{"id":1414,"date":"2015-10-07T02:00:07","date_gmt":"2015-10-07T02:00:07","guid":{"rendered":"http:\/\/fisicaevestibular.com.br\/novo\/?page_id=1414"},"modified":"2024-08-23T14:22:36","modified_gmt":"2024-08-23T14:22:36","slug":"resolucao-comentada-dos-exercicios-de-vestibulares-sobre-leis-de-kepler-da-gravitacao-universal","status":"publish","type":"page","link":"https:\/\/fisicaevestibular.com.br\/novo\/mecanica\/gravitacao\/leis-de-kepler-gravitacao-universal\/resolucao-comentada-dos-exercicios-de-vestibulares-sobre-leis-de-kepler-da-gravitacao-universal\/","title":{"rendered":"Leis de Kepler da Gravita\u00e7\u00e3o Universal – Resolu\u00e7\u00e3o"},"content":{"rendered":"

Resolu\u00e7\u00e3o comentada dos exerc\u00edcios de vestibulares sobre<\/b><\/span><\/span><\/span><\/p>\n

leis de Kepler da Gravita\u00e7\u00e3o Universal<\/b><\/span><\/span><\/span><\/p>\n

 <\/p>\n

 <\/p>\n

01-<\/b><\/span><\/span><\/span> C (vide teoria)<\/b><\/span><\/span><\/span><\/p>\n

O2-<\/b><\/span><\/span><\/span> A (vide teoria)<\/b><\/span><\/span><\/span><\/p>\n

03-<\/b><\/span><\/span><\/span> (01 + 02 +08)=11(vide teoria)<\/b><\/span><\/span><\/span><\/p>\n

04-<\/b><\/span><\/span><\/span> E (vide teoria)<\/b><\/span><\/span><\/span><\/p>\n

05- <\/b><\/span><\/span><\/span>Regra de tr\u00eas\u00a0 —\u00a0 1 volta \u2013 12 meses\u00a0—\u00a0\u00a0\u00a0\u00bc volta – T\u00a0 —\u00a0 T=3 meses R- D<\/b><\/span><\/span><\/span><\/p>\n

06-<\/b><\/span><\/span><\/span> a) De 1<\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/sup><\/span> de Abril a 30 de Maio = 60dias\u00a0 —\u00a0 regra de tr\u00eas\u00a0 —\u00a0 365 dias \u2013 6,98.10<\/b><\/span><\/span><\/span>22<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 60 dias\u00a0 — A\u00a0 —\u00a0 A=69,8.6.10<\/b><\/span><\/span><\/span>22<\/b><\/span><\/span><\/sup><\/span>\/365\u00a0 —\u00a0 A\u00bb1,14.10<\/b><\/span><\/span><\/span>22<\/b><\/span><\/span><\/sup><\/span>m<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/b><\/span><\/span><\/span><\/p>\n

b) Segunda lei de Kepler (lei das \u00e1reas)<\/b><\/span><\/span><\/span><\/p>\n

07-<\/b><\/span><\/span><\/span> T<\/b><\/span><\/span><\/span>OP<\/b><\/span><\/span><\/sub><\/span>=T<\/b><\/span><\/span><\/span>MN<\/b><\/span><\/span><\/sub><\/span>, pela segunda lei de Kepler, e V<\/b><\/span><\/span><\/span>MN<\/b><\/span><\/span><\/sub><\/span>(af\u00e9lio)<V<\/b><\/span><\/span><\/span>OP<\/b><\/span><\/span><\/sub><\/span>(peri\u00e9lio), como conseq\u00fc\u00eancia dessa lei.<\/b><\/span><\/span><\/span><\/p>\n

R- B<\/b><\/span><\/span><\/span><\/p>\n

08-<\/b><\/span><\/span><\/span> Energia cin\u00e9tica=movimento=velocidade R- E<\/b><\/span><\/span><\/span><\/p>\n

