e7a).<\/b><\/span><\/span><\/span> <\/b><\/p>\n

Um <\/b><\/span><\/span><\/span>litro corresponde ao volume interno de um recipiente em forma de cubo com 1dm de aresta (1 L=<\/b><\/span><\/span><\/span><\/p>\n

1 dm<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>).<\/b><\/span><\/span><\/span><\/p>\n

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

Sistema Internaconal de Unidades (SI)<\/b><\/span><\/span><\/span><\/p>\n

Com o <\/b><\/span><\/span><\/span>decorrer do tempo<\/b><\/span><\/span><\/span>, com as <\/b><\/span><\/span><\/span>rela\u00e7\u00f5es comerciais entre os pa\u00edses aumentando<\/b><\/span><\/span><\/span> foi <\/b><\/span><\/span><\/span>necess\u00e1rio<\/b><\/span><\/span><\/span> obter um <\/b><\/span><\/span><\/span>sistema de unidades mais abrangente, \u00fatil e complexo, <\/b><\/span><\/span><\/span>o <\/b><\/span><\/span><\/span>Sistema Internacional de<\/b><\/span><\/span><\/span> <\/b>Unidades (SI)<\/b><\/span><\/span><\/span> <\/b>que, no <\/b><\/span><\/span><\/span>Brasil,<\/b><\/span><\/span><\/span> foi adotado em 1962 sendo <\/b><\/span><\/span><\/span>posteriormente,<\/b><\/span><\/span><\/span> em 1998 <\/b><\/span><\/span><\/span>ratificado pelo CONMETRO que tornou seu uso obrigat\u00f3rio em todo Brasil.<\/b><\/span><\/span><\/span><\/p>\n

Nela <\/b><\/span><\/span><\/span>constam<\/b><\/span><\/span><\/span> as <\/b><\/span><\/span><\/span>sete grandezas de base adotadas pela Organiza\u00e7\u00e3o Internacional de Normatiza\u00e7\u00e3o (ISO) <\/b><\/span><\/span><\/span>que s\u00e3o: <\/b><\/span><\/span><\/span>comprimento, massa , tempo, corrente el\u00e9trica, temperatura termodin\u00e2mica, quantidade de mat\u00e9ria e intensidade luminosa, cujas caracter\u00edsticas est\u00e3o na tabela ao lado .<\/b><\/span><\/span><\/span>\"\"<\/p>\n

Todas as outras grandezas s\u00e3o descritas como grandezas derivadas e s\u00e3o medidas utilizando unidades derivadas, que s\u00e3o definidas como produtos de pot\u00eancias de unidades de base. Exemplos de grandezas derivadas e de unidades derivadas est\u00e3o listadas na tabela abaixo.<\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Unidades derivadas com nomes especiais no SI<\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

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

\"\"<\/p>\n

\"\"<\/p>\n

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

Regras para se escrever unidades do SI<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

As <\/b><\/span><\/span><\/span>unidades do Sistema Internacional de Unidades (SI) <\/b><\/span><\/span><\/span>podem ser <\/b><\/span><\/span><\/span>escritas por seus nomes <\/b><\/span><\/span><\/span>ou <\/b><\/span><\/span><\/span>representadas por seus s\u00edmbolos. Exemplos:<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Os <\/b><\/span><\/span><\/span>nomes das unidades do SI devem ser escritos <\/b><\/span><\/span><\/span>em<\/b><\/span><\/span><\/span> <\/b>letra\u00a0<\/b><\/span><\/span><\/span>min\u00fascula<\/b><\/span><\/span><\/span>,<\/b><\/span><\/span><\/span> <\/b>mesmo<\/b><\/span><\/span><\/span> sendo nomes de pessoas.\u00a0 <\/b><\/span><\/span><\/span><\/p>\n

Exemplos: <\/b><\/span><\/span><\/span>quilograma<\/b><\/span><\/span><\/span>; <\/b><\/span><\/span><\/span>newton, metro quadrado, kelvin, etc<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Os <\/b><\/span><\/span><\/span>s\u00edmbolos das unidades <\/b><\/span><\/span><\/span>s\u00e3o impressos em caracteres romanos (direitos). <\/b><\/span><\/span><\/span><\/p>\n

Em geral os <\/b><\/span><\/span><\/span>s\u00edmbolos das unidades s\u00e3o escritos em min\u00fasculas, <\/b><\/span><\/span><\/span>mas, se o <\/b><\/span><\/span><\/span>nome da unidade deriva <\/b><\/span><\/span><\/span>de um <\/b><\/span><\/span><\/span>nome pr\u00f3prio, a primeira letra do s\u00edmbolo \u00e9 mai\u00fascula. <\/b><\/span><\/span><\/span><\/p>\n

O <\/b><\/span><\/span><\/span>nome da unidade propriamente dita come\u00e7a sempre por uma min\u00fascula, <\/b><\/span><\/span><\/span>salvo se se <\/b><\/span><\/span><\/span>trata do primeiro nome de uma frase ou do nome \u201cgrau Celsius\u201d.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Escrevendo<\/b><\/span><\/span><\/span> os nomes no plural\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>os <\/b><\/span><\/span><\/span>nomes das unidades do SI <\/b><\/span><\/span><\/span>n\u00e3o <\/b><\/span><\/span><\/span>coincidem<\/b><\/span><\/span><\/span> com as <\/b><\/span><\/span><\/span>regras gramaticais,<\/b><\/span><\/span><\/span> ou seja, <\/b><\/span><\/span><\/span>os <\/b><\/span><\/span><\/span>s\u00edmbolos das unidades ficam invari\u00e1veis no plural.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Escrevendo os <\/b><\/span><\/span><\/span>s\u00edmbolos das unidades\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 o <\/b><\/span><\/span><\/span>acento t\u00f4nico <\/b><\/span><\/span><\/span>recai <\/b><\/span><\/span><\/span>sobre a unidade <\/b><\/span><\/span><\/span>e n\u00e3o sobre o <\/b><\/span><\/span><\/span>prefixo.<\/b><\/span><\/span><\/span><\/p>\n

