{"id":1283,"date":"2015-09-02T01:10:40","date_gmt":"2015-09-02T01:10:40","guid":{"rendered":"http:\/\/fisicaevestibular.com.br\/novo\/?page_id=1283"},"modified":"2024-08-23T13:16:56","modified_gmt":"2024-08-23T13:16:56","slug":"resolucao-comentada-dos-exercicios-de-vestibulares-sobre-associacao-de-molas","status":"publish","type":"page","link":"https:\/\/fisicaevestibular.com.br\/novo\/mecanica\/dinamica\/mhs\/associacao-de-molas\/resolucao-comentada-dos-exercicios-de-vestibulares-sobre-associacao-de-molas\/","title":{"rendered":"Associa\u00e7\u00e3o de molas – Resolu\u00e7\u00e3o"},"content":{"rendered":"

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

Associa\u00e7\u00e3o de molas<\/b><\/span><\/span><\/span><\/p>\n

 <\/p>\n

1-\u00a0Per\u00edodo T da mola da figura 1\u00a0 —\u00a0 T = 2\u03c0\u221am\/k<\/b><\/span><\/span><\/span><\/p>\n

Como as molas est\u00e3o associadas em paralelo, a constante el\u00e1stica da mola equivalente, que, substituindo as duas produz o mesmo efeito ser\u00e1 k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>\u00a0= k + k\u00a0 —\u00a0 k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>\u00a0=2k e seu per\u00edodo ser\u00e1 T\u2019 = 2\u03c0\u221am\/2k\u00a0 —\u00a0 T\u2019 = 2\u03c0\u221am\/k.1\/\u221a2.<\/b><\/span><\/span><\/span><\/p>\n

T\/T\u2019 =\u00a0\u221a2\u00a0 —\u00a0 T\u2019 = T\/\u221a2\u00a0 — racionalizando\u00a0 —\u00a0 T\u2019= T\u221a2\/2\u00a0 Resposta C<\/b><\/span><\/span><\/span><\/p>\n

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

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

a) Como as duas molas de constantes k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0est\u00e3o em para, a mola equivalente ter\u00e1 constante k<\/b><\/span><\/span><\/span>e1<\/b><\/span><\/span><\/sub><\/span>\u00a0=30 + 30 = 60N\/m. Ent\u00e3o teremos:<\/b><\/span><\/span><\/span><\/p>\n

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

As duas molas acima est\u00e3o em s\u00e9rie, ent\u00e3o a mola equivalente ter\u00e1 constante k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>, dada por: 1\/k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>\u00a0= 1\/60 + 1\/30\u00a0 —\u00a0\u00a0<\/b><\/span><\/span><\/span>k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>\u00a0= 20N\/m,<\/b><\/span><\/span><\/span><\/p>\n

que \u00e9 a constante el\u00e1stica total\u00a0 equivalente do conjunto.<\/b><\/span><\/span><\/span><\/p>\n

b) T = 2p\u221am\/k\u00a0 —\u00a0 T = 2p\u221a20\/9 \/20\u00a0 —\u00a0 T = 2.3.1\/3\u00a0 —\u00a0 T=2s\u00a0 —\u00a0 f=1\/T\u00a0 —\u00a0\u00a0<\/b><\/span><\/span><\/span><\/span>f=0,5Hz<\/b><\/span><\/span><\/span><\/span><\/p>\n

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

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

As 3 molas de constantes k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0est\u00e3o em paralelo e ser\u00e3o substitu\u00eddas por uma \u00fanica mola de constante k<\/b><\/span><\/span><\/span>e1<\/b><\/span><\/span><\/sub><\/span>=3k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>.<\/b><\/span><\/span><\/span><\/p>\n

As duas molas de constantes k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0tamb\u00e9m est\u00e3o em paralelo e ser\u00e3o substitu\u00eddas por um \u00fanica mola de constante k<\/b><\/span><\/span><\/span>e2<\/b><\/span><\/span><\/sub><\/span>=2k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span><\/p>\n

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

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

A mola resultante das duas acima, que est\u00e3o em s\u00e9rie, ter\u00e1 k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>, tal que: 1\/k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>\u00a0= 1\/3k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0+ 1\/2k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0 —\u00a0 1\/k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>\u00a0= 2k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0+ 3k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0\/ 6k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>.<\/b><\/span><\/span><\/span><\/p>\n

K<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>\u00a0= 6k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>.k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0\/ 2k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0+ 3k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span><\/p>\n

O per\u00edodo desse sistema vale\u00a0 —\u00a0 T = 2p\u00d6m\/6k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>.k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0\/ 2k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0+ 3k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0 —\u00a0 T = 2p\u00d6m(2k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0+ 3k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>)\/6k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>.k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span><\/p>\n

F = 1\/T = 1\/2p\u221a6k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>.k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0\/ m(2k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0+ 3k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>)<\/b><\/span><\/span><\/span><\/p>\n

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

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

a) A mola inteira (mola equivalente) tem constante el\u00e1stica k\u2019=10N\/m sendo que 1\/k\u2019= 1\/k + 1\/k +1\/k, onde k \u00e9 a constante el\u00e1stica de cada parte.<\/b><\/span><\/span><\/span><\/p>\n

