(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.2' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 132222, 3101] NotebookOptionsPosition[ 113791, 2785] NotebookOutlinePosition[ 118093, 2866] CellTagsIndexPosition[ 118013, 2861] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", "FirstSlide", CellTags-> "SlideShowHeader",ExpressionUUID->"ae861ed8-5fa1-4088-84fd-2651cc7b368a"], Cell["Introdu\[CCedilla]\[ATilde]o \[AGrave] F\[IAcute]sica Computacional I \ (2023.2)", "Title", CellChangeTimes->{ 3.559948400406288*^9, {3.699269571766136*^9, 3.699269573578824*^9}, { 3.7099080105038958`*^9, 3.7099080108133383`*^9}, {3.727189496068931*^9, 3.7271894962748613`*^9}, {3.83932299294512*^9, 3.839323000605739*^9}, { 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Por exemplo, \ para obter ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "3"], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"m", "=", "1"}], "n"], FractionBox["1", RowBox[{ SuperscriptBox["x", "n"], "+", SuperscriptBox["y", "m"]}]]}]}], ","}], TraditionalForm]], ExpressionUUID->"513b6b25-df1d-416d-9392-fc8269f64f0d"], " escrevemos" }], "Text", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.8414825461989927`*^9, 3.841482606253091*^9}, { 3.841482643294322*^9, 3.841482736076749*^9}, {3.8414828811351233`*^9, 3.841482906340209*^9}},ExpressionUUID->"3010aa34-6070-4ecd-a024-\ 1bc12e2563f8"], Cell[BoxData[ RowBox[{ RowBox[{"Sum", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"(", RowBox[{ RowBox[{"x", "^", "n"}], "+", RowBox[{"y", "^", "m"}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"m", ",", "1", ",", "n"}], "}"}]}], "]"}], " "}]], "Input", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.8414827641209393`*^9, 3.8414827943542223`*^9}, 3.8414829063410892`*^9, 3.8414841985223923`*^9},ExpressionUUID->"55eb87f7-0f77-4b75-8a39-\ 2283aa89b768"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"32be91fb-777d-400b-86fa-9a049cede895"], Cell[TextData[{ "O valor de uma s\[EAcute]rie infinita, desde que convergente, pode ser \ calculado. 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Em vez disso, podemos utilizar a fun\ \[CCedilla]\[ATilde]o ", ButtonBox["NSum", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/NSum.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/NSum.html"], ": " }], "Text", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.841484186493013*^9, 3.841484381387475*^9}, 3.841484465313518*^9},ExpressionUUID->"6c0faee5-8020-4b1d-bdf5-\ 36f5dc396e89"], Cell[BoxData[ RowBox[{ RowBox[{"NSum", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"n", "^", "2"}], ")"}], "/", RowBox[{"(", RowBox[{"1", "+", RowBox[{"2", "*", RowBox[{"Exp", "[", "n", "]"}]}]}], ")"}]}], ",", RowBox[{"{", RowBox[{"n", ",", "1", ",", "\[Infinity]"}], "}"}]}], "]"}], " "}]], "Input", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.8414843869098454`*^9, 3.8414843974914837`*^9}, { 3.841484465314561*^9, 3.841484469795952*^9}}, CellLabel->"",ExpressionUUID->"110159e1-9b94-4808-9677-4fb0ed4768ec"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ccbea514-18b8-498d-b295-7981bb17390b"], Cell[CellGroupData[{ Cell["S\[EAcute]ries de pot\[EHat]ncias", "Section", CellChangeTimes->{{3.840726344024035*^9, 3.840726345080748*^9}, { 3.841484630157443*^9, 3.8414846332283907`*^9}},ExpressionUUID->"290d3ab0-51b2-499e-87ab-\ d974fd0d75af"], Cell[TextData[{ "Representa\[CCedilla]\[OTilde]es em s\[EAcute]ries de pot\[EHat]ncias de \ uma fun\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "907fcaa9-debb-4951-b565-8b498e5b71e2"], " em torno de ", Cell[BoxData[ FormBox[ RowBox[{"x", "=", SubscriptBox["x", "0"]}], TraditionalForm]],ExpressionUUID-> "8720839b-cdc7-4a7d-aec2-6307d4df22a0"], " s\[ATilde]o obtidas no Mathematica utilizando a fun\[CCedilla]\[ATilde]o \ ", StyleBox[ButtonBox["Series", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Series.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Series.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Por exemplo, para obter a expans\[ATilde]o at\[EAcute] ordem ", Cell[BoxData[ FormBox[ SuperscriptBox["u", "11"], TraditionalForm]],ExpressionUUID-> "b996442e-c82a-405e-a6ac-cccf859fd4a4"], " da fun\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"exp", "(", RowBox[{"sen", "(", "u", ")"}], ")"}], TraditionalForm]],ExpressionUUID-> "b9f904a1-d453-4b13-8453-3c0154da0efc"], " em torno de ", Cell[BoxData[ FormBox[ RowBox[{"u", "=", "0"}], TraditionalForm]],ExpressionUUID-> "00f494fe-d5e4-4bb9-a7aa-ac6a954ef6c8"], ":" }], "Text", CellChangeTimes->{{3.8407263528327703`*^9, 3.840726386414*^9}, { 3.840726458714198*^9, 3.840726482804037*^9}, {3.8414919756375732`*^9, 3.841492059474931*^9}, {3.841492206713057*^9, 3.841492278824851*^9}, { 3.841492387323824*^9, 3.841492387570656*^9}},ExpressionUUID->"90a8a30a-154b-41f4-a84e-\ 4ed101c61e2f"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Series", "[", RowBox[{ RowBox[{"Exp", "[", RowBox[{"Sin", "[", "u", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"u", ",", "0", ",", "11"}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8414922857850857`*^9, 3.8414923021935463`*^9}, { 3.8414923393586397`*^9, 3.841492366734119*^9}, 3.841492400712227*^9, 3.84149287322731*^9, {3.841494540461071*^9, 3.8414945609978437`*^9}, 3.841494617611292*^9}, CellLabel->"",ExpressionUUID->"596087b8-5767-493b-a121-4394304d0777"], Cell[TextData[{ "Na sa\[IAcute]da, ", StyleBox["O[u]", FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox[ SuperscriptBox["", "12"], TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "39ca0a93-3b1f-4927-af0a-418f8390587b"], " indica termos de ordem igual ou superior a ", Cell[BoxData[ FormBox[ SuperscriptBox["u", "12"], TraditionalForm]],ExpressionUUID-> "27616ff3-c923-4648-b2d5-622318935d0b"], ". Para eliminar essa indica\[CCedilla]\[ATilde]o da sa\[IAcute]da (o que \ \[EAcute] necess\[AAcute]rio, por exemplo, para extrair valores \ num\[EAcute]ricos), podemos utilizar a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Normal", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Normal.