Exemplos: <\/b><\/span><\/span><\/span>megametro; hectolitro; centigrama; micrometro\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>exce\u00e7\u00f5es:<\/b><\/span><\/span><\/span> quil\u00f4metro; dec\u00e2metro; hect\u00f4metro; dec\u00edmetro; cent\u00edmetro e mil\u00edmetro.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Nenhum s\u00edmbolo deve ser seguido de ponto.<\/b><\/span><\/span><\/span><\/p>\n

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

\"\" hora<\/b><\/span><\/span><\/span> \"\" correto \"\" <\/b><\/span><\/span><\/span>(h)\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>errado<\/b><\/span><\/span><\/span> (h.;hr.);\u00a0<\/b><\/span><\/span><\/span><\/p>\n

\"\" segundo<\/b><\/span><\/span><\/span> \"\" correto <\/b><\/span><\/span><\/span>(s) \"\"<\/b><\/span><\/span><\/span> <\/b>errado<\/b><\/span><\/span><\/span> (s.;seg.);\u00a0<\/b><\/span><\/span><\/span><\/p>\n

\"\"\u00a0<\/span>metro<\/b><\/span><\/span><\/span> \"\" correto <\/b><\/span><\/span><\/span>(m) \"\"<\/b><\/span><\/span><\/span> <\/b>errado <\/b><\/span><\/span><\/span>(m.;mtr.;mt.);\u00a0 <\/b><\/span><\/span><\/span><\/p>\n

\"\" minuto<\/b><\/span><\/span><\/span> \"\" correto (<\/b><\/span><\/span><\/span>min)<\/b><\/span><\/span><\/span> \"\" <\/b><\/span><\/span><\/span>errado<\/b><\/span><\/span><\/span> (min.; minut);<\/b><\/span><\/span><\/span><\/p>\n

\"\"\u00a0<\/span>quilograma<\/b><\/span><\/span><\/span> \"\" correto <\/b><\/span><\/span><\/span>(kg) \"\"<\/b><\/span><\/span><\/span> <\/b>errado<\/b><\/span><\/span><\/span> (kg.;kgr.).<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

O <\/b><\/span><\/span><\/span>s\u00edmbolo das unidades SI \u00e9 invar\u00e1vel <\/b><\/span><\/span><\/span>e, <\/b><\/span><\/span><\/span>no plural n\u00e3o deve ser seguido de \u201cs\u201d.<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span><\/p>\n

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

12 horas \"\"<\/b><\/span><\/span><\/span> (12h e n\u00e3o 12 hs); <\/b><\/span><\/span><\/span><\/p>\n

5 minutos \"\"<\/b><\/span><\/span><\/span> (5 min e n\u00e3o 5mins); <\/b><\/span><\/span><\/span><\/p>\n

6 metros \"\"<\/b><\/span><\/span><\/span> (6m e n\u00e3o 6ms).<\/b><\/span><\/span><\/span><\/p>\n

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

Quando voc\u00ea for <\/b><\/span><\/span><\/span>escrever uma unidade composta n\u00e3o misture nome com s\u00edmbolo.<\/b><\/span><\/span><\/span><\/p>\n

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

Metro por segundo ao quadrado e n\u00e3o metro\/s<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>; km\/h e n\u00e3o km\/hora.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\u00a0<\/span>O <\/b><\/span><\/span><\/span>grama \u00e9 de g\u00eanero masculino <\/b><\/span><\/span><\/span>e, portanto, voc\u00ea deve <\/b><\/span><\/span><\/span>escrever fazendo a concord\u00e2ncia de forma correta.\u00a0 <\/b><\/span><\/span><\/span><\/p>\n

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

Seis quilogramas; oito miligramas; duzentos e vinte e um gramas.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\u00a0<\/span>O <\/b><\/span><\/span><\/span>prefixo quilo <\/b><\/span><\/span><\/span>deve ser <\/b><\/span><\/span><\/span>escrito de maneira correta.\u00a0 <\/b><\/span><\/span><\/span><\/p>\n

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

Quilolitro e n\u00e3o kilolitro; quil\u00f4metro e n\u00e3o kil\u00f4metro; quilograma e n\u00e3o kilograma.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\u00a0<\/span>Nas <\/b><\/span><\/span><\/span>medidas de tempo<\/b><\/span><\/span><\/span> voc\u00ea deve <\/b><\/span><\/span><\/span>escrever os s\u00edmbolos da maneira correta.<\/b><\/span><\/span><\/span><\/p>\n

\u00a0Exemplos: <\/b><\/span><\/span><\/span><\/p>\n

5h20min e n\u00e3o 5,20h ou 5h 20\u2019<\/b><\/span><\/span><\/span>; <\/b><\/span><\/span><\/span><\/p>\n

2h24min5s e n\u00e3o 2h24\u20195\u2019\u2019.<\/b><\/span><\/span><\/span><\/p>\n

Observa\u00e7\u00e3o: <\/b><\/span><\/span><\/span>os <\/b><\/span><\/span><\/span>s\u00edmbolos (\u2019) e (\u2019 \u2019) <\/b><\/span><\/span><\/span>s\u00e3o unidades de <\/b><\/span><\/span><\/span>\u00e2ngulo plano e n\u00e3o de tempo.<\/b><\/span><\/span><\/span><\/p>\n

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

An\u00e1lise Dimensional<\/b><\/span><\/span><\/span><\/p>\n

A <\/b><\/span><\/span><\/span>an\u00e1lise dimensional no campo da f\u00edsica <\/b><\/span><\/span><\/span>est\u00e1 <\/b><\/span><\/span><\/span>relacionada com as unidades de medida <\/b><\/span><\/span><\/span>das <\/b><\/span><\/span><\/span>grandezas f\u00edsicas<\/b><\/span><\/span><\/span> e \u00e9 <\/b><\/span><\/span><\/span>\u00fatil na resolu\u00e7\u00e3o de equa\u00e7\u00f5es que relacionam essas grandezas f\u00edsicas. <\/b><\/span><\/span><\/span><\/p>\n