\"\"<\/p>\n

1\/k\u2019=3\/k\u00a0 —\u00a0 1\/12 = 3\/k\u00a0 —\u00a0\u00a0<\/b><\/span><\/span><\/span>k =36N\/m<\/b><\/span><\/span><\/span><\/p>\n

b) Paralelo\u00a0 —\u00a0 k\u00ade=36 + 36 +36\u00a0 —\u00a0 k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>=108N\/m\u00a0 —\u00a0 T=2\u03c0\u221am\/k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>\u00a0 — \u00a0T=2\u03c0\u221a0,1\/108\u00a0 —\u00a0<\/b><\/span><\/span><\/span>T\u00a0=\u00a06.10<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span>.\u03c0\u00a0s<\/b><\/span><\/span><\/span><\/p>\n

c) S\u00e9rie\u00a0 —\u00a0 k<\/b><\/span><\/span><\/span>e<\/b><\/span><\/span><\/sub><\/span>=12N\/m\u00a0 —\u00a0 T=2\u03c0\u221am\/k<\/b><\/span><\/span><\/span>e\u00a0\u00a0<\/b><\/span><\/span><\/sub><\/span>—\u00a0 T=2\u03c0\u221a0,1\/12\u00a0 —\u00a0\u00a0<\/b><\/span><\/span><\/span>T=\u00a018.10<\/b><\/span><\/span><\/span>-2<\/b><\/span><\/span><\/sup><\/span>.\u03c0\u00a0s<\/b><\/span><\/span><\/span><\/p>\n

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

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

a) Peso de cada massa\u00a0 —\u00a0 P=mg\u00a0 —\u00a0 P=0,01.10\u00a0 —\u00a0 P=0,1N. Como as molas s\u00e3o ideais, suas massas s\u00e3o desprez\u00edveis.<\/b><\/span><\/span><\/span><\/p>\n

Observe que a mola 1 est\u00e1 sujeita \u00e0 for\u00e7a F=0,3N (s\u00e3o as 3 massas que est\u00e3o deformando-a)<\/b><\/span><\/span><\/span><\/p>\n

F<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>=k<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>.x<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0 —\u00a0 0,3=0,1.x<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0 —\u00a0\u00a0<\/b><\/span><\/span><\/span>x<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>\u00a0= 3cm<\/b><\/span><\/span><\/span><\/p>\n

A mola 2 est\u00e1 sujeita \u00e0 F=0,2N (apenas duas massas est\u00e3o deformando-a)<\/b><\/span><\/span><\/span><\/p>\n

F<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>=k<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>.x<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0 —\u00a0 0,2=0,1.x<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>\u00a0 —\u00a0\u00a0<\/b><\/span><\/span><\/span>x<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>= 2cm<\/b><\/span><\/span><\/span><\/p>\n

Mola 3\u00a0 —\u00a0 F<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sub><\/span>=k<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sub><\/span>.x<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sub><\/span>\u00a0 —\u00a0 0,1=0,1.x<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sub><\/span>\u00a0 —\u00a0\u00a0<\/b><\/span><\/span><\/span>x<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sub><\/span>= 1cm<\/b><\/span><\/span><\/span><\/p>\n

b) mola 1 \u2013\u00a0<\/b><\/span><\/span><\/span>L<\/b><\/span><\/span><\/span>1<\/b><\/span><\/span><\/sub><\/span>= 23cm<\/b><\/span><\/span><\/span><\/p>\n

\u00a0\u00a0\u00a0 mola 2 \u2013\u00a0<\/b><\/span><\/span><\/span>L<\/b><\/span><\/span><\/span>2<\/b><\/span><\/span><\/sub><\/span>= 22cm<\/b><\/span><\/span><\/span><\/p>\n

\u00a0\u00a0\u00a0 mola 3 \u2013\u00a0<\/b><\/span><\/span><\/span>L<\/b><\/span><\/span><\/span>3<\/b><\/span><\/span><\/sub><\/span>= 21cm<\/b><\/span><\/span><\/span><\/p>\n

c)\u00a0<\/b><\/span><\/span><\/span>6 cm<\/b><\/span><\/span><\/span><\/p>\n

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

Resolu\u00e7\u00e3o comentada dos exerc\u00edcios de vestibulares sobre Associa\u00e7\u00e3o de molas   1-\u00a0Per\u00edodo T da mola da figura 1\u00a0 —\u00a0 T = 2\u03c0\u221am\/k Como as molas est\u00e3o associadas em paralelo, a constante el\u00e1stica da mola equivalente, que, substituindo as duas produz o mesmo efeito ser\u00e1 ke\u00a0= k + k\u00a0 —\u00a0 ke\u00a0=2k e seu per\u00edodo ser\u00e1 T\u2019 = 2\u03c0\u221am\/2k\u00a0 —\u00a0 T\u2019 =<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":1279,"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-1283","page","type-page","status-publish","hentry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/1283","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=1283"}],"version-history":[{"count":4,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/1283\/revisions"}],"predecessor-version":[{"id":10836,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/1283\/revisions\/10836"}],"up":[{"embeddable":true,"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/pages\/1279"}],"wp:attachment":[{"href":"https:\/\/fisicaevestibular.com.br\/novo\/wp-json\/wp\/v2\/media?parent=1283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}