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Normal.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.8414924229837103`*^9, 3.841492509158279*^9}, { 3.841492691240909*^9, 3.841492752619896*^9}, {3.841492800608882*^9, 3.841492800613748*^9}, {3.8414945887061663`*^9, 3.841494599213222*^9}},ExpressionUUID->"d6b399aa-6d55-4ad1-b9b3-\ abb4e9c8399b"], Cell[BoxData[ RowBox[{"Normal", "[", "%", "]"}]], "Input", CellChangeTimes->{{3.841492804684682*^9, 3.841492808442698*^9}, { 3.8414945646410847`*^9, 3.841494581519225*^9}}, CellLabel->"",ExpressionUUID->"9fa0a135-e1bd-4e03-a441-b5d531998f78"] }, Closed]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"77a830d5-b503-409c-be8c-446d5377cbde"], Cell[TextData[{ "Nas vers\[OTilde]es mais recentes do Mathematica, a \ fun\[CCedilla]\[ATilde]o ", StyleBox["Series", FontFamily->"Source Code Pro", FontWeight->"Regular"], " consegue expandir uma fun\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "40055936-8f60-495d-b7e0-68e63971d673"], " em torno de ", Cell[BoxData[ FormBox[ RowBox[{"x", "\[Rule]", "\[Infinity]"}], TraditionalForm]],ExpressionUUID-> "bd3600d9-eed3-43cc-9f3c-cdb35f8282b3"], ". Vamos ilustrar essa possibilidade com o exemplo da aula passada sobre o \ potencial el\[EAcute]trico de um disco uniformemente carregado de raio ", Cell[BoxData[ FormBox["a", TraditionalForm]],ExpressionUUID-> "f9c23af1-2bc2-491c-8e53-4b0b53cac187"], " e carga total ", Cell[BoxData[ FormBox["q", TraditionalForm]],ExpressionUUID-> "da3c5ea3-050d-4652-86f5-0bc3f607ab6b"], ", calculado ao longo do eixo ", Cell[BoxData[ FormBox["z", TraditionalForm]],ExpressionUUID-> "30460ddb-2b40-4089-a269-0fef5808da1c"], ". Nesse caso, o potencial \[EAcute] dado por ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"V", "(", "z", ")"}], "=", RowBox[{ FractionBox["q", RowBox[{"2", "\[Pi]", " ", SuperscriptBox["a", "2"], SubscriptBox["\[Epsilon]", "0"]}]], RowBox[{"(", RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox["a", "2"], "+", SuperscriptBox["z", "2"]}]], "-", "z"}], ")"}]}]}], TraditionalForm]],ExpressionUUID->"ebdf95b5-26a0-453d-98d8-340c08dcc6b6"], ". Vamos mostrar que a grandes dist\[AHat]ncias do disco o potencial \ reproduz aquele de uma carga puntiforme." }], "Text", CellChangeTimes->{{3.840728496206863*^9, 3.8407285405878477`*^9}, { 3.840729503945147*^9, 3.84072957314462*^9}, {3.841493156226722*^9, 3.841493159780838*^9}, {3.841493390808278*^9, 3.841493626060704*^9}, { 3.841493656886868*^9, 3.841493713409244*^9}, {3.841494119403997*^9, 3.841494126589694*^9}, {3.8730331072622547`*^9, 3.8730331072676697`*^9}},ExpressionUUID->"36db2722-f94d-4ef1-987d-\ 9611e27345b4"], Cell[BoxData[ RowBox[{ RowBox[{"V", "[", "z", "]"}], ":=", RowBox[{ RowBox[{"q", "/", RowBox[{"(", RowBox[{"2", "*", "\[Pi]", "*", RowBox[{"a", "^", "2"}], "*", SubscriptBox["\[Epsilon]", "0"]}], ")"}]}], "*", RowBox[{"(", RowBox[{ RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"a", "^", "2"}], "+", RowBox[{"z", "^", "2"}]}], "]"}], "-", "z"}], ")"}], " "}]}]], "Input",\ CellChangeTimes->{{3.841493628698719*^9, 3.841493642645*^9}, { 3.84149372429417*^9, 3.84149375494352*^9}, {3.841494129015389*^9, 3.8414941444420233`*^9}, 3.841494237843163*^9}, CellLabel->"",ExpressionUUID->"6c7e1480-cdeb-4159-92db-76142ed729d1"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Series", "[", RowBox[{ RowBox[{"V", "[", "z", "]"}], ",", RowBox[{"{", RowBox[{"z", ",", "\[Infinity]", ",", "4"}], "}"}]}], "]"}], " "}]], "Input", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.840728544658496*^9, 3.840728562568491*^9}, 3.8412167118554487`*^9, {3.8414931616260557`*^9, 3.841493208934765*^9}, { 3.841493363485342*^9, 3.841493363576949*^9}, {3.8414937654671583`*^9, 3.8414937718289547`*^9}, {3.841494238472169*^9, 3.841494249041967*^9}}, CellLabel->"",ExpressionUUID->"60dab9ce-72c7-44e7-8318-68f358e26ab8"], Cell["\<\ Note que a expans\[ATilde]o cont\[EAcute]m pot\[EHat]ncias negativas, com as \ quais o Mathematica consegue lidar. Caso queiramos apenas o termo respons\ \[AAcute]vel pela maior contribui\[CCedilla]\[ATilde]o, podemos utilizar a \ sintaxe abaixo: \ \>", "Text", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.84149417642558*^9, 3.841494249043638*^9}, { 3.8414943148112717`*^9, 3.841494336940538*^9}, {3.841494470445211*^9, 3.8414944761261787`*^9}},ExpressionUUID->"5f774720-ac83-4352-a9fa-\ 0fdf2dd3a36d"], Cell[BoxData[ RowBox[{ RowBox[{"Series", "[", RowBox[{ RowBox[{"V", "[", "z", "]"}], ",", RowBox[{"z", "\[Rule]", " ", "\[Infinity]"}]}], "]"}], " "}]], "Input", CellGroupingRules->{"GroupTogetherGrouping", 10000.}, CellChangeTimes->{{3.841493354239213*^9, 3.841493358435561*^9}, { 3.8414941841945057`*^9, 3.841494187372115*^9}, {3.8414942353926353`*^9, 3.841494249044692*^9}}, CellLabel->"",ExpressionUUID->"037cd0cf-f5ff-4e0e-b7e4-6c075b2f88d5"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"916aafb6-db56-46e9-abc4-ac7b00e07c2a"], Cell[TextData[{ "Vamos ilustrar a obten\[CCedilla]\[ATilde]o de s\[EAcute]ries de fun\ \[CCedilla]\[OTilde]es de v\[AAcute]rias vari\[AAcute]veis com o caso geral \ do potencial de um disco: ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"V", "(", RowBox[{"x", ",", "y", ",", "z"}], ")"}], "=", RowBox[{ FractionBox["q", RowBox[{"4", SuperscriptBox["\[Pi]", "2"], SuperscriptBox["a", "2"], SubscriptBox["\[Epsilon]", "0"]}]], RowBox[{ SubsuperscriptBox["\[Integral]", "0", "a"], RowBox[{ StyleBox["dr", FontSlant->"Italic"], RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", "\[Pi]"}]], RowBox[{ StyleBox["d\[Phi]", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], RowBox[{"r", "/", SuperscriptBox[ RowBox[{"[", RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", RowBox[{"r", " ", "cos", " ", "\[Phi]"}]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"y", "-", RowBox[{"r", " ", "sin", " ", "\[Phi]"}]}], ")"}], "2"], "+", SuperscriptBox["z", "2"]}], "]"}], RowBox[{"1", "/", "2"}]]}]}]}]}]}]}]}], TraditionalForm]], ExpressionUUID->"4d2a27fd-91a6-409a-bdcc-d8bd599ea43f"], " " }], "Text", CellChangeTimes->{{3.8410460233083143`*^9, 3.8410460412733793`*^9}, { 3.8414947218276873`*^9, 3.841494862183041*^9}, {3.8414950184999313`*^9, 3.8414950759148483`*^9}, {3.841512344981151*^9, 3.841512400542829*^9}, { 3.84159445169516*^9, 3.8415944912259007`*^9}},ExpressionUUID->"6026ccad-e8e5-4659-852a-\ 29f271e84752"], Cell[TextData[{ "Come\[CCedilla]amos expandindo o integrando em s\[EAcute]rie, supondo que o \ ponto em que queremos calcular o potencial esteja pr\[OAcute]ximo do eixo ", Cell[BoxData[ FormBox["z", TraditionalForm]],ExpressionUUID-> "6a01a0d7-483b-4c2d-a197-1d0ece290887"], " mas longe do disco. " }], "Text", CellChangeTimes->{{3.841495092716528*^9, 3.841495136864716*^9}, { 3.841495220002638*^9, 3.84149522517741*^9}, {3.8415122501164703`*^9, 3.841512255616736*^9}},ExpressionUUID->"2432d87d-accc-41bf-bf07-\ a9da126b9fae"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"v", "[", RowBox[{"x_", ",", "y_", ",", "z_"}], "]"}], ":=", RowBox[{"r", "/", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "-", RowBox[{"r", "*", RowBox[{"Cos", "[", "\[Phi]", "]"}]}]}], ")"}], "^", "2"}], "+", RowBox[{ RowBox[{"(", RowBox[{"y", "-", RowBox[{"r", "*", RowBox[{"Sin", "[", "\[Phi]", "]"}]}]}], ")"}], "^", "2"}], "+", RowBox[{"z", "^", "2"}]}], "]"}]}]}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"Series", "[", RowBox[{ RowBox[{"v", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "\[Infinity]", ",", "3"}], 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{3.84149755047639*^9, 3.841497550632381*^9}, { 3.841497722349032*^9, 3.841497730947398*^9}, 3.8414979774385567`*^9, 3.841512210596977*^9, 3.841594053146576*^9, {3.841594128772107*^9, 3.841594167045197*^9}, 3.841594519746603*^9}, CellLabel->"",ExpressionUUID->"b864c67e-fa04-4a54-801c-e2fd30442de3"], Cell["\<\ Perceba que novamente o termo mais importante corresponde ao potencial de uma \ carga puntiforme.\ \>", "Text", CellGroupingRules->{"GroupTogetherGrouping", 10002.}, CellChangeTimes->{{3.841497844268312*^9, 3.841497881920944*^9}, 3.8415122105982933`*^9, 3.841594053147478*^9, {3.841594128773155*^9, 3.8415941670462112`*^9}},ExpressionUUID->"c7190c4f-4d08-4948-aac3-\ a4dea56fe7a1"] }, Closed]] }, Closed]] }, Closed]] }, Open ]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"e743604b-1c12-44a3-bfe6-87bd824714a7"], Cell[TextData[{ "Se quisermos obter o coeficiente de uma pot\[EHat]ncia espec\[IAcute]fica \ na expans\[ATilde]o em s\[EAcute]rie, podemos utilizar a fun\[CCedilla]\ \[ATilde]o ", StyleBox[ButtonBox["SeriesCoefficient", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/SeriesCoefficient.html"], None}, ButtonNote-> "https://reference.wolfram.com/language/ref/SeriesCoefficient.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Por exemplo, para obter o coeficiente do termo ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], RowBox[{ SuperscriptBox["y", "3"], "/", SuperscriptBox["z", "7"]}]}], TraditionalForm]],ExpressionUUID-> "7ac0a394-3186-4a53-afaf-7e809bcbcb32"], " na expans\[ATilde]o do integrando do slide anterior, utilizamos " }], "Text", CellChangeTimes->{{3.841668041823781*^9, 3.841668100347762*^9}, { 3.841668156976202*^9, 3.841668156982239*^9}, {3.841668293038082*^9, 3.841668359311798*^9}, {3.841668440698966*^9, 3.841668454016659*^9}, 3.841743913306546*^9},ExpressionUUID->"0555e125-aa6a-41fa-8d14-\ 3d01e83ab6ea"], Cell[BoxData[ RowBox[{ RowBox[{"SeriesCoefficient", "[", RowBox[{ RowBox[{"v", "[", RowBox[{"x", ",", "y", ",", "z"}], "]"}], ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "2"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", "0", ",", "3"}], "}"}], ",", RowBox[{"{", RowBox[{"z", ",", "\[Infinity]", ",", "7"}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.841668363947023*^9, 3.84166838311263*^9}, { 3.8416684587710238`*^9, 3.841668460923037*^9}, {3.8416685389059353`*^9, 3.841668572910033*^9}}, CellLabel->"",ExpressionUUID->"98a83e91-e521-4e25-8f55-f2185dd134af"], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"f87f8a58-294f-48b2-919c-217c7c061a25"], Cell[CellGroupData[{ Cell["Solu\[CCedilla]\[ATilde]o de equa\[CCedilla]\[OTilde]es diferenciais \ ordin\[AAcute]rias", "Section", CellChangeTimes->{{3.841045904890222*^9, 3.8410459071729403`*^9}, { 3.8415945700753517`*^9, 3.841594579154108*^9}},ExpressionUUID->"04ad38b9-d9a2-41fd-8e33-\ 1937fabbaee9"], Cell[TextData[{ "O Mathematica pode resolver equa\[CCedilla]\[OTilde]es diferenciais tanto \ simbolicamente quanto numericamente. Por enquanto, vamos nos restringir a \ equa\[CCedilla]\[OTilde]es diferenciais ordin\[AAcute]rias, e explorar \ principalmente aquelas equa\[CCedilla]\[OTilde]es diferenciais associadas ao \ movimento de part\[IAcute]culas puntiformes, obtidas a partir da segunda lei \ de Newton, ou seja,\n", Cell[BoxData[ FormBox[ RowBox[{ TemplateBox[<|"boxes" -> FormBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox[ StyleBox["d", "TI"], "2"], OverscriptBox[ StyleBox["r", "TI"], "\[RightVector]"]}], RowBox[{ StyleBox["d", "TI"], SuperscriptBox[ StyleBox["t", "TI"], "2"]}]], "\[LongEqual]", FractionBox[ RowBox[{ StyleBox["d", "TI"], OverscriptBox[ StyleBox["v", "TI"], "\[RightVector]"]}], RowBox[{ StyleBox["d", "TI"], StyleBox["t", "TI"]}]], "\[LongEqual]", FractionBox[ OverscriptBox[ StyleBox["F", "TI"], "\[RightVector]"], StyleBox["m", "TI"]]}], TraditionalForm], "errors" -> {}, "input" -> "\\frac{d^2\\vec{r}}{dt^2}=\\frac{d\\vec{v}}{dt}=\\frac{\\vec{F}}{m}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], ","}], TraditionalForm]],ExpressionUUID-> "6c1f8944-4e58-4e39-b8b8-ded4ae3dccf6"], "\nem que ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ OverscriptBox[ StyleBox["r", "TI"], "\[RightVector]"], TraditionalForm], "errors" -> {}, "input" -> "\\vec{r}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "c6c62185-5248-41e6-ba7d-c4c9a1871075"], " \[EAcute] o vetor posi\[CCedilla]\[ATilde]o de uma part\[IAcute]cula, ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ OverscriptBox[ StyleBox["v", "TI"], "\[RightVector]"], TraditionalForm], "errors" -> {}, "input" -> "\\vec{v}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "91ca3cd8-37ea-47e5-a2a3-6dcdec1f61a2"], " \[EAcute] seu vetor velocidade, ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["t", "TI"], TraditionalForm], "errors" -> {}, "input" -> "t", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "93386282-85d3-4547-a1d7-05b0004bd863"], " \[EAcute] o tempo, ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["m", "TI"], TraditionalForm], "errors" -> {}, "input" -> "m", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "f7428eb5-12c5-49aa-b5c5-e1b998fbe5c1"], " \[EAcute] a massa da part\[IAcute]cula e ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ OverscriptBox[ StyleBox["F", "TI"], "\[RightVector]"], TraditionalForm], "errors" -> {}, "input" -> "\\vec{F}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "9e2c8c11-c35e-4e8d-ae96-b4d3a87d26b4"], " \[EAcute] o vetor for\[CCedilla]a resultante atuando sobre a \ part\[IAcute]cula. " }], "Text", CellChangeTimes->{{3.8410665589778347`*^9, 3.8410665736356688`*^9}, { 3.841069327433757*^9, 3.841069346878233*^9}, {3.841069386413529*^9, 3.841069386416679*^9}, {3.841594656623073*^9, 3.841594740625969*^9}, { 3.84159480987049*^9, 3.841594850575725*^9}, {3.841595007660842*^9, 3.8415951795460787`*^9}, {3.841595302126652*^9, 3.841595303791597*^9}, { 3.841595522223703*^9, 3.841595525957761*^9}, {3.903767991819138*^9, 3.9037680079544773`*^9}, {3.90376869896763*^9, 3.9037687228192806`*^9}, { 3.9037688212932405`*^9, 3.9037688819971585`*^9}},ExpressionUUID->"17483290-9726-40dc-901a-\ 0239adb96926"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"819adaeb-b750-4384-8bd8-23b33ce04733"], Cell[TextData[{ "A solu\[CCedilla]\[ATilde]o simb\[OAcute]lica de equa\[CCedilla]\[OTilde]es \ diferenciais \[EAcute] obtida a partir da fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["DSolve", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/DSolve.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/DSolve.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ". " }], "Text", CellChangeTimes->{{3.841069824495468*^9, 3.8410698723684464`*^9}, { 3.841126648040525*^9, 3.841126703885021*^9}, {3.841126787731592*^9, 3.841126789520269*^9}, {3.841594762980836*^9, 3.84159480454916*^9}, { 3.841595555500298*^9, 3.8415956047536497`*^9}},ExpressionUUID->"6f2af07a-0a3d-478a-8393-\ 33ce571b5160"] }, Open ]], Cell[TextData[{ "Vamos considerar inicialmente a queda, a partir do repouso, de um corpo \ sujeito a uma for\[CCedilla]a de resist\[EHat]ncia do ar proporcional \ \[AGrave] velocidade, e calcular a depend\[EHat]ncia da velocidade do corpo \ com o tempo. A equa\[CCedilla]\[ATilde]o diferencial a resolver \[EAcute] ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", " ", RowBox[{"v", "/", " ", "d"}], " ", "t"}], "=", RowBox[{ RowBox[{"-", "g"}], "-", RowBox[{"b", " ", "v", " "}]}]}], TraditionalForm]],ExpressionUUID-> "a479c09e-1816-4e92-bb29-922e4558c144"], ", sendo ", Cell[BoxData[ FormBox["b", TraditionalForm]],ExpressionUUID-> "8ef58bee-c9b3-4b71-944e-e4df8cb2e216"], " um par\[AHat]metro constante. A sintaxe \[EAcute]" }], "Text", CellChangeTimes->{{3.841595611965324*^9, 3.841595705598823*^9}, { 3.841595757427567*^9, 3.841595774068248*^9}, {3.841745923183269*^9, 3.8417459407774963`*^9}},ExpressionUUID->"3a27b55d-d132-4eed-9dba-\ 99207a1c1cc3"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"v", "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{ RowBox[{"-", "g"}], "-", RowBox[{"b", "*", RowBox[{"v", "[", "t", "]"}]}]}]}], ",", RowBox[{"v", "[", "t", "]"}], ",", "t"}], "]"}], " "}]], "Input", CellChangeTimes->{{3.841595717079739*^9, 3.841595751187438*^9}, { 3.8415957843116083`*^9, 3.8415957844937696`*^9}, {3.841595947169833*^9, 3.841595953374267*^9}, 3.841775580424871*^9}, CellLabel->"",ExpressionUUID->"f721ddc2-0b42-48f3-92f4-431c51aae09d"], Cell[TextData[{ "A solu\[CCedilla]\[ATilde]o \[EAcute] apresentada na forma de uma lista \ aninhada com uma regra de substitui\[CCedilla]\[ATilde]o, e h\[AAcute] uma \ constante de integra\[CCedilla]\[ATilde]o ", Cell[BoxData[ TemplateBox[{"1"}, "C"]], CellChangeTimes->{{3.841595776557761*^9, 3.841595785246605*^9}}, ExpressionUUID->"50a6d9f3-efbe-438b-9f8f-fba22ea2cb43"], " presente. Para impedir que ela apare\[CCedilla]a, basta informarmos a \ condi\[CCedilla]\[ATilde]o inicial, transformando o primeiro argumento de ", StyleBox["DSolve", FontFamily->"Source Code Pro", FontWeight->"Regular"], " em uma lista:" }], "Text", CellChangeTimes->{{3.8415957997963133`*^9, 3.8415959406121807`*^9}, { 3.841595994792634*^9, 3.841596015200276*^9}, {3.841745956225107*^9, 3.841745957663875*^9}},ExpressionUUID->"ae3200d7-a27d-46a1-ba80-\ 2254d22944ad"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"v", "'"}], "[", "t", "]"}], "\[Equal]", RowBox[{ RowBox[{"-", "g"}], "-", RowBox[{"b", "*", RowBox[{"v", "[", "t", "]"}]}]}]}], ",", RowBox[{ RowBox[{"v", "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"v", "[", "t", "]"}], ",", "t"}], "]"}], " ", "//", "Simplify", " "}]}]], "Input", CellChangeTimes->{{3.841595975848894*^9, 3.8415959856023083`*^9}, { 