Ela <\/b><\/span><\/span><\/span>facilita a memoriza\u00e7\u00e3o das equa\u00e7\u00f5es matem\u00e1ticas dos fen\u00f4menos f\u00edsicos.<\/b><\/span><\/span><\/span><\/p>\n

Qualquer <\/b><\/span><\/span><\/span>equa\u00e7\u00e3o matem\u00e1tica <\/b><\/span><\/span><\/span>que <\/b><\/span><\/span><\/span>representa uma situa\u00e7\u00e3o f\u00edsica <\/b><\/span><\/span><\/span>qualquer precisa, <\/b><\/span><\/span><\/span>al\u00e9m de um n\u00famero que a quantifique de uma<\/b><\/span><\/span><\/span> <\/b>unidade de medida que fa\u00e7a a classifica\u00e7\u00e3o qualitativa<\/b><\/span><\/span><\/span>.<\/b><\/span><\/span><\/span>\"\"<\/p>\n

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

An\u00e1lise de uma grandeza f\u00edsica<\/b><\/span><\/span><\/span><\/p>\n

Todas<\/b><\/span><\/span><\/span> as <\/b><\/span><\/span><\/span>grandezas f\u00edsicas <\/b><\/span><\/span><\/span>podem ser <\/b><\/span><\/span><\/span>analisadas dimensionalmente<\/b><\/span><\/span><\/span> atrav\u00e9s de <\/b><\/span><\/span><\/span>tr\u00eas unidades <\/b><\/span><\/span><\/span>que voc\u00ea deve <\/b><\/span><\/span><\/span>tomar como par\u00e2metros <\/b><\/span><\/span><\/span>que s\u00e3o: <\/b><\/span><\/span><\/span>Comprimento (L), Tempo (T) e massa (M).<\/b><\/span><\/span><\/span> <\/b><\/p>\n

As <\/b><\/span><\/span><\/span>outras unidades de medidas <\/b><\/span><\/span><\/span>de <\/b><\/span><\/span><\/span>grandezas derivadas ter\u00e3o suas unidades de medidas provenientes dessas tr\u00eas.\u00a0<\/b><\/span><\/span><\/span><\/p>\n

Costuma-se <\/b><\/span><\/span><\/span>adotar as grandezas fundamentais do S.I.<\/b><\/span><\/span><\/span> para se <\/b><\/span><\/span><\/span>escreverem as equa\u00e7\u00f5es dimensionais.<\/b><\/span><\/span><\/span><\/p>\n

Representa\u00e7\u00e3o de uma grandeza dimensional<\/b><\/span><\/span><\/span><\/p>\n

Uma <\/b><\/span><\/span><\/span>grandeza dimensional <\/b><\/span><\/span><\/span>\u00e9 <\/b><\/span><\/span><\/span>representada por <\/b><\/span><\/span><\/span>colchetes<\/b><\/span><\/span><\/span> <\/b>e voc\u00ea <\/b><\/span><\/span><\/span>pode expressa-la em fun\u00e7\u00e3o das grandezas fundamentais. <\/b><\/span><\/span><\/span><\/p>\n

Por exemplo, <\/b><\/span><\/span><\/span>uma grandeza f\u00edsica (F), <\/b><\/span><\/span><\/span>que <\/b><\/span><\/span><\/span>depende da massa, do comprimento e do tempo, tem sua equa\u00e7\u00e3o dimensional escrita da seguinte maneira:<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Grandeza adimensional<\/b><\/span><\/span><\/span><\/p>\n

Uma <\/b><\/span><\/span><\/span>grandeza<\/b><\/span><\/span><\/span> \u00e9 dita\u00a0<\/b><\/span><\/span><\/span>adimensional<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>se ela \u00e9 <\/b><\/span><\/span><\/span>desprovida de unidades, <\/b><\/span><\/span><\/span>ou seja,\u00a0 o <\/b><\/span><\/span><\/span>resultado final da <\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

dimens\u00e3o \u00e9 unit\u00e1rio. <\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Explica\u00e7\u00e3o dimensional \"\"<\/b><\/span><\/span><\/span> Uma <\/b><\/span><\/span><\/span>grandeza G \u00e9 uma grandeza adimensional<\/b><\/span><\/span><\/span> se G = v.t\/\u2206S, sendo <\/b><\/span><\/span><\/span>v<\/b><\/span><\/span><\/span> a <\/b><\/span><\/span><\/span>velocidade,<\/b><\/span><\/span><\/span> <\/b>t<\/b><\/span><\/span><\/span> o <\/b><\/span><\/span><\/span>tempo<\/b><\/span><\/span><\/span> e <\/b><\/span><\/span><\/span>\u2206S<\/b><\/span><\/span><\/span> o <\/b><\/span><\/span><\/span>comprimento.<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span><\/p>\n

Assim,\u00a0<\/b><\/span><\/span><\/span>[G] =(m\/s)t\/\u2206S <\/b><\/span><\/span><\/span>= (L\/T).T\/L = <\/b><\/span><\/span><\/span>L.T<\/b><\/span><\/span><\/span>-1<\/b><\/span><\/span><\/sup><\/span>\u00a0.T .L<\/b><\/span><\/span><\/span>-1<\/b><\/span><\/span><\/sup><\/span>\u00a0<\/b><\/span><\/span><\/span> \"\"\u00a0 <\/b><\/span><\/span><\/span>[G] = 1<\/b><\/span><\/span><\/span><\/p>\n

Como proceder para calcular a f\u00f3rmula dimensionals de uma grandeza dimensional derivada<\/b><\/span><\/span><\/span><\/p>\n