09-<\/b><\/span><\/span><\/span> T<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/R<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>3<\/b><\/span><\/span><\/sup><\/span>= T<\/b><\/span><\/span><\/span>P<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/R<\/b><\/span><\/span><\/span>P<\/b><\/span><\/span><\/sub><\/span>3<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 (1)<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span> ano\/(1)<\/b><\/span><\/span><\/span>3 <\/b><\/span><\/span><\/sup><\/span>ua = T<\/b><\/span><\/span><\/span>P<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/(2,5)<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>ua\u00a0 —\u00a0 T<\/b><\/span><\/span><\/span>p<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>=(2,5)<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 T<\/b><\/span><\/span><\/span>P<\/b><\/span><\/span><\/sub><\/span>=2,5.\u221a(25\/10)\u00a0 —\u00a0 <\/b><\/span><\/span><\/span>\u00a0\u00a0<\/b><\/span><\/span><\/sup><\/span>T<\/b><\/span><\/span><\/span>P<\/b><\/span><\/span><\/sub><\/span> \u00bb<\/b><\/span><\/span><\/span> <\/b>4 anos terrestres\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/b><\/span><\/span><\/span><\/p>\n

b) Como quanto maior a dist\u00e2ncia do planeta ao Sol, maior \u00e9 o ano desse planeta (terceira lei de Kepler, lei dos per\u00edodos), portanto ser\u00e1 mais curto.<\/b><\/span><\/span><\/span><\/p>\n

10- <\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/R<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>3<\/b><\/span><\/span><\/sup><\/span>=T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/R<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>3<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 (24)<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/(42)<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>.10<\/b><\/span><\/span><\/span>9<\/b><\/span><\/span><\/sup><\/span>=T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/(105)<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>.10<\/b><\/span><\/span><\/span>6<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 T=3\u221a2\u00a0 —\u00a0 R- A<\/b><\/span><\/span><\/span><\/p>\n

11-<\/b><\/span><\/span><\/span> 32<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/1<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>=32<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>.8<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/R<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 R<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>=(2<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>)<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 R=4u<\/b><\/span><\/span><\/span><\/p>\n

12-<\/b><\/span><\/span><\/span> C (vide teoria)<\/b><\/span><\/span><\/span><\/p>\n

13-<\/b><\/span><\/span><\/span> <\/b>Observe a figura abaixo\u00a0 —\u00a0 M: massa do Sol\u00a0 —\u00a0 m: massa do planeta\u00a0 —\u00a0 r: raio da \u00f3rbita\u00a0 —\u00a0 \"\": velocidade orbital do planeta;<\/b><\/span><\/span><\/span><\/p>\n

\"\": for\u00e7a gravitacional;\u00a0 —\u00a0 \"\": for\u00e7a resultante centr\u00edpeta\u00a0 —\u00a0 3\u00aa lei de Kepler afirma que: \u201co quadrado do per\u00edodo de transla\u00e7\u00e3o<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

(T) do planeta \u00e9 diretamente proporcional ao cubo do raio de sua \u00f3rbita\u201d\u00a0 — T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span> = k r<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 como o movimento \u00e9 circular uniforme, a for\u00e7a gravitacional comporta-se como resultante centr\u00edpeta\u00a0 —\u00a0 F<\/b><\/span><\/span><\/span>G<\/b><\/span><\/span><\/sub><\/span>=R<\/b><\/span><\/span><\/span>C<\/b><\/span><\/span><\/sub><\/span>\u00a0 —\u00a0 GMm\/r<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>=mV<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/r (I)\u00a0 —\u00a0 V=\u0394S\/\u0394t=2\u03c0r\/T\u00a0 —\u00a0 V<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>=4\u03c0<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>r<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span> (II)\u00a0 —\u00a0 (II) em (I)\u00a0 —\u00a0 4\u03c0<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>r<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>=GM\/r\u00a0 —\u00a0 r<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>\/T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>=GM\/4\u03c0<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>=4\u03c0<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>r<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>\/GM\u00a0 —\u00a0 observe que G,M e \u03c0 s\u00e3o todos constantes constituindo uma s\u00f3 constante k\u00a0 — <\/b><\/span><\/span><\/span>\u00a0T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>=kr<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span><\/p>\n

14-<\/b><\/span><\/span><\/span> <\/b>As tr\u00eas afirma\u00e7\u00f5es s\u00e3o, nessa mesma ordem, a 1\u00aa, 2\u00aa e 3\u00aa Leis de Kepler\u00a0 —\u00a0 <\/b><\/span><\/span><\/span>R- E<\/b><\/span><\/span><\/span><\/p>\n

15-<\/b><\/span><\/span><\/span> <\/b>I. Correta\u00a0 —\u00a0 \u00a0um ano luz corresponde \u00e0 dist\u00e2ncia percorrida pela luz em ano, <\/b><\/span><\/span><\/span>no v\u00e1cuo<\/b><\/span><\/span><\/span>.<\/b><\/span><\/span><\/span><\/p>\n

