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(Adaptada) Na aula de Resistência dos Materiais Aplicada do professor Zezo no início do semestre letivo, o conteúdo de Torção foi abordado com os alunos tanto para elementos de estruturas metálicas como em concreto armado.
O momento de inércia polar no trecho AB, corresponde a:

","status","pending_classification","hasRelatedVideo","relatedVideos",[109],{"_110":111,"_129":130,"_52":145,"_180":181,"_198":146},"content",{"_44":112,"_58":113,"_114":-5,"_115":116,"_117":118,"_119":-5,"_46":120,"_123":124,"_125":124,"_126":-5,"_127":128},"468e3e62-35e0-47be-a15f-68dd43bb32bb","Óptica Geométrica - Parte 1","description","content_type_id","1051eee7-c62e-4b2c-9900-77a29fcec080","resource_id","fa6aaf42-4cc4-4d76-8915-7a67081f7e9a","user_id",{"_121":122},"external_id","4963","created_at","2025-08-20T12:27:41.608438+00:00","updated_at","content_group_id","slug","optica-geometrica-parte-1","prometheusData",{"_131":132,"_138":139,"_140":141,"_142":143,"_28":144},"provider",{"_44":133,"_134":135,"_136":137},"06f120be-2194-4548-8336-7ba44ffbae14","key","grupo_q_proteus","kind","proteus","reference","d80487a0-0f98-42c8-99bb-551165ba3068","state","completed","thumbnail_url","https://proteus.grupoq.io/videos/d80487a0-0f98-42c8-99bb-551165ba3068/thumbnailsthumb.0000000.jpg","https://proteus.grupoq.io/videos/d80487a0-0f98-42c8-99bb-551165ba3068/levels.m3u8",[146],{"_44":147,"_58":148,"_60":149,"_150":151,"_152":-5,"_123":153,"_125":153,"_46":154,"_127":149,"_161":162},"9772e958-72c6-4499-ad47-877cf94d670d","Exatas","exatas","tag_group_id","c1ce2cf2-b217-4cc7-8182-81c56bd3522d","parent_id","2025-04-29T01:56:45.284172+00:00",{"_155":156,"_157":158,"_159":160},"childrenCount",18,"content_count",0,"question_count",135489,"children",[163],{"_44":164,"_58":165,"_60":166,"_150":151,"_152":147,"_123":167,"_125":167,"_46":168,"_127":171,"_161":172},"ca6fee0e-79c6-4f1c-b500-ee081a071949","Mecânica","mecanica","2025-04-29T02:19:12.828432+00:00",{"_155":169,"_157":158,"_159":170},10,19948,"mecanica-2",[173],{"_44":174,"_58":59,"_60":61,"_150":151,"_152":164,"_123":175,"_125":175,"_46":176,"_127":178,"_161":179},"ac55bdba-154d-42e7-bc25-72dde822663d","2025-04-29T03:44:29.139762+00:00",{"_155":158,"_157":158,"_159":177},3376,"resistencia-dos-materiais-4",[],"video",{"_44":118,"_13":113,"_114":-5,"_182":183,"_184":185,"_28":-5,"_186":-5,"_131":185,"_187":188,"_142":-5,"_189":-5,"_190":-5,"_191":-5,"_192":-5,"_46":193,"_194":158,"_195":-5,"_104":196,"_123":197,"_125":197},"duration",20,"source_type","prometheus","iframe_url","provider_video_id","1cd3d633-d690-4f4b-9995-9b0e44994e35","format","quality","file_path","file_size",{"_121":122},"views_count","last_played_at","draft","2025-08-19T11:56:20.758718+00:00","tag","similarQuestions",[201,231,249],{"_40":202,"_42":43,"_44":203,"_78":204,"_101":219,"_102":220,"_52":221},"112831da-be9c-458e-94cb-e8292d0dba12","3c0b9441-b878-4896-8bac-fa4ca61e3f13",[205,208,211,214,217],{"_44":206,"_82":83,"_84":207},"265bd71a-6749-41f5-b291-67efd646ca64","y'' + 2y' - 3y = 0",{"_44":209,"_82":88,"_84":210},"6d3c3a1d-8e80-4ff3-b97a-d5b5bca91940","6y'' - y' - y = 0",{"_44":212,"_82":91,"_84":213},"4168ac67-64fd-46ed-a69c-a01dbfcde7ea","y'' - 2y' + y = 0",{"_44":215,"_82":95,"_84":216},"cc7bc8df-6995-4264-ac5a-32ea9b4ed7db","9y'' + 6y' + y = 0",{"_44":202,"_82":99,"_84":218},"y'' - 2y' + 6y = 0","Somente em y''−2y'+6y=0 o polinômio característico r²−2r+6=0 tem Δ<0, gerando raízes complexas conjugadas (1±i√5) e, portanto, solução em termos de exponencial modulando seno e cosseno. Nos demais itens (a–d) o discriminante é ≥0, produzindo raízes reais (distintas ou repetidas).","

Resolva as seguintes EDOs determinando a solução geral e, quando houver condições iniciais do PVI, a solução particular.