Primeiro voc\u00ea deve conhecer a f\u00f3rmula dimensional das grandezas f\u00edsicas dimensionais fornecidas na tabela abaixo.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Consideram-se <\/b><\/span><\/span><\/span>unidades derivadas do SI apenas aquelas que podem ser expressas <\/b><\/span><\/span><\/span>atrav\u00e9s das <\/b><\/span><\/span><\/span>unidades b\u00e1sicas (fundamentais) do SI, expressas na tabela acima.<\/b><\/span><\/span><\/span><\/p>\n

Exemplos de <\/b><\/span><\/span><\/span>como proceder no c\u00e1lculo das dimens\u00f5es de uma grandeza dimensional derivada <\/b><\/span><\/span><\/span>ou de uma <\/b><\/span><\/span><\/span>grandeza expressa por uma fun\u00e7\u00e3o matem\u00e1tica:<\/b><\/span><\/span><\/span><\/p>\n

Calcule a f\u00f3rmula dimensional das seguintes grandezas f\u00edsicas derivadas:<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Peso (P)<\/b><\/span><\/span><\/span> \"\" <\/b><\/span><\/span><\/span>unidade no SI \"\" <\/b><\/span><\/span><\/span>[newton (N)]<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

Press\u00e3o (P<\/b><\/span><\/span><\/span>r<\/b><\/span><\/span><\/sub><\/span>) \"\" unidade no SI \"\" [pascal (P<\/b><\/span><\/span><\/span>a<\/b><\/span><\/span><\/sub><\/span>)] <\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

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

\"\"<\/p>\n

Energia, Trabalho (W)<\/b><\/span><\/span><\/span> \"\" <\/b><\/span><\/span><\/span>unidade no SI \"\"<\/b><\/span><\/span><\/span> <\/b>[joule J)] <\/b><\/span><\/span><\/span> <\/b><\/p>\n

\u00a0\"\"<\/span>Trabalho \u00e9 varia\u00e7\u00e3o de energia\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>W = F.d = mad = <\/b><\/span><\/span><\/span>(kg).(m\/s<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>).m\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 \u00a0<\/b><\/span><\/span><\/span>[W] = (M).(L.T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>).L<\/b><\/span><\/span><\/span> \"\"<\/b><\/span><\/span><\/span><\/p>\n

[W] = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span><\/p>\n

ou<\/b><\/span><\/span><\/span>, energia cin\u00e9tica<\/b><\/span><\/span><\/span> \"\" <\/b><\/span><\/span><\/span>E<\/b><\/span><\/span><\/span>c<\/b><\/span><\/span><\/sub><\/span> = mV<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/2 =<\/b><\/span><\/span><\/span> m.(\u0394S\/\u0394t)\/ 2 = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span> \/ T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span> \"\"<\/b><\/span><\/span><\/span> <\/b>[E<\/b><\/span><\/span><\/span>c<\/b><\/span><\/span><\/sub><\/span>] = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2 \"\"<\/b><\/span><\/span><\/sup><\/span><\/p>\n

ou ainda, <\/b><\/span><\/span><\/span>energia potencial gravitacional \"\" E<\/b><\/span><\/span><\/span>p<\/b><\/span><\/span><\/sub><\/span> =mgh = <\/b><\/span><\/span><\/span>(kg).(m\/s<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>).m\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 \u00a0<\/b><\/span><\/span><\/span>[E<\/b><\/span><\/span><\/span>p<\/b><\/span><\/span><\/sub><\/span>] = (M).(L.T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>).L<\/b><\/span><\/span><\/span> \"\"<\/b><\/span><\/span><\/span><\/p>\n

[E<\/b><\/span><\/span><\/span>p<\/b><\/span><\/span><\/sub><\/span>] = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span><\/p>\n

\"\"<\/p>\n

Segundo <\/b><\/span><\/span><\/span>Einstein,<\/b><\/span><\/span><\/span> uma <\/b><\/span><\/span><\/span>luz de frequ\u00eancia f <\/b><\/span><\/span><\/span>pode ser considerada como sendo <\/b><\/span><\/span><\/span>constitu\u00edda de f\u00f3tons, part\u00edculas com massa em <\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>repouso nula e com energia E = hf, <\/b><\/span><\/span><\/span>sendo <\/b><\/span><\/span><\/span>h <\/b><\/span><\/span><\/span>a <\/b><\/span><\/span><\/span>constante de Planck. <\/b><\/span><\/span><\/span>Calcule a <\/b><\/span><\/span><\/span>dimens\u00e3o da constante h.<\/b><\/span><\/span><\/span>\"\"<\/p>\n

E = hf\u00a0 \"\"<\/b><\/span><\/span><\/span> <\/b>h = E\/f \"\"<\/b><\/span><\/span><\/span> f = 1\/T \"\" E = (F.d)\/(1\/T) = m.a.d.T<\/b><\/span><\/span><\/span>-1 <\/b><\/span><\/span><\/sup><\/span>= kg.m\/s<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.m.s<\/b><\/span><\/span><\/span>-1<\/b><\/span><\/span><\/sup><\/span>\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>h = kg.m<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.s<\/b><\/span><\/span><\/span>-1<\/b><\/span><\/span><\/sup><\/span>\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>[h] = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-1<\/b><\/span><\/span><\/sup><\/span><\/p>\n