II. Correta.<\/b><\/span><\/span><\/span><\/p>\n

III. Errada\u00a0 —\u00a0 m\/s<\/b><\/span><\/span><\/span>2 <\/b><\/span><\/span><\/sup><\/span>\u00a0\u00e9 uma das unidades de acelera\u00e7\u00e3o<\/b><\/span><\/span><\/span><\/p>\n

R- D<\/b><\/span><\/span><\/span><\/p>\n

16-<\/b><\/span><\/span><\/span> <\/b>Da 3\u00aa lei de Kepler: o quadrado do per\u00edodo de transla\u00e7\u00e3o (ano do planeta) \u00e9 diretamente proporcional ao cubo do raio m\u00e9dio da \u00f3rbita\u00a0 — T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span> = k r<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>, podemos concluir que quanto mais distante do Sol orbitar o planeta, mais longo \u00e9 seu ano\u00a0 —\u00a0 portanto, os chamados planetas internos, Merc\u00fario e V\u00eanus, t\u00eam anos mais curtos do que o ano terrestre\u00a0 — <\/b><\/span><\/span><\/span>\u00a0R- D <\/b><\/span><\/span><\/span>\u00a0<\/b><\/span><\/span><\/span><\/p>\n

17-<\/b><\/span><\/span><\/span> <\/b>1. Falsa\u00a0 —\u00a0 quando a Terra vai do af\u00e9lio para o peri\u00e9lio, aumenta o m\u00f3dulo da velocidade, e quando vai do peri\u00e9lio para o af\u00e9lio, diminui o m\u00f3dulo da velocidade.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

2. Verdadeira\u00a0 —\u00a0 de acordo com o princ\u00edpio da a\u00e7\u00e3o-rea\u00e7\u00e3o (3\u00aa lei de Newton), a\u00e7\u00e3o e rea\u00e7\u00e3o t\u00eam sempre a mesma intensidade.<\/b><\/span><\/span><\/span><\/p>\n

R- C<\/b><\/span><\/span><\/span><\/p>\n

18-<\/b><\/span><\/span><\/span> <\/b>Observe pela figura abaixo que, se o efeito da gravita\u00e7\u00e3o deixasse de existir, o planeta sairia pela tangente entrando em<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

movimento retil\u00edneo uniforme, percorrendo sempre a mesma dist\u00e2ncia b em temos iguais, portanto varrendo a mesma \u00e1rea\u00a0 —\u00a0 \u00a0A = b\/2\u00a0 —<\/b><\/span><\/span><\/span>\u00a0 R- A<\/b><\/span><\/span><\/span><\/p>\n

19-<\/b><\/span><\/span><\/span> <\/b>1. Falsa. O campo magn\u00e9tico da agulha existe, por\u00e9m, para que essa agulha sofra alguma deflex\u00e3o, ela tem que sofrer influ\u00eancia de outro campo magn\u00e9tico.<\/b><\/span><\/span><\/span><\/p>\n

2. Verdadeira. Se a agulha da b\u00fassola n\u00e3o sofre deflex\u00e3o, \u00e9 porque ela n\u00e3o est\u00e1 em presen\u00e7a de algum campo magn\u00e9tico, sendo, portanto, nulo o campo magn\u00e9tico na Lua.\u00a0<\/b><\/span><\/span><\/span><\/p>\n

R- B<\/b><\/span><\/span><\/span><\/p>\n

20-<\/b><\/span><\/span><\/span> <\/b>O ano terrestre \u00e9 o per\u00edodo de transla\u00e7\u00e3o da Terra em torno do Sol (tempo que a Terra demora para efetuar uma volta completa ao redor do Sol)\u00a0 —\u00a0 se a nova \u00f3rbita fica mais perto do Sol\u00a0 —\u00a0 \u00a0r\u2019 < r\u00a0 —\u00a0 usando a 3\u00aa lei de Kepler\u00a0 —\u00a0 (T\u2019\/T)<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>=(r\u2019\/r)<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 observe nessa express\u00e3o que se r\u2019< r\u00a0 —\u00a0 T\u2019< T\u00a0 —\u00a0 o que torna o ano terrestre mais curto\u00a0 —\u00a0 o per\u00edodo aproximado do ciclo lunar \u00e9 T = 27 dias\u00a0 —\u00a0 o novo per\u00edodo \u00e9 T\u2019 = 80 dias\u00a0 —\u00a0 usando novamente a 3\u00aa lei de Kepler\u00a0 —\u00a0 (r\u2019\/r)<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>=(80\/27)<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 r\u2019\u2248\"\"\u00a0 —\u00a0 r\u2019\u22482r\u00a0 —\u00a0 <\/b><\/span><\/span><\/span>R- B<\/b><\/span><\/span><\/span><\/p>\n