3.841596070928871*^9, 3.841596087194428*^9}, {3.841596294007408*^9, 3.8415962946416397`*^9}, 3.84166003855014*^9}, CellLabel->"",ExpressionUUID->"46b66f26-107d-4102-9d1c-1fec71128d42"], Cell[TextData[{ "Da solu\[CCedilla]\[ATilde]o conclu\[IAcute]mos, como esperado, que existe \ uma velocidade terminal de m\[OAcute]dulo ", Cell[BoxData[ FormBox[ RowBox[{"g", "/", "b"}], TraditionalForm]],ExpressionUUID-> "3172306c-2cfa-4aad-a746-98d6f64b889e"], ", que tamb\[EAcute]m podemos observar tra\[CCedilla]ando o gr\[AAcute]fico \ de ", Cell[BoxData[ FormBox[ RowBox[{"v", "(", "t", ")"}], TraditionalForm]],ExpressionUUID-> "5ebb5a72-2d3b-416c-8438-d1b9e53251d4"], ":" }], "Text", CellChangeTimes->{{3.841596175008204*^9, 3.841596278136402*^9}},ExpressionUUID->"067ae3b6-5855-4d22-8424-\ a081007c8aa7"], Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"v", "[", "t", "]"}], "/.", "sol"}], "/.", RowBox[{"{", RowBox[{ RowBox[{"g", "\[Rule]", " ", "10"}], ",", RowBox[{"b", "\[Rule]", " ", "1"}]}], "}"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.841596121917713*^9, 3.841596154568227*^9}, { 3.841596307125616*^9, 3.841596308053091*^9}, 3.841660039584216*^9}, CellLabel->"",ExpressionUUID->"3c291639-6a50-4c2a-9cbd-9aef58237291"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"1ebb86e5-ae3a-41c4-a4f9-56967928527d"], Cell["\<\ Para determinarmos a altura da part\[IAcute]cula como \ fun\[CCedilla]\[ATilde]o do tempo, temos que resolver uma equa\[CCedilla]\ \[ATilde]o diferencial de segunda ordem:\ \>", "Text", CellChangeTimes->{{3.84113284494243*^9, 3.841133028657627*^9}, { 3.84113314872999*^9, 3.841133148730081*^9}, {3.841133185708596*^9, 3.841133198171258*^9}, {3.84113329867115*^9, 3.841133332401669*^9}, { 3.841133583738096*^9, 3.841133631504698*^9}, {3.841133797487454*^9, 3.841133829345244*^9}, {3.841133890525779*^9, 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"C"]], CellChangeTimes->{3.841596759573064*^9},ExpressionUUID-> "78e8dc5b-4658-4fed-891a-ea63a289e25c"], ", que n\[ATilde]o estariam presentes se estabelec\[EHat]ssemos as condi\ \[CCedilla]\[OTilde]es iniciais para a velocidade e a altura da \ part\[IAcute]cula:" }], "Text", CellChangeTimes->{{3.841596774310128*^9, 3.841596785390606*^9}, { 3.841596826522129*^9, 3.841596903966227*^9}},ExpressionUUID->"3be554f0-0b54-4457-845c-\ cb656f003b7c"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"soly", "=", RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"y", "''"}], "[", "t", "]"}], "\[Equal]", RowBox[{ RowBox[{"-", "g"}], "-", RowBox[{"b", "*", RowBox[{ RowBox[{"y", "'"}], "[", "t", "]"}]}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"y", "'"}], "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"y", "[", "0", "]"}], "\[Equal]", "h"}]}], "}"}], ",", RowBox[{"y", "[", "t", "]"}], ",", "t"}], "]"}], "//", "Simplify", " "}]}]], "Input", CellChangeTimes->{{3.841596906814436*^9, 3.841596958087111*^9}, 3.8416600469205303`*^9, 3.841775702683448*^9, 3.8417757481797*^9}, CellLabel->"",ExpressionUUID->"e787534c-b2ff-4f2a-a263-3f284f3ac54e"], Cell[TextData[{ "Tra\[CCedilla]ando o gr\[AAcute]fico de ", Cell[BoxData[ FormBox[ RowBox[{"y", "(", "t", ")"}], TraditionalForm]],ExpressionUUID-> "7b5d14d0-b286-4f5b-98e9-dbcbd4c41564"], ", notamos mais uma vez o efeito da velocidade terminal: " }], "Text", CellChangeTimes->{{3.841597030377881*^9, 3.841597063193411*^9}},ExpressionUUID->"77a56d04-140a-4ddd-b8be-\ 439f2fc25fc6"], Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"y", "[", "t", "]"}], "/.", "soly"}], "/.", RowBox[{"{", RowBox[{ RowBox[{"g", "\[Rule]", " ", "10"}], ",", RowBox[{"b", "\[Rule]", " ", "1"}], ",", RowBox[{"h", "\[Rule]", " ", "100"}]}], "}"}]}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.841596968177726*^9, 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Para isso, \[EAcute] preciso \ \[OpenCurlyDoubleQuote]eliminar\[CloseCurlyDoubleQuote] as chaves na solu\ \[CCedilla]\[ATilde]o, o que pode ser feito com a sintaxe abaixo:\ \>", "Text", CellChangeTimes->{{3.841135311404216*^9, 3.84113537789122*^9}, { 3.841135429961483*^9, 3.841135429965598*^9}, {3.8416598271766653`*^9, 3.8416598899883223`*^9}, {3.841659996177533*^9, 3.8416600087213087`*^9}, 3.8416601149561977`*^9},ExpressionUUID->"5dda8998-ea32-493e-ace1-\ f46c0168046d"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"altura", "[", "t_", "]"}], ":=", RowBox[{ RowBox[{"y", "[", "t", "]"}], "/.", RowBox[{"soly", "[", RowBox[{"[", "1", "]"}], "]"}], " "}]}]], "Input", CellChangeTimes->{{3.841659938844657*^9, 3.841659946734606*^9}, { 3.841659979961254*^9, 3.841659980541143*^9}, {3.841660063909317*^9, 3.841660081051421*^9}, 3.841660235708366*^9}, CellLabel->"",ExpressionUUID->"f68378fd-9512-41e6-baf2-98ca94cb01b4"], Cell[TextData[{ "Temos que dar para a fun\[CCedilla]\[ATilde]o um nome diferente de ", StyleBox["y[", FontFamily->"Source Code Pro", FontWeight->"Regular"], StyleBox["t", FontFamily->"Source Code Pro", FontWeight->"Regular", FontSlant->"Italic"], StyleBox["_]", FontFamily->"Source Code Pro", FontWeight->"Regular"], StyleBox[" para evitar problemas com uma defini\[CCedilla]\[ATilde]o \ recursiva.", FontWeight->"Regular"] }], "ItemParagraph", CellChangeTimes->{{3.841660144948983*^9, 3.8416602189287443`*^9}},ExpressionUUID->"f3f0470b-566b-43be-9b73-\ d80d962ed3b8"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"altura", "[", "t", "]"}], " "}]], "Input", CellChangeTimes->{{3.841659948971552*^9, 3.841659952484067*^9}, { 3.841660071607079*^9, 3.841660086989134*^9}, 3.841660236673728*^9}, CellLabel->"",ExpressionUUID->"e9c89890-c066-421c-9550-8bdc950b7534"], Cell[TextData[{ "Com essa defini\[CCedilla]\[ATilde]o, fica mais f\[AAcute]cil efetuar c\ \[AAcute]lculos. Por exemplo, podemos investigar o limite do resultado quando \ a resist\[EHat]ncia do ar se torna desprez\[IAcute]vel, utilizando a fun\ \[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Limit", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Limit.