\"\"<\/p>\n

Se uma <\/b><\/span><\/span><\/span>equa\u00e7\u00e3o que traduz uma lei f\u00edsica \u00e9 <\/b><\/span><\/span><\/span>homog\u00eanea,<\/b><\/span><\/span><\/span> as <\/b><\/span><\/span><\/span>parcelas <\/b><\/span><\/span><\/span>que <\/b><\/span><\/span><\/span>constituem<\/b><\/span><\/span><\/span> os <\/b><\/span><\/span><\/span>dois<\/b><\/span><\/span><\/span><\/p>\n

membros da igualdade fornecem os<\/b><\/span><\/span><\/span>\"\"<\/p>\n

mesmos s\u00edmbolos dimensionais. <\/b><\/span><\/span><\/span><\/p>\n

Assim , na <\/b><\/span><\/span><\/span>express\u00e3o<\/b><\/span><\/span><\/span> da <\/b><\/span><\/span><\/span>energia relativ\u00edstica de Einstein E = m<\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/sub><\/span>.c<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>, <\/b><\/span><\/span><\/span>onde\u00a0<\/b><\/span><\/span><\/span>E<\/b><\/span><\/span><\/span>\u00a0representa a energia relativ\u00edstica da part\u00edcula em repouso,\u00a0<\/b><\/span><\/span><\/span>m<\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/sub><\/span>\u00a0a massa de repouso da part\u00edcula e\u00a0<\/b><\/span><\/span><\/span>c<\/b><\/span><\/span><\/span>\u00a0a velocidade da luz no v\u00e1cuo, <\/b><\/span><\/span><\/span>as <\/b><\/span><\/span><\/span>unidades dimensionais de\u00a0<\/b><\/span><\/span><\/span>E (calculadas acima <\/b><\/span><\/span><\/span>[E<\/b><\/span><\/span><\/span>c<\/b><\/span><\/span><\/sub><\/span>] = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span>)<\/b><\/span><\/span><\/span> <\/b>\u00a0<\/b><\/span><\/span><\/span>e de <\/b><\/span><\/span><\/span>E = <\/b><\/span><\/span><\/span>m<\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/sub><\/span>.c<\/b><\/span><\/span><\/span>2\u00a0<\/b><\/span><\/span><\/sup><\/span>devem ser iguais. Prove essa igualdade.<\/b><\/span><\/span><\/span><\/p>\n

Energia, Trabalho (W)<\/b><\/span><\/span><\/span> \"\" <\/b><\/span><\/span><\/span>unidade no SI \"\"<\/b><\/span><\/span><\/span> <\/b>[joule J)] <\/b><\/span><\/span><\/span> \"\"\u00a0 <\/b><\/span><\/span><\/span>W = F.d = mad = <\/b><\/span><\/span><\/span>(kg).(m\/s<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>).m\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 \u00a0<\/b><\/span><\/span><\/span>[W] = (M).(L.T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>).L<\/b><\/span><\/span><\/span> \"\" <\/b><\/span><\/span><\/span>[W] = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span> \"\"<\/b><\/span><\/span><\/span> <\/b>ou<\/b><\/span><\/span><\/span>, energia cin\u00e9tica<\/b><\/span><\/span><\/span> \"\" <\/b><\/span><\/span><\/span>E<\/b><\/span><\/span><\/span>c<\/b><\/span><\/span><\/sub><\/span> = mV<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/2 =<\/b><\/span><\/span><\/span> m.(\u0394S\/\u0394t)\/ 2 = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span> \/ T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span> \"\"<\/b><\/span><\/span><\/span> <\/b>[E<\/b><\/span><\/span><\/span>c<\/b><\/span><\/span><\/sub><\/span>] = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2 \"\"<\/b><\/span><\/span><\/sup><\/span> <\/b>ou ainda, <\/b><\/span><\/span><\/span>energia potencial gravitacional \"\" E<\/b><\/span><\/span><\/span>p<\/b><\/span><\/span><\/sub><\/span> =mgh = <\/b><\/span><\/span><\/span>(kg).(m\/s<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>).m\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 \u00a0<\/b><\/span><\/span><\/span>[E<\/b><\/span><\/span><\/span>p<\/b><\/span><\/span><\/sub><\/span>] = (M).(L.T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>).L<\/b><\/span><\/span><\/span> \"\"<\/b><\/span><\/span><\/span><\/p>\n

[E<\/b><\/span><\/span><\/span>p<\/b><\/span><\/span><\/sub><\/span>] = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span><\/p>\n

Energia relativ\u00edstica \"\"<\/b><\/span><\/span><\/span> <\/b>E<\/b><\/span><\/span><\/span>r<\/b><\/span><\/span><\/sub><\/span> = m<\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/sub><\/span>.c<\/b><\/span><\/span><\/span>2 \"\"<\/b><\/span><\/span><\/sup><\/span> <\/b>[ m<\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/sub><\/span>] <\/b><\/span><\/span><\/span>= kg <\/b><\/span><\/span><\/span>= M<\/b><\/span><\/span><\/span>\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>c = m\/s\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0\u00a0 <\/b><\/span><\/span><\/span>c<\/b><\/span><\/span><\/span>2 <\/b><\/span><\/span><\/sup><\/span>= (m\/s)<\/b><\/span><\/span><\/span>2 <\/b><\/span><\/span><\/sup><\/span>= m<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.s<\/b><\/span><\/span><\/span>-2 \"\"<\/b><\/span><\/span><\/sup><\/span><\/p>\n

[c<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>] = L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span>\u00a0 \"\"<\/b><\/span><\/span><\/span> <\/b>[ m.c<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>] = M.L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span>\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 portanto <\/b><\/span><\/span><\/span>W, Ec, Ep, E<\/b><\/span><\/span><\/span>r<\/b><\/span><\/span><\/sub><\/span> e m.c<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0<\/b><\/span><\/span><\/span>possuem <\/b><\/span><\/span><\/span>s\u00edmbolos <\/b><\/span><\/span><\/span>dimensionalmente iguais e s\u00e3o homog\u00eaneas.<\/b><\/span><\/span><\/span><\/p>\n

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

\"\" Quando voc\u00ea est\u00e1 <\/b><\/span><\/span><\/span>estudando um determinado fen\u00f4meno, <\/b><\/span><\/span><\/span>voc\u00ea <\/b><\/span><\/span><\/span>analisa as vari\u00e1veis que participam dele<\/b><\/span><\/span><\/span>,<\/b><\/span><\/span><\/span> <\/b>de modo <\/b><\/span><\/span><\/span>que elas forne\u00e7am dados relevantes sobre o\u00a0 mesmo. <\/b><\/span><\/span><\/span><\/p>\n