21-<\/b><\/span><\/span><\/span> R- C <\/b><\/span><\/span><\/span>\u00a0—\u00a0 veja teoria<\/b><\/span><\/span><\/span><\/p>\n

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

22-<\/b><\/span><\/span><\/span>01. Correta.<\/b><\/span><\/span><\/span><\/p>\n

02. Falsa\u00a0 — Galileu Galilei afirmou que Sol ocupava o centro do Universo com os planetas girando ao seu redor.<\/b><\/span><\/span><\/span><\/p>\n

04- Falsa\u00a0 — Kepler conclui que as \u00f3rbitas eram el\u00edpticas baseado nas observa\u00e7\u00f5es e anota\u00e7\u00f5es de Tycho Brahe.<\/b><\/span><\/span><\/span><\/p>\n

08- Falsa\u00a0 —\u00a0 \u00e9 exatamente o contr\u00e1rio (veja fisicaevestibular-lei das \u00e1reas de Kepler).<\/b><\/span><\/span><\/span><\/p>\n

16. Falsa\u00a0 —\u00a0 para velocidades pr\u00f3ximas \u00e0s da luz s\u00e3o v\u00e1lidas as leis de Einstein.<\/b><\/span><\/span><\/span><\/p>\n

32- Verdadeira.<\/b><\/span><\/span><\/span><\/p>\n

Corretas: 01 e 32\u00a0 —\u00a0\u00a0<\/b><\/span><\/span><\/span>Soma=33<\/b><\/span><\/span><\/span>.<\/b><\/span><\/span><\/span><\/p>\n

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

\u00a023- <\/b><\/span><\/span><\/span>Pode ser resolvida pela segunda lei de Kepler (lei das \u00e1reas) de enunciado: \u201c O segmento de reta imagin\u00e1rio que une o centro do Sol ao centro do planeta descreve \u00e1reas proporcionais\u00a0aos tempos gastos para percorr\u00ea-las\u201d<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Ent\u00e3o:<\/b><\/span><\/span><\/span><\/p>\n

A<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\/\u2206t<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0~A<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\/\u2206t<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>=constante=K<\/b><\/span><\/span><\/span><\/p>\n

Essa constante K depende do planeta e recebe o nome de velocidade areolar<\/b><\/span><\/span><\/span><\/p>\n

Observe na express\u00e3o acima que quando A<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>=A<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0\u00a0 —\u00a0 \u2206t<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>=\u00a0\u2206t<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>,\u00a0ou seja, para o arco maior 34, ser percorrido no mesmo intervalo de tempo que o arco menor12, a velocidade em 3,4 (mais perto do Sol –\u00a0peri\u00e9lio) deve ser maior que a velocidade em 1,2 (mais afastado do Sol \u2013\u00a0af\u00e9lio).Portanto\u00a0os planetas aceleram do af\u00e9lio para o peri\u00e9lio e retardam do peri\u00e9lio para o af\u00e9lio\u00a0 —\u00a0 como a for\u00e7a resultante sobre o cometa \u00e9 o produto de sua massa pela sua acelera\u00e7\u00e3o e como a acelera\u00e7\u00e3o \u00e9 maior no af\u00e9lio (ponto P), a for\u00e7a resultante sobre ele no af\u00e9lio (ponto P) dever\u00e1 ser maior\u00a0 —\u00a0 ou ainda, a intensidade da for\u00e7a resultante sobre o cometa\u00a0 corresponde \u00e0 for\u00e7a de atra\u00e7\u00e3o gravitacional entre ele e o Sol, fornecida por\u00a0\u00a0 F<\/b><\/span><\/span><\/span>R<\/b><\/span><\/span><\/sub><\/span>=G.M.m\/r<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 veja que, como G,M e m s\u00e3o constantes, a intensidade da for\u00e7a resultante sobre o cometa \u00e9 inversamente proporcional \u00e0 dist\u00e2ncia r e, como no af\u00e9lio r \u00e9 menor, nele a for\u00e7a resultante ter\u00e1 maior intensidade.<\/b><\/span><\/span><\/span><\/p>\n