(a) y'' + 2y' - 3y = 0

(b) 6y'' - y' - y = 0

(d) 9y'' + 6y' + y = 0

(e) y'' - 2y' + 6y = 0

(f) 4y'' + 9y = 0

(g) 6y'' - 5y' + y = 0; \\, y(0) = 4; \\, y'(0) = 0

(h) y'' + 4y' + 4y = 0; \\, y(-1) = 2; \\, y'(-1) = 1

(i) y'' + y = 0; \\, y\bigg(\\frac{\\pi}{3}\\bigg) = 2; \\, y'\\bigg(\\frac{\\pi}{3}\\bigg) = -4

(j) y'' - 2y' - 3y = 3e^{2t}

(k) y'' + 9y = t^2 e^{3t} + 6

(l) 2y'' + 3y' + y = t^2 + 3 \\sin(t)

(m) y'' + y' - 2y = 2t; \\, y(0) = 0; \\, y'(0) = 1

(n) y'' - 2y' + y = te^t + 4; \\, y(0) = 1; \\, y'(0) = 1

(o) y'' + 4y = 3 \\sin(2t); \\, y(0) = 2; \\, y'(0) = -1

",[222],{"_44":223,"_58":224,"_60":225,"_150":151,"_152":226,"_123":227,"_125":227,"_46":228,"_127":225},"6516bbbd-f2a3-4710-8382-702240b17836","Mecânica na Engenharia de Petróleo","mecanica-na-engenharia-de-petroleo","b787e438-0ab3-454c-ab43-5d3d14c2720d","2025-04-29T03:44:35.328997+00:00",{"_155":229,"_157":158,"_159":230},4,12302,{"_40":232,"_42":43,"_44":233,"_78":234,"_101":246,"_102":247,"_52":248},"6831a9b9-c9a4-4a76-bd7a-2ce98bfea875","93a32dd4-5fc6-4ccb-a76f-4724bf113742",[235,238,241,243],{"_44":236,"_82":83,"_84":237},"13e3f476-a0a5-421f-b47e-0969e8e9a7e0","

Verdadeiro, verdadeiro, falso

",{"_44":239,"_82":88,"_84":240},"c9a20855-d84b-4860-8b35-9784818282cf","

Falso, verdadeiro, verdadeiro

",{"_44":232,"_82":91,"_84":242},"

Verdadeiro, falso, verdadeiro

",{"_44":244,"_82":95,"_84":245},"3c98539b-1af9-4d44-aecd-c266754a2075","

Falso, falso, falso

","85,5 % decomposto → 14,5 % restante.\nN/N0 = 0,145 ⇒ t = ln(1/0,145)/(ln2/5730) ≈ 1,6×10⁴ anos.\nItens prováveis: (1) resta 14,5 % do C-14 (V); (2) idade ≈ 1,2×10⁴ anos ou 3 meias-vidas (F); (3) t½ do C-14 = 5730 anos (V). Sequência: V-F-V, correspondendo à alternativa c.","

Qual é a idade da madeira na qual 85,5\\% do C-14 se decompos?