Estas <\/b><\/span><\/span><\/span>vari\u00e1veis<\/b><\/span><\/span><\/span> se <\/b><\/span><\/span><\/span>denominam<\/b><\/span><\/span><\/span>\u00a0grandezas f\u00edsicas\u00a0<\/b><\/span><\/span><\/span>que correspondem a\u00a0<\/b><\/span><\/span><\/span>tudo que voc\u00ea pode contar, medir, pesar, enfim, enumerar.<\/b><\/span><\/span><\/span><\/p>\n

\"\" A<\/b><\/span><\/span><\/span> medi\u00e7\u00e3o \u00e9 a <\/b><\/span><\/span><\/span>t\u00e9cnica por meio da qual voc\u00ea atribui um n\u00famero <\/b><\/span><\/span><\/span>a uma grandeza f\u00edsica e, para avali\u00e1-la, voc\u00ea deve compar\u00e1-la com outra similar tomada como padr\u00e3o, denominada <\/b><\/span><\/span><\/span>unidade.<\/b><\/span><\/span><\/span><\/p>\n

\"\" Sistema Internaconal de Unidades (SI)<\/b><\/span><\/span><\/span><\/p>\n

Nele <\/b><\/span><\/span><\/span>constam<\/b><\/span><\/span><\/span> as <\/b><\/span><\/span><\/span>sete grandezas de base adotadas pela Organiza\u00e7\u00e3o Internacional de Normatiza\u00e7\u00e3o (ISO) <\/b><\/span><\/span><\/span>que s\u00e3o: <\/b><\/span><\/span><\/span>comprimento, massa , tempo, corrente el\u00e9trica, temperatura termodin\u00e2mica, quantidade de mat\u00e9ria e intensidade luminosa, cujas caracter\u00edsticas est\u00e3o na tabela ao lado .<\/b><\/span><\/span><\/span>\"\"<\/p>\n

\"\"<\/p>\n

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

\"\" Conhecer as Regras para se escrever as unidades do SI, enunciadas na teoria acima.<\/b><\/span><\/span><\/span><\/p>\n

\"\" Qualquer <\/b><\/span><\/span><\/span>equa\u00e7\u00e3o matem\u00e1tica <\/b><\/span><\/span><\/span>que <\/b><\/span><\/span><\/span>representa uma situa\u00e7\u00e3o f\u00edsica <\/b><\/span><\/span><\/span>qualquer precisa, <\/b><\/span><\/span><\/span>al\u00e9m de um n\u00famero que a quantifique de uma<\/b><\/span><\/span><\/span> <\/b>unidade de medida que fa\u00e7a a classifica\u00e7\u00e3o qualitativa<\/b><\/span><\/span><\/span>.<\/b><\/span><\/span><\/span>\"\"<\/p>\n

\"\" Todas<\/b><\/span><\/span><\/span> as <\/b><\/span><\/span><\/span>grandezas f\u00edsicas <\/b><\/span><\/span><\/span>podem ser <\/b><\/span><\/span><\/span>analisadas dimensionalmente<\/b><\/span><\/span><\/span> atrav\u00e9s de <\/b><\/span><\/span><\/span>tr\u00eas unidades <\/b><\/span><\/span><\/span>que voc\u00ea deve <\/b><\/span><\/span><\/span>tomar como par\u00e2metros <\/b><\/span><\/span><\/span>que s\u00e3o: <\/b><\/span><\/span><\/span>Comprimento (L), Tempo (T) e massa (M).<\/b><\/span><\/span><\/span> <\/b><\/p>\n

As <\/b><\/span><\/span><\/span>outras unidades de medidas <\/b><\/span><\/span><\/span>de <\/b><\/span><\/span><\/span>grandezas derivadas ter\u00e3o suas unidades de medidas provenientes dessas tr\u00eas.\u00a0<\/b><\/span><\/span><\/span><\/p>\n

Costuma-se <\/b><\/span><\/span><\/span>adotar as grandezas fundamentais do S.I.<\/b><\/span><\/span><\/span> para se <\/b><\/span><\/span><\/span>escreverem as equa\u00e7\u00f5es dimensionais.<\/b><\/span><\/span><\/span><\/p>\n

\"\" Uma <\/b><\/span><\/span><\/span>grandeza dimensional <\/b><\/span><\/span><\/span>\u00e9 <\/b><\/span><\/span><\/span>representada por <\/b><\/span><\/span><\/span>colchetes<\/b><\/span><\/span><\/span> <\/b>e voc\u00ea <\/b><\/span><\/span><\/span>pode expressa-la em fun\u00e7\u00e3o das grandezas fundamentais. <\/b><\/span><\/span><\/span><\/p>\n

Por exemplo, <\/b><\/span><\/span><\/span>uma grandeza f\u00edsica (F), <\/b><\/span><\/span><\/span>que <\/b><\/span><\/span><\/span>depende da massa, do comprimento e do tempo, tem sua equa\u00e7\u00e3o dimensional escrita da seguinte maneira:<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\"<\/p>\n

\"\" Analisar detalhadamente <\/b><\/span><\/span><\/span>os exerc\u00edcios acima sobre <\/b><\/span><\/span><\/span>como <\/b><\/span><\/span><\/span>calcular a f\u00f3rmula dimensional de algumas grandezas f\u00edsicas derivadas:<\/b><\/span><\/span><\/span><\/p>\n

\"\" F\u00f3rmulas dimensionais de algumas grandezas f\u00edsicas derivadas<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Selecionei alguns exerc\u00edcios para que voc\u00ea possa se aprofundar no assunto <\/b><\/span><\/span><\/span><\/p>\n

Grandezas f\u00edsicas,<\/b><\/span><\/span><\/span> <\/b>Sistema de Unidades e An\u00e1lise Dimensional<\/b><\/span><\/span><\/span><\/p>\n