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Limit.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ":" }], "Text", CellChangeTimes->{{3.841660284967141*^9, 3.841660326771578*^9}, { 3.841660359495207*^9, 3.8416603595031967`*^9}},ExpressionUUID->"93c827d5-3a6c-43e5-9cc3-\ 009b8ecb5a79"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Limit", "[", RowBox[{ RowBox[{"altura", "[", "t", "]"}], ",", RowBox[{"b", "\[Rule]", " ", "0"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.8416603766443043`*^9, 3.8416603939271173`*^9}}, CellLabel->"",ExpressionUUID->"5c4340ad-7666-4456-b4ac-2e91d49faca0"], Cell["\<\ Note que ter\[IAcute]amos problemas se tent\[AAcute]ssemos calcular essa \ mesma fun\[CCedilla]\[ATilde]o utilizando o operador de substitui\[CCedilla]\ \[ATilde]o:\ \>", "Text", CellChangeTimes->{{3.873033543720537*^9, 3.873033575781986*^9}, { 3.9037664669282427`*^9, 3.9037664710734367`*^9}},ExpressionUUID->"88f4ddcc-c3db-4159-9ad3-\ 8461b38a37db"], Cell[BoxData[ RowBox[{ RowBox[{"altura", "[", "t", "]"}], "/.", RowBox[{"b", "\[Rule]", " ", "0"}]}]], "Input", CellChangeTimes->{{3.8730335081475077`*^9, 3.873033515572081*^9}}, CellLabel->"",ExpressionUUID->"32536b0b-46f1-4d72-93c2-b87fdefa5daa"] }, Closed]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"119c62fe-d5f6-44a0-b33a-8472a3087ae5"], Cell[TextData[{ "A fun\[CCedilla]\[ATilde]o ", StyleBox["DSolve", FontFamily->"Source Code Pro", FontWeight->"Regular"], " tamb\[EAcute]m lida com sistemas de equa\[CCedilla]\[OTilde]es \ diferenciais ordin\[AAcute]rias acopladas. Vamos ilustrar essa capacidade com \ o problema de duas part\[IAcute]culas conectadas a molas, conforme mostrado \ na figura abaixo." }], "Text", CellChangeTimes->{{3.841136770347685*^9, 3.841136774565173*^9}, { 3.841136843990129*^9, 3.841136876727298*^9}, {3.841660470953413*^9, 3.841660483923648*^9}, {3.841660714313951*^9, 3.8416607813751802`*^9}, { 3.873033644334306*^9, 3.873033644340686*^9}},ExpressionUUID->"78e3fc46-3867-434c-a05f-\ 83755920fe29"] }, Open ]], Cell[BoxData[ GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzt3Qm8/fW0P34lDUqIZCjJUNyiKJJSGpCMDaIBhcrQQCqihJskSTKngSZD klLpKirJdOsSZYquDA2XLi663Ov+Pn/Pt//Kp0973p99zt7nrNfjsb7fc/bZ w2e/P+/1Xq81vNd7jZfut92eS97lLnd53bJ/+2e73V+/+f77737w9vf62y/P 3/d1r9xr3z1e/ox9D9hjrz32f+JL7/q3Bz/zN/nR356/1N/+rxKJRCKRSCQS iUQikUgkEolEIpFIJBKJRCKRSCQSiUQikUgkEolEIpFIJBKJRCKRSCQSiUQi kUgkEolEIpFIJBKJRCKRSCQSiRHw//7f/7td/u///q+Inwd5nef+9a9/Hfg1 iUSiPTR1N3RxkNeFrqfuJhKJRCLRLthVNvkvf/lL9cc//rH6zW9+U/3qV7+q 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"SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"16bc25f6-f740-430b-a6f5-437fca000d5b"], Cell[TextData[{ "Introduzindo ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ RowBox[{"\[Omega]", "\[LongEqual]", SqrtBox[ RowBox[{ StyleBox["k", "TI"], "/", StyleBox["m", "TI"]}]]}], TraditionalForm], "errors" -> {}, "input" -> "\\omega=\\sqrt{k/m}", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "6d3c17a4-0ef3-4e60-bd0c-4d95be93797c"], " e supondo que o sistema seja abandonado do repouso, apenas com a massa da \ esquerda deslocada de uma quantidade ", Cell[BoxData[ FormBox[ TemplateBox[<|"boxes" -> FormBox[ StyleBox["A", "TI"], TraditionalForm], "errors" -> {}, "input" -> "A", "state" -> "Boxes"|>, "TeXAssistantTemplate"], TraditionalForm]],ExpressionUUID-> "cce965cc-9cc1-46a1-9844-86e06eb792da"], ", temos" }], "Text", CellChangeTimes->{{3.841666248194399*^9, 3.841666305513163*^9}, { 3.841666553701706*^9, 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CellLabel->"",ExpressionUUID->"29549a82-799b-4745-a2ba-bc7e01250ea3"], Cell[TextData[{ "Se quisermos definir fun\[CCedilla]\[OTilde]es para as solu\[CCedilla]\ \[OTilde]es, temos que nos referir \[AGrave]s regras de transforma\[CCedilla]\ \[ATilde]o separadamente, o que pode ser feito com o aux\[IAcute]lio da fun\ \[CCedilla]\[ATilde]o ", StyleBox["Part", FontFamily->"Source Code Pro", FontWeight->"Regular"], " da seguinte forma:" }], "Text", CellChangeTimes->{{3.841666753818289*^9, 3.841666808909021*^9}, { 3.8416668461692753`*^9, 3.8416668675103292`*^9}},ExpressionUUID->"79b61e53-da17-48b0-b5a5-\ 624f20b76948"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"sol12", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}], " "}], "\[IndentingNewLine]", RowBox[{"sol12", "[", RowBox[{"[", RowBox[{"1", ",", "2"}], "]"}], "]"}]}], "Input", CellChangeTimes->{{3.841666811840892*^9, 3.841666841242731*^9}, 3.841671591213456*^9}, CellLabel->"",ExpressionUUID->"1db29d83-e20c-4184-8ceb-e021623abf9b"], Cell["Logo, vamos definir", "Text", CellChangeTimes->{{3.841666906481804*^9, 3.841666912026922*^9}},ExpressionUUID->"0f30e5c8-4e73-482b-90c5-\ 186d9d75373a"], Cell[BoxData[{ RowBox[{ RowBox[{"pos1", "[", "t_", "]"}], ":=", RowBox[{ RowBox[{"x1", "[", "t", "]"}], "/.", RowBox[{"sol12", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}]}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"pos2", "[", "t_", "]"}], ":=", RowBox[{ RowBox[{"x2", "[", "t", "]"}], "/.", RowBox[{"sol12", "[", RowBox[{"[", RowBox[{"1", ",", "2"}], "]"}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.841666914768224*^9, 3.8416669443603973`*^9}}, CellLabel-> "In[124]:=",ExpressionUUID->"816e5cc5-aac7-42de-afca-70dea146016f"] }, Closed]] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"01609085-f87c-413c-b899-dfeb2c66cdd4"], Cell[TextData[{ "Vamos fazer gr\[AAcute]ficos das solu\[CCedilla]\[OTilde]es, supondo ", Cell[BoxData[ FormBox[ RowBox[{"\[Omega]", "=", RowBox[{"2", "\[Pi]"}]}], TraditionalForm]],ExpressionUUID-> "62eb9120-5cd2-4b7b-a1fd-5125665a9ffb"], " e ", Cell[BoxData[ FormBox[ RowBox[{"A", "=", "5"}], TraditionalForm]],ExpressionUUID-> "3e233d21-81f4-4173-871e-4547f470df22"], " (em unidades arbitr\[AAcute]rias):" }], "Text", CellChangeTimes->{{3.841667073350079*^9, 3.841667074603415*^9}, { 3.84166711829928*^9, 3.84166722107796*^9}},ExpressionUUID->"da834854-950c-494b-82db-\ 9feefff28cba"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"subs", "=", RowBox[{"{", RowBox[{ RowBox[{"\[Omega]", "\[Rule]", " ", RowBox[{"2", "*", "\[Pi]"}]}], ",", RowBox[{"A", "\[Rule]", " ", "5"}]}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"pos1", "[", "t", "]"}], "/.", "subs"}], ",", RowBox[{ RowBox[{"pos2", "[", "t", "]"}], "/.", "subs"}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "5"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.841667313832353*^9, 3.841667337521984*^9}, { 3.841667858401917*^9, 3.841667858575582*^9}}, CellLabel->"",ExpressionUUID->"6beb3b9a-2e3c-4067-b48f-268801d6d5f8"], Cell[TextData[{ "Note que as solu\[CCedilla]\[OTilde]es s\[ATilde]o superposi\[CCedilla]\ \[OTilde]es de movimentos harm\[OHat]nicos com duas frequ\[EHat]ncias \ distintas, ", Cell[BoxData[ FormBox["\[Omega]", TraditionalForm]],ExpressionUUID-> "cb87fb67-865a-43d6-86d6-cd8d2758a121"], " e ", Cell[BoxData[ FormBox[ RowBox[{ SqrtBox["3"], "\[Omega]"}], TraditionalForm]],ExpressionUUID-> "8c5bc9d7-c23b-4751-85ee-58dd976dc00b"], ". Esses s\[ATilde]o os chamados modos normais de \ oscila\[CCedilla]\[ATilde]o do sistema, e correspondem a situa\[CCedilla]\ \[OTilde]es em que as duas massas movem-se com a mesma frequ\[EHat]ncia e com \ uma fase relativa fixa. Por simetria, essas situa\[CCedilla]\[OTilde]es devem \ corresponder a abandonar o sistema do repouso com (i) ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1"], "=", SubscriptBox["x", "2"]}], TraditionalForm]],ExpressionUUID-> "7ef08a92-d02c-44de-8cc5-c2aac3884da0"], " ou (ii) ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "1"], "=", RowBox[{"-", SubscriptBox["x", "2"]}]}], TraditionalForm]],ExpressionUUID-> "c898bdf9-a912-4a37-bc43-1a4d83ab0b9e"], ", o que pode ser confirmado pela solu\[CCedilla]\[ATilde]o \ expl\[IAcute]cita das equa\[CCedilla]\[OTilde]es de movimento:" }], "Text", CellChangeTimes->{{3.841667410046423*^9, 3.8416674962433357`*^9}, { 3.8416676291977158`*^9, 3.841667764670178*^9}, {3.841667949104847*^9, 3.84166795048777*^9}, 3.841746287347541*^9},ExpressionUUID->"697130b9-45d6-4629-a029-\ 4f1b2f46d2b1"], Cell[BoxData[ RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{"eq1", ",", "eq2", ",", RowBox[{ RowBox[{"x1", "[", "0", "]"}], "\[Equal]", "A"}], ",", RowBox[{ RowBox[{"x2", "[", "0", "]"}], "\[Equal]", "A"}], ",", RowBox[{ RowBox[{ RowBox[{"x1", "'"}], "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"x2", "'"}], "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x1", "[", "t", "]"}], ",", RowBox[{"x2", "[", "t", "]"}]}], "}"}], ",", "t"}], "]"}], "//", "FullSimplify", " ", RowBox[{"(*", " ", RowBox[{"Primeiro", " ", "modo", " ", "normal"}], " ", "*)"}], " "}]], "Input", CellChangeTimes->{{3.84166778103025*^9, 3.8416678240179358`*^9}}, CellLabel->"",ExpressionUUID->"324aa243-2ce0-47bd-bd3e-809d181feecd"], Cell[BoxData[ RowBox[{ RowBox[{"DSolve", "[", RowBox[{ RowBox[{"{", RowBox[{"eq1", ",", "eq2", ",", RowBox[{ RowBox[{"x1", "[", "0", "]"}], "\[Equal]", "A"}], ",", RowBox[{ RowBox[{"x2", "[", "0", "]"}], "\[Equal]", RowBox[{"-", "A"}]}], ",", RowBox[{ RowBox[{ RowBox[{"x1", "'"}], "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{ RowBox[{"x2", "'"}], "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"x1", "[", "t", "]"}], ",", RowBox[{"x2", "[", "t", "]"}]}], "}"}], ",", "t"}], "]"}], "//", "FullSimplify", " ", RowBox[{"(*", " ", RowBox[{"Segundo", " ", "modo", " ", "normal"}], " ", "*)"}], " "}]], "Input", CellChangeTimes->{{3.841667803282383*^9, 3.841667822698217*^9}}, CellLabel->"",ExpressionUUID->"163fd959-a303-4c16-8141-8861c69fe12c"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"584b9b4b-210e-49c3-80fd-6f6337a9667b"], Cell[TextData[{ "Equa\[CCedilla]\[OTilde]es diferenciais que n\[ATilde]o podem ser \ resolvidas analiticamente requerem o uso da fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["NDSolve", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/NDSolve.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/NDSolve.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", cuja sintaxe vamos exemplificar a seguir." }], "Text", CellChangeTimes->{{3.841669462729906*^9, 3.8416695380476313`*^9}, { 3.8416695786135902`*^9, 3.841669578623136*^9}},ExpressionUUID->"7426d614-b864-4444-8e54-\ 412a17cab964"], Cell[TextData[{ "Considere primeiro um p\[EHat]ndulo simples, de comprimento ", Cell[BoxData[ FormBox["L", TraditionalForm]],ExpressionUUID-> "b2f22fcb-ffca-48b8-9339-687f688c1c00"], ", mas n\[ATilde]o restrito ao limite de pequenas \ oscila\[CCedilla]\[OTilde]es. Nesse caso, a equa\[CCedilla]\[ATilde]o de \ movimento para o \[AHat]ngulo ", Cell[BoxData[ FormBox["\[Theta]", TraditionalForm]],ExpressionUUID-> "b6863ac9-cda2-4933-a8a5-29bc21d8836c"], " de desvio em rela\[CCedilla]\[ATilde]o \[AGrave] vertical \[EAcute] ", Cell[BoxData[ FormBox[ RowBox[{ OverscriptBox["\[Theta]", ".."], "=", RowBox[{ RowBox[{"-", SuperscriptBox["\[Omega]", "2"]}], "sen\[Theta]"}]}], TraditionalForm]], ExpressionUUID->"cc8f0446-58ce-4086-9216-3dbd42f96dcf"], ", com ", Cell[BoxData[ FormBox[ RowBox[{"\[Omega]", "=", SqrtBox[ RowBox[{"g", "/", "L"}]]}], TraditionalForm]],ExpressionUUID-> "4dac9a98-8402-4e20-8666-0a410f35aa95"], ", que n\[ATilde]o admite solu\[CCedilla]\[ATilde]o anal\[IAcute]tica \ fechada. A solu\[CCedilla]\[ATilde]o num\[EAcute]rica, supondo ", Cell[BoxData[ FormBox[ RowBox[{"\[Omega]", "=", RowBox[{"2", "\[Pi]"}]}], TraditionalForm]],ExpressionUUID-> "60f6dc23-ec86-437f-83e6-832d67e1b2e0"], " rad e um p\[EHat]ndulo partindo do repouso, com um \[AHat]ngulo ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["\[Theta]", "0"], "=", RowBox[{"\[Pi]", "/", "3"}]}], TraditionalForm]],ExpressionUUID-> "d093a58f-5916-4ee1-931b-dfe7cacb5baf"], " rad, \[EAcute] obtida entre ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", "0"}], TraditionalForm]],ExpressionUUID-> "b23d6667-73c2-4605-8c6e-e93b438a4ab6"], " e ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", "10"}], TraditionalForm]],ExpressionUUID-> "2b9734b4-02db-4a4b-b249-012a32268c94"], " s pela sintaxe abaixo:" }], "Text", CellChangeTimes->{{3.841669591644033*^9, 3.841669632382187*^9}, { 3.841669698339326*^9, 3.841669787328947*^9}, {3.841669907978394*^9, 3.841669944106264*^9}, {3.841669977519938*^9, 3.8416700859061737`*^9}, { 3.841670124696618*^9, 3.841670211142284*^9}, {3.841670269407209*^9, 3.841670359272195*^9}, {3.841670447213931*^9, 3.841670483068635*^9}, { 3.9037667660234604`*^9, 3.903766771912467*^9}},ExpressionUUID->"c7b31c4c-bc0c-4e59-8b1b-\ daa14cbecb36"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sol", "=", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[Theta]", "''"}], "[", "t", "]"}], "\[Equal]", RowBox[{ RowBox[{"-", SuperscriptBox["\[Omega]", "2"]}], "*", RowBox[{"Sin", "[", RowBox[{"\[Theta]", "[", "t", "]"}], "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"\[Theta]", "[", "0", "]"}], "\[Equal]", "\[Theta]0"}]}], "}"}], "/.", RowBox[{"{", RowBox[{ RowBox[{"\[Omega]", "\[Rule]", " ", RowBox[{"2", "*", "\[Pi]"}]}], ",", RowBox[{"\[Theta]0", "\[Rule]", " ", RowBox[{"\[Pi]", "/", "3"}]}]}], "}"}]}], ",", "\[Theta]", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.841670360606423*^9, 3.841670442287109*^9}, { 3.8416704949638767`*^9, 3.841670561561328*^9}, {3.841670596733412*^9, 3.8416706060884123`*^9}, {3.841670815526223*^9, 3.841670816014935*^9}, 3.841671651582242*^9}, CellLabel->"",ExpressionUUID->"83371c59-a816-4f29-94e8-186d11522aea"], Cell[TextData[{ "Note que o Mathematica retorna uma solu\[CCedilla]\[ATilde]o em forma de \ uma \[OpenCurlyDoubleQuote]fun\[CCedilla]\[ATilde]o de interpola\[CCedilla]\ \[ATilde]o\[CloseCurlyDoubleQuote], que permite obter uma estimativa para a \ solu\[CCedilla]\[ATilde]o em qualquer ponto do intervalo entre ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", "0"}], TraditionalForm]],ExpressionUUID-> "ab38e804-8e47-4d99-b30b-88cc2b052a6d"], " e ", Cell[BoxData[ FormBox[ RowBox[{"t", "=", "10"}], TraditionalForm]],ExpressionUUID-> "86071b7f-13d9-4631-ae1d-e7a4097ce239"], " s. (A \[OpenCurlyDoubleQuote]interpola\[CCedilla]\[ATilde]o\ \[CloseCurlyDoubleQuote] vem do fato de que a solu\[CCedilla]\[ATilde]o num\ \[EAcute]rica \[EAcute] obtida estritamente para um conjunto discreto de \ pontos, fora dos quais h\[AAcute] um ajuste, geralmente polinomial, para \ fornecer novas estimativas.)" }], "ItemParagraph", CellChangeTimes->{{3.8416706594474916`*^9, 3.8416707970912533`*^9}},ExpressionUUID->"fc86d22c-54b3-4da5-a73b-\ a0e874880796"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"825f4589-e9ea-4bcc-aece-b2a31ce0f6d2"], Cell["\<\ Assim como antes, podemos definir uma fun\[CCedilla]\[ATilde]o a partir da \ solu\[CCedilla]\[ATilde]o:\ \>", "Text", CellChangeTimes->{{3.841670839684544*^9, 3.841670854995302*^9}},ExpressionUUID->"ee43ac72-665d-48e6-bf56-\ c804c5807ffe"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"angulo", "[", "t_", "]"}], ":=", RowBox[{ RowBox[{"\[Theta]", "[", "t", "]"}], "/.", RowBox[{"sol", "[", RowBox[{"[", "1", "]"}], "]"}], " "}]}]], "Input", CellChangeTimes->{{3.841670857533699*^9, 3.8416708681809797`*^9}, { 3.841670913683281*^9, 3.841670920403982*^9}, 3.841671647802556*^9, 3.8416832566674833`*^9}, CellLabel->"",ExpressionUUID->"2def031e-13c2-4fcf-adfc-6087fa829b78"], Cell[CellGroupData[{ Cell["\<\ Vamos produzir um gr\[AAcute]fico do \[AHat]ngulo como \ fun\[CCedilla]\[ATilde]o do tempo, colocando tamb\[EAcute]m o comportamento \ harm\[OHat]nico caso valesse o limite de pequenas \ oscila\[CCedilla]\[OTilde]es:\ \>", "Text", CellChangeTimes->{{3.841670940789277*^9, 3.841670963625909*^9}, { 3.841670999750203*^9, 3.8416710294724913`*^9}},ExpressionUUID->"778602b5-4190-440d-84a0-\ 1a49ab97add1"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"angulo", "[", "t", "]"}], ",", RowBox[{ RowBox[{"\[Pi]", "/", "3"}], "*", RowBox[{"Cos", "[", RowBox[{"2", "*", "\[Pi]", "*", "t"}], "]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.841670976840888*^9, 3.8416709848105173`*^9}, { 3.841671074876195*^9, 3.841671101802663*^9}, 3.84167164402295*^9}, CellLabel->"",ExpressionUUID->"5c802070-369f-474e-84d0-35e0df9425bc"], Cell["\<\ Vemos claramente que o per\[IAcute]odo \[OpenCurlyDoubleQuote]verdadeiro\ \[CloseCurlyDoubleQuote] \[EAcute] maior do que aquele previsto pela aproxima\ \[CCedilla]\[ATilde]o de pequenas oscila\[CCedilla]\[OTilde]es. Essa tend\ \[EHat]ncia se acentua \[AGrave] medida que o \[AHat]ngulo inicial aumenta:\ \>", "Text", CellChangeTimes->{{3.841671119671062*^9, 3.841671305572225*^9}, { 3.8416743339519672`*^9, 3.8416743358750877`*^9}},ExpressionUUID->"0d042f85-d4f3-428b-b105-\ fb0a9cd8288e"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"subs", "=", RowBox[{"{", RowBox[{ RowBox[{"\[Omega]", "\[Rule]", " ", RowBox[{"2", "*", "\[Pi]"}]}], ",", RowBox[{"\[Theta]0", "\[Rule]", " ", RowBox[{"\[Pi]", "-", "0.01"}]}]}], "}"}]}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"sol", "=", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[Theta]", "''"}], "[", "t", "]"}], "\[Equal]", RowBox[{ RowBox[{"-", SuperscriptBox["\[Omega]", "2"]}], "*", RowBox[{"Sin", "[", RowBox[{"\[Theta]", "[", "t", "]"}], "]"}]}]}], ",", RowBox[{ RowBox[{ RowBox[{"\[Theta]", "'"}], "[", "0", "]"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"\[Theta]", "[", "0", "]"}], 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