",[222],{"_40":250,"_42":43,"_44":251,"_78":252,"_101":267,"_102":268,"_52":269},"3ce109b1-6cdb-4b4d-8816-a6af4d74bed0","c478a46c-46a8-4942-ae14-2e79def3dda2",[253,256,259,261,264],{"_44":254,"_82":83,"_84":255},"a5a16a42-3e58-4c76-80f2-c291aa2d8e49","Sim",{"_44":257,"_82":88,"_84":258},"5e6d0254-696c-4da0-b572-82e3a19f433e","Não",{"_44":250,"_82":91,"_84":260},"Indeterminado",{"_44":262,"_82":95,"_84":263},"35403f68-a880-429d-97f4-db507c58de38","Todos",{"_44":265,"_82":99,"_84":266},"2e06d948-d37e-4350-afc7-ced04941bcaf","Nenhum","Os polinômios e números fornecidos estão corrompidos (símbolos “????”), impossibilitando aplicar Ruffini e concluir se são raízes. Sem dados completos, o problema é indeterminado.","Emplea la regla de Ruffini para dictaminar si los siguientes números son o no raíces de los polinomios citados:
a) ????3 de ????4????5
b) ????2 de ????2????????2
c) ????1 de 2????????1
d) ????1 de 2????2????",[222],[222,271,280,286,290,291,292,300,305,313,314,319,325],{"_44":272,"_58":273,"_60":274,"_150":151,"_152":275,"_123":167,"_125":167,"_46":276,"_127":279},"861ad829-750d-4a44-9569-2f18b430b4df","Engenharia Mecatrônica","engenharia-mecatronica","eb9315c5-bd04-4116-aa46-02c4fde320c0",{"_155":277,"_157":158,"_159":278},5,30614,"engenharia-mecatronica-1",{"_44":281,"_58":59,"_60":61,"_150":151,"_152":282,"_123":283,"_125":283,"_46":284,"_127":285},"8850e507-a691-4d45-a7c7-16221e9c5af2","88fe96b3-2a17-4198-90d5-ddda3d6f5d45","2025-04-29T04:00:13.215003+00:00",{"_155":158,"_157":158,"_159":177},"resistencia-dos-materiais-6",{"_44":282,"_58":165,"_60":166,"_150":151,"_152":272,"_123":227,"_125":227,"_46":287,"_127":289},{"_155":277,"_157":158,"_159":288},25741,"mecanica-3",{"_44":147,"_58":148,"_60":149,"_150":151,"_152":-5,"_123":153,"_125":153,"_46":154,"_127":149},{"_44":174,"_58":59,"_60":61,"_150":151,"_152":164,"_123":175,"_125":175,"_46":176,"_127":178},{"_44":293,"_58":294,"_60":295,"_150":151,"_152":275,"_123":167,"_125":167,"_46":296,"_127":299},"b6e37f32-f8e5-4025-bf0f-69ffc00fb331","Engenharia Civil","engenharia-civil",{"_155":297,"_157":158,"_159":298},21,45865,"engenharia-civil-3",{"_44":226,"_58":301,"_60":302,"_150":151,"_152":275,"_123":167,"_125":167,"_46":303,"_127":304},"Engenharia de Petróleo","engenharia-de-petroleo",{"_155":277,"_157":158,"_159":230},"engenharia-de-petroleo-1",{"_44":306,"_58":307,"_60":308,"_150":151,"_152":293,"_123":309,"_125":309,"_46":310,"_127":312},"c5bf57d4-c3d8-4365-b53e-085fa14a3a95","Estruturas","estruturas","2025-04-29T03:44:32.303369+00:00",{"_155":277,"_157":158,"_159":311},8852,"estruturas-1",{"_44":164,"_58":165,"_60":166,"_150":151,"_152":147,"_123":167,"_125":167,"_46":168,"_127":171},{"_44":315,"_58":59,"_60":61,"_150":151,"_152":223,"_123":316,"_125":316,"_46":317,"_127":318},"d1e4567b-f68a-49e1-b3e5-3888343d7871","2025-04-29T04:00:14.680867+00:00",{"_155":158,"_157":158,"_159":177},"resistencia-dos-materiais-7",{"_44":320,"_58":321,"_60":61,"_150":151,"_152":306,"_123":322,"_125":322,"_46":323,"_127":324},"da8d2ecf-e949-4afc-966f-98b4b367fac1","Resistencia dos Materiais","2025-04-29T04:00:06.9766+00:00",{"_155":158,"_157":158,"_159":177},"resistencia-dos-materiais-5",{"_44":275,"_58":326,"_60":327,"_150":151,"_152":-5,"_123":328,"_125":328,"_46":329,"_127":327},"Engenharia","engenharia","2025-04-29T01:55:38.635246+00:00",{"_155":330,"_157":158,"_159":331},29,351586,"tagsTree",[146,334],{"_44":275,"_58":326,"_60":327,"_150":151,"_152":-5,"_123":328,"_125":328,"_46":329,"_127":327,"_161":335},[336,342,348],{"_44":272,"_58":273,"_60":274,"_150":151,"_152":275,"_123":167,"_125":167,"_46":276,"_127":279,"_161":337},[338],{"_44":282,"_58":165,"_60":166,"_150":151,"_152":272,"_123":227,"_125":227,"_46":287,"_127":289,"_161":339},[340],{"_44":281,"_58":59,"_60":61,"_150":151,"_152":282,"_123":283,"_125":283,"_46":284,"_127":285,"_161":341},[],{"_44":293,"_58":294,"_60":295,"_150":151,"_152":275,"_123":167,"_125":167,"_46":296,"_127":299,"_161":343},[344],{"_44":306,"_58":307,"_60":308,"_150":151,"_152":293,"_123":309,"_125":309,"_46":310,"_127":312,"_161":345},[346],{"_44":320,"_58":321,"_60":61,"_150":151,"_152":306,"_123":322,"_125":322,"_46":323,"_127":324,"_161":347},[],{"_44":226,"_58":301,"_60":302,"_150":151,"_152":275,"_123":167,"_125":167,"_46":303,"_127":304,"_161":349},[350],{"_44":223,"_58":224,"_60":225,"_150":151,"_152":226,"_123":227,"_125":227,"_46":228,"_127":225,"_161":351},[352],{"_44":315,"_58":59,"_60":61,"_150":151,"_152":223,"_123":316,"_125":316,"_46":317,"_127":318,"_161":353},[],"videoDislikes",{},"videoLikes",{},"actionData","errors"]

Comentários