01-(UERJ-RJ <\/b><\/span><\/span><\/span>Admita que, em um <\/b><\/span><\/span><\/span>determinado lago, a cada 40 cm de profundidade, <\/b><\/span><\/span><\/span>a <\/b><\/span><\/span><\/span>intensidade de luz \u00e9 reduzida em 20%, de acordo com a equa\u00e7\u00e3o<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

na qual <\/b><\/span><\/span><\/span>I \u00e9 a intensidade da luz em uma profundidade h, em cent\u00edmetros, e I<\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/sub><\/span>\u00a0\u00e9 a intensidade na superf\u00edcie.<\/b><\/span><\/span><\/span>\"\"<\/p>\n

Um <\/b><\/span><\/span><\/span>nadador verificou, ao mergulhar nesse lago, <\/b><\/span><\/span><\/span>que a <\/b><\/span><\/span><\/span>intensidade da luz, em um ponto P<\/b><\/span><\/span><\/span>, \u00e9 de <\/b><\/span><\/span><\/span>32% daquela observada<\/b><\/span><\/span><\/span><\/p>\n

na superf\u00edcie. <\/b><\/span><\/span><\/span>A <\/b><\/span><\/span><\/span>profundidade do ponto P, em metros<\/b><\/span><\/span><\/span>,<\/b><\/span><\/span><\/span><\/p>\n

considerando log2 = 0,3, <\/b><\/span><\/span><\/span>equivale a:<\/b><\/span><\/span><\/span><\/p>\n

a) <\/b><\/span><\/span><\/span>0,64\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<\/b><\/span><\/span><\/span><\/p>\n

b) <\/b><\/span><\/span><\/span>1,8\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<\/b><\/span><\/span><\/span><\/p>\n

c) <\/b><\/span><\/span><\/span>2,0\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<\/b><\/span><\/span><\/span><\/p>\n

d) <\/b><\/span><\/span><\/span>3,2\u00a0<\/b><\/span><\/span><\/span><\/p>\n

Resolu\u00e7\u00e3o:\u00a0<\/b><\/span><\/span><\/span><\/p>\n

Observe que <\/b><\/span><\/span><\/span>32% da luminosidade da superf\u00edcie vale\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>I = (32\/100).l\u00ado\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>I = l<\/b><\/span><\/span><\/span>0<\/b><\/span><\/span><\/sub><\/span>.0,8<\/b><\/span><\/span><\/span>h\/40<\/b><\/span><\/span><\/sup><\/span>\u00a0\u00a0\"\"<\/b><\/span><\/span><\/span><\/p>\n

(32\/100)l0 = l0.0,8h<\/b><\/span><\/span><\/span>\u00a0h\/40<\/b><\/span><\/span><\/sup><\/span>\u00a0 \"\" l<\/b><\/span><\/span><\/span>0<\/b><\/span><\/span><\/sub><\/span>.0,8<\/b><\/span><\/span><\/span>h\/40<\/b><\/span><\/span><\/sup><\/span>\u00a0\u00a0 \"\"\u00a0 (32\/100)l0 = l0.0,8h<\/b><\/span><\/span><\/span>\u00a0h\/40<\/b><\/span><\/span><\/sup><\/span>\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>cancelando I0\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>(32\/100) = 0,8<\/b><\/span><\/span><\/span>\u00a0h\/40<\/b><\/span><\/span><\/sup><\/span>. <\/b><\/span><\/span><\/span><\/p>\n

Fazendo o logaritmo em ambos os lados da igualdade\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>log(32\/100) = log[0,8<\/b><\/span><\/span><\/span>\u00a0h\/40<\/b><\/span><\/span><\/sup><\/span>)].\u00a0 \u00a0<\/b><\/span><\/span><\/span><\/p>\n

Propriedades do log\u00a0 \"\"\u00a0\u00a0 log32 – log100 = (h\/40).log0,8\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 log2<\/b><\/span><\/span><\/span>5<\/b><\/span><\/span><\/sup><\/span>\u00a0– log10\u00b2 = (h\/40).log(8\/10)\u00a0 \"\"<\/b><\/span><\/span><\/span><\/p>\n

5log2 – 2log10 = (h\/40).(log8 – log10)\u00a0 \"\" <\/b><\/span><\/span><\/span>5.0,3 – 2 = (h\/40).(log2\u00b3 – 1)\u00a0 \"\"\u00a0 1,5 – 2 = (h\/40).(3log2 – 1)\u00a0 \"\"\u00a0 – 0,5 = (h\/40).(3.0,3 – 1)\u00a0 \"\"\u00a0<\/b><\/span><\/span><\/span> – 0,5 = (h\/40).(- 0,1)\u00a0 \"\"\u00a0 (- 0,5)\/(- 0,1) = h\/40\u00a0 \"\"\u00a0 5 = h\/40\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>h = 5.40\u00a0 \"\"\u00a0 h = 200 cm=2,0m\u00a0 \"\"\u00a0 <\/b><\/span><\/span><\/span>R- C<\/b><\/span><\/span><\/span><\/p>\n

02<\/b><\/span><\/span><\/span>–<\/b><\/span><\/span><\/span>Um <\/b><\/span><\/span><\/span>projetista de m\u00e1quinas de lavar roupas <\/b><\/span><\/span><\/span>estava interessado em <\/b><\/span><\/span><\/span>determinar o volume de \u00e1gua utilizado por uma dada lavadora de roupas <\/b><\/span><\/span><\/span>durante o seu funcionamento, de modo a <\/b><\/span><\/span><\/span>otimizar a economia de \u00e1gua por parte do aparelho. <\/b><\/span><\/span><\/span>Ele percebeu que o <\/b><\/span><\/span><\/span>volume V<\/b><\/span><\/span><\/span> de \u00e1gua necess\u00e1rio para uma lavagem <\/b><\/span><\/span><\/span>depende da massa m das roupas<\/b><\/span><\/span><\/span> a serem lavadas, <\/b><\/span><\/span><\/span>do<\/b><\/span><\/span><\/span> <\/b>intervalo de tempo \u2206t <\/b><\/span><\/span><\/span>que esta m\u00e1quina leva para encher de \u00e1gua <\/b><\/span><\/span><\/span>e da <\/b><\/span><\/span><\/span>press\u00e3o P da \u00e1gua na tubula\u00e7\u00e3o <\/b><\/span><\/span><\/span>que alimenta esta m\u00e1quina de lavar.<\/b><\/span><\/span><\/span>\"\"<\/p>\n