24- <\/b><\/span><\/span><\/span>Cop\u00e9rnico prop\u00f4s o modelo helioc\u00eantrico que afirmava que a Terra girava ao redor do Sol,\u00a0 contrariando os dogmas da igreja cat\u00f3lica\u00a0\u00a0 — Cop\u00e9rnico nada afirmou sobre as formas das \u00f3rbitas, mais tarde declaradas como el\u00edpticas pela Primeira Lei de Kepler\u00a0 —\u00a0 Einstein prop\u00f4s a teoria da relatividade modificando o conceito espa\u00e7o tempo e revolucionando a f\u00edsica\u00a0 —\u00a0 suas<\/b><\/span><\/span><\/span><\/p>\n

Id\u00e9ias foram questionadas na Alemanha devido ao fato de ser judeu\u00a0 —\u00a0 enviou uma carta ao presidente dos Estados Unidos na qual contestava a utiliza\u00e7\u00e3o de suas teorias para fins b\u00e9licos\u00a0 —\u00a0\u00a0<\/b><\/span><\/span><\/span>R- D<\/b><\/span><\/span><\/span><\/p>\n

25-<\/b><\/span><\/span><\/span> a) Verdadeira\u00a0 —\u00a0 Segunda lei de Kepler (lei das \u00e1reas)\u00a0 —\u00a0 \u201c O segmento de reta imagin\u00e1rio que une o centro do Sol ao centro do planeta descreve \u00e1reas proporcionais\u00a0 aos tempos gastos para percorr\u00ea-las\u201d<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

Essa constante K depende do planeta e recebe o nome de velocidade areolar\u00a0 —\u00a0 observe na express\u00e3o acima que quando A<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0= A<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0<\/b><\/span><\/span><\/span><\/p>\n

—\u00a0 \u2206t<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0= \u2206t<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>,\u00a0ou seja, para o arco maior 34, ser percorrido no mesmo intervalo de tempo que o arco menor12, a velocidade em 3,4 (mais perto do Sol –\u00a0peri\u00e9lio) deve ser maior que a velocidade em 1,2 (mais afastado do Sol \u2013\u00a0af\u00e9lio),\u00a0 —\u00a0 portanto\u00a0os planetas aceleram do af\u00e9lio para o peri\u00e9lio e retardam do peri\u00e9lio para o af\u00e9lio.<\/b><\/span><\/span><\/span><\/p>\n

b) Falsa\u00a0 —\u00a0 as \u00f3rbitas s\u00e3o el\u00edpticas com o Sol ocupando um dos focos da elipse\u00a0 —\u00a0 primeira lei de Kepler.<\/b><\/span><\/span><\/span><\/p>\n

c) Falsa\u00a0 — \u00a0\u00a0\u00a0O ver\u00e3o n\u00e3o ocorre quando a Terra est\u00e1 no peri\u00e9lio e nem o inverno quando ela est\u00e1 no af\u00e9lio\u00a0 —\u00a0 as esta\u00e7\u00f5es ocorrem devido ao fato de \u00e1reas da Terra, devido \u00e0 inclina\u00e7\u00e3o da mesma, receberem mais ou menos luz do Sol durante seu movimento de transla\u00e7\u00e3o.\u00a0<\/b><\/span><\/span><\/span><\/p>\n

d) Falsa\u00a0 —\u00a0 veja justificativa a.<\/b><\/span><\/span><\/span><\/p>\n

e) Falsa\u00a0 —\u00a0 s\u00e3o v\u00e1lidas para todo o Universo.<\/b><\/span><\/span><\/span><\/p>\n

R- A\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/b><\/span><\/span><\/span><\/p>\n

26- <\/b><\/span><\/span><\/span>Substituindo os dados fornecidos na express\u00e3o da terceira lei de Kepler, tamb\u00e9m fornecida\u00a0 —\u00a0 \u00a0(T<\/b><\/span><\/span><\/span>j<\/b><\/span><\/span><\/sub><\/span>\/T<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>)<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0= (R<\/b><\/span><\/span><\/span>J<\/b><\/span><\/span><\/sub><\/span>\/R<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>)<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 (T<\/b><\/span><\/span><\/span>J<\/b><\/span><\/span><\/sub><\/span>\/1)<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0=(7,5.10<\/b><\/span><\/span><\/span>11<\/b><\/span><\/span><\/sup><\/span>\/1,5.10<\/b><\/span><\/span><\/span>11<\/b><\/span><\/span><\/sup><\/span>)<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 T<\/b><\/span><\/span><\/span>J<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0= 5<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>=125\u00a0 —\u00a0 T<\/b><\/span><\/span><\/span>J<\/b><\/span><\/span><\/sub><\/span>\u00a0= \u221a(125)\u00a0 —\u00a0 T<\/b><\/span><\/span><\/span>J<\/b><\/span><\/span><\/sub><\/span>=11,180 anos terrestres\u00a0 —\u00a0<\/b><\/span><\/span><\/span>\u00a0R- C<\/b><\/span><\/span><\/span><\/p>\n