Assim, ele <\/b><\/span><\/span><\/span>expressou o volume de \u00e1gua atrav\u00e9s da fun\u00e7\u00e3o <\/b><\/span><\/span><\/span>V = k<\/b><\/span><\/span><\/span> <\/b>m<\/b><\/span><\/span><\/span>a<\/b><\/span><\/span><\/sup><\/span>\u00a0(\u2206t)<\/b><\/span><\/span><\/span>b<\/b><\/span><\/span><\/sup><\/span>\u00a0P<\/b><\/span><\/span><\/span>n<\/b><\/span><\/span><\/sup><\/span>, <\/b><\/span><\/span><\/span>onde <\/b><\/span><\/span><\/span>k<\/b><\/span><\/span><\/span> \u00e9 uma constante adimensional e a, b e n s\u00e3o coeficientes a serem determinados.<\/b><\/span><\/span><\/span><\/p>\n

Calcule os <\/b><\/span><\/span><\/span>valores de <\/b><\/span><\/span><\/span>a, b e n<\/b><\/span><\/span><\/span> para que a equa\u00e7\u00e3o seja dimensionalmente correta.<\/b><\/span><\/span><\/span><\/p>\n

Resolu\u00e7\u00e3o:\u00a0<\/b><\/span><\/span><\/span><\/p>\n

[V] = K.M<\/b><\/span><\/span><\/span>a<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>b<\/b><\/span><\/span><\/sup><\/span>.(F\/A)<\/b><\/span><\/span><\/span>n <\/b><\/span><\/span><\/sup><\/span>= K.M<\/b><\/span><\/span><\/span>a<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>b<\/b><\/span><\/span><\/sup><\/span>.(M.T<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span>.L<\/b><\/span><\/span><\/span>-1<\/b><\/span><\/span><\/sup><\/span>)<\/b><\/span><\/span><\/span>n<\/b><\/span><\/span><\/sup><\/span>\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 [V] = [L.L.L]<\/b><\/span><\/span><\/span> <\/b>= [L]<\/b><\/span><\/span><\/span>3 \"\"<\/b><\/span><\/span><\/sup><\/span> <\/b>[L]<\/b><\/span><\/span><\/span>3 <\/b><\/span><\/span><\/sup><\/span>= K.M<\/b><\/span><\/span><\/span>a<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>b<\/b><\/span><\/span><\/sup><\/span>.M<\/b><\/span><\/span><\/span>n<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2n<\/b><\/span><\/span><\/sup><\/span>.L<\/b><\/span><\/span><\/span>-1n<\/b><\/span><\/span><\/sup><\/span> \"\" <\/b><\/span><\/span><\/sub><\/span>L<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>\/<\/b><\/span><\/span><\/span><\/p>\n

L<\/b><\/span><\/span><\/span>-1n<\/b><\/span><\/span><\/sup><\/span> <\/b>= K.M<\/b><\/span><\/span><\/span>a<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>b<\/b><\/span><\/span><\/sup><\/span>.M<\/b><\/span><\/span><\/span>n<\/b><\/span><\/span><\/sup><\/span>.T<\/b><\/span><\/span><\/span>-2n<\/b><\/span><\/span><\/sup><\/span> \"\" <\/b><\/span><\/span><\/sub><\/span>(L<\/b><\/span><\/span><\/span>3+n<\/b><\/span><\/span><\/sup><\/span>).(T<\/b><\/span><\/span><\/span>2n<\/b><\/span><\/span><\/sup><\/span>).M<\/b><\/span><\/span><\/span>0<\/b><\/span><\/span><\/sup><\/span> = L<\/b><\/span><\/span><\/span>0<\/b><\/span><\/span><\/sup><\/span>. K.(M<\/b><\/span><\/span><\/span>a + n<\/b><\/span><\/span><\/sup><\/span>).T<\/b><\/span><\/span><\/span>b<\/b><\/span><\/span><\/sup><\/span>\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>3 + n = 0<\/b><\/span><\/span><\/span>\u00a0 \"\"\u00a0<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>n = – 3\u00a0<\/b><\/span><\/span><\/span>\u00a0\"\"\u00a0\u00a0 <\/b><\/span><\/span><\/span>2n = b\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>2.<\/b><\/span><\/span><\/span><\/p>\n

(-3) = b\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span>b = – 6\u00a0<\/b><\/span><\/span><\/span>\u00a0\"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>a + n = 0\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>a \u2013 3 = 0\u00a0 \"\"<\/b><\/span><\/span><\/span>\u00a0 <\/b><\/span><\/span><\/span>a = 3.<\/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":"

Introdu\u00e7\u00e3o \u00e0 F\u00edsica Sistema de Unidades \u2013 An\u00e1lise Dimensional \u00a0 Grandezas F\u00edsicas Quando voc\u00ea est\u00e1 estudando um determinado fen\u00f4meno, voc\u00ea analisa as vari\u00e1veis que participam dele, de modo que elas forne\u00e7am dados relevantes sobre o\u00a0 mesmo. Estas vari\u00e1veis se denominam\u00a0grandezas f\u00edsicas\u00a0que correspondem a\u00a0tudo que voc\u00ea pode contar, medir, pesar, enfim, enumerar. Escolhida a grandeza f\u00edsica a ser analisada no fen\u00f4meno<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":129,"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-2456","page","type-page","status-publish","hentry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2456","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=2456"}],"version-history":[{"count":2,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2456\/revisions"}],"predecessor-version":[{"id":2465,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/2456\/revisions\/2465"}],"up":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/129"}],"wp:attachment":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/media?parent=2456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}