Observa\u00e7\u00e3o: Na realidade, a terceira lei de Kepler afirma que T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/R<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>=K\u2019=constante\u00a0 —\u00a0\u00a0 observe que a medida que R aumenta, T tamb\u00e9m aumenta, o que significa que\u00a0quanto mais afastado o planeta estiver do Sol maior ser\u00e1 seu ano (tempo que demora para dar um volta completa ao redor do Sol)\u00a0 —\u00a0 para dois planetas quaisquer como,\u00a0 por exemplo, Terra e Marte, vale a rela\u00e7\u00e3o\u00a0 T<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/R<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>3<\/b><\/span><\/span><\/sup><\/span>=T<\/b><\/span><\/span><\/span>M<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/R<\/b><\/span><\/span><\/span>M<\/b><\/span><\/span><\/sub><\/span>3\u00a0<\/b><\/span><\/span><\/sup><\/span>\u00a0—\u00a0 ao efetuar um volta completa ao redor do Sol num per\u00edodo (ano) T um planeta percorre \u2206S=2\u03c0R e sua velocidade orbital vale V=\u2206S\/T\u00a0 —\u00a0 T=2\u03c0R\/V, que substitu\u00edda em T<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/R<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>=K\u2019\u00a0 fornece\u00a0 4\u03c0<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>R<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/VR<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sup><\/span>=K\u2019\u00a0 —\u00a0 V=4\u03c0<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\/K\u2019R\u00a0 —\u00a0 V=constante\/R\u00a0 —\u00a0 V \u00e9 inversamente proporcional a R ou seja, quanto mais afastado o\u00a0sat\u00e9lite ou planeta estiver, menor ser\u00e1 sua velocidade orbital.<\/b><\/span><\/span><\/span><\/p>\n

27-<\/b><\/span><\/span><\/span><\/p>\n

Aplicando a lei dos cossenos\u00a0 —\u00a0 d<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0 = d<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0+ d<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0\u2013 2.d<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>.d<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>.cos120<\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/sup><\/span>\u00a0 —\u00a0 d=\u221a(R<\/b><\/span><\/span><\/span>J<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0+ R<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0\u2013<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

2.R<\/b><\/span><\/span><\/span>j<\/b><\/span><\/span><\/sub><\/span>.R<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>.cos120<\/b><\/span><\/span><\/span>o<\/b><\/span><\/span><\/sup><\/span>)\u00a0 —\u00a0 d=\u221a(R<\/b><\/span><\/span><\/span>J<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0+ R<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0-2.R<\/b><\/span><\/span><\/span>J<\/b><\/span><\/span><\/sub><\/span>.R<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>.(-1\/2)\u00a0 —\u00a0 d=\u221a(R<\/b><\/span><\/span><\/span>J<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0+ R<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>2<\/b><\/span><\/span><\/sup><\/span>\u00a0+ R<\/b><\/span><\/span><\/span>j<\/b><\/span><\/span><\/sub><\/span>.R<\/b><\/span><\/span><\/span>T<\/b><\/span><\/span><\/sub><\/span>)\u00a0 —\u00a0<\/b><\/span><\/span><\/span>\u00a0R- D<\/b><\/span><\/span><\/span><\/p>\n

 <\/p>\n

Voltar para os Exerc\u00edcios<\/span><\/a><\/h3>\n","protected":false},"excerpt":{"rendered":"

Resolu\u00e7\u00e3o comentada dos exerc\u00edcios de vestibulares sobre leis de Kepler da Gravita\u00e7\u00e3o Universal     01- C (vide teoria) O2- A (vide teoria) 03- (01 + 02 +08)=11(vide teoria) 04- E (vide teoria) 05- Regra de tr\u00eas\u00a0 —\u00a0 1 volta \u2013 12 meses\u00a0—\u00a0\u00a0\u00a0\u00bc volta – T\u00a0 —\u00a0 T=3 meses R- D 06- a) De 1o de Abril a 30 de<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":1410,"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-1414","page","type-page","status-publish","hentry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/1414","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=1414"}],"version-history":[{"count":3,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/1414\/revisions"}],"predecessor-version":[{"id":10881,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/1414\/revisions\/10881"}],"up":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/1410"}],"wp:attachment":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/media?parent=1414"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}