(* 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[ 94732, 2416] NotebookOptionsPosition[ 81333, 2207] NotebookOutlinePosition[ 85483, 2286] CellTagsIndexPosition[ 85404, 2281] 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}, { 3.8792310850616074`*^9, 3.879231085319373*^9}, {3.906042392089398*^9, 3.906042392205077*^9}},ExpressionUUID->"92508663-47f1-4deb-9615-\ d2c271e8e453"], Cell["\<\ Aula 07: gr\[AAcute]ficos tridimensionais; gr\[AAcute]ficos interativos; \ anima\[CCedilla]\[OTilde]es\ \>", "Subtitle", CellChangeTimes->{{3.699269582406044*^9, 3.699269584285553*^9}, { 3.8393232649484053`*^9, 3.839323284264048*^9}, 3.840031268565436*^9, { 3.8407028843722677`*^9, 3.840702924713172*^9}, {3.840714600599773*^9, 3.8407146281949797`*^9}, {3.841157909295815*^9, 3.841157911477744*^9}, { 3.841480259563218*^9, 3.8414802737602463`*^9}, {3.8415945589945517`*^9, 3.841594560546528*^9}, {3.842262188714292*^9, 3.8422622084045753`*^9}, { 3.842288735025601*^9, 3.8422887376233883`*^9}, {3.842288769619337*^9, 3.8422887810036163`*^9}, {3.843294467913928*^9, 3.843294500290168*^9}, { 3.843553405919444*^9, 3.8435534093620787`*^9}, {3.8792310881409645`*^9, 3.8792310883059664`*^9}, {3.9060423952808247`*^9, 3.9060423966098013`*^9}},ExpressionUUID->"14f8f51b-58e9-4033-aea1-\ fd611ac95bb9"] }, Open ]], Cell[CellGroupData[{ Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"ee3ea2a1-973b-4e9a-a704-0d8a3c30d1be"], Cell["Gr\[AAcute]ficos tridimensionais", "Section", CellChangeTimes->{ 3.699269664116198*^9, 3.70454748091461*^9, {3.704547516075143*^9, 3.7045475190812902`*^9}, {3.706294276733098*^9, 3.706294304716371*^9}, { 3.839323324298633*^9, 3.839323331059084*^9}, 3.840031288455207*^9, { 3.840714636688951*^9, 3.840714639748958*^9}, {3.841057130911109*^9, 3.84105713513932*^9}, {3.841480515514193*^9, 3.84148051572683*^9}, { 3.84226221732572*^9, 3.842262223457004*^9}, {3.843294513408573*^9, 3.843294517052754*^9}},ExpressionUUID->"7da6da63-6378-4f26-b58d-\ d42a7250afa4"], Cell[TextData[{ "Dada uma fun\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"f", "(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]],ExpressionUUID-> "637bdd88-9699-4c6c-8035-cf4e81ad3c67"], ", invocar a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Plot3D", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Plot3D.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Plot3D.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], " do Mathematica na forma ", StyleBox["Plot3D[f,{x,xmin,xmax},{y,ymin,ymax}]", FontFamily->"Source Code Pro", FontWeight->"Regular"], " produz um gr\[AAcute]fico tridimensional de ", Cell[BoxData[ FormBox["f", TraditionalForm]],ExpressionUUID-> "3a08a011-02c3-49c5-8076-afd75209800a"], " como fun\[CCedilla]\[ATilde]o das vari\[AAcute]veis ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "31bd3db3-084e-45f4-ac6d-c9f3b620bc09"], " de ", StyleBox["xmin", FontFamily->"Source Code Pro", FontWeight->"Regular"], " at\[EAcute] ", StyleBox["xmax", FontFamily->"Source Code Pro", FontWeight->"Regular"], " e ", Cell[BoxData[ FormBox["y", TraditionalForm]],ExpressionUUID-> "b5c7659a-0dd6-4b58-8879-5c57aa2a21f3"], " de ", StyleBox["ymin", FontFamily->"Source Code Pro", FontWeight->"Regular"], " at\[EAcute] ", StyleBox["ymax", FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Vamos ilustrar com o gr\[AAcute]fico do potencial el\[EAcute]trico de um \ dipolo, ou seja, duas cargas puntiformes ", Cell[BoxData[ FormBox[ RowBox[{"+", "Q"}], TraditionalForm]],ExpressionUUID-> "cf9ebf8a-6554-4852-883e-9cd219cc2167"], " e ", Cell[BoxData[ FormBox[ RowBox[{"-", "Q"}], TraditionalForm]],ExpressionUUID-> "dc94a22e-19ae-4c7e-935d-16dcbed10d1b"], " situadas em ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{"(", RowBox[{ RowBox[{"-", "b"}], ",", "0"}], ")"}]}], TraditionalForm]], ExpressionUUID->"ca724cfb-92d2-4286-9b58-8bd430267c6f"], " e ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], "=", RowBox[{"(", RowBox[{"b", ",", "0"}], ")"}]}], TraditionalForm]],ExpressionUUID-> "6e047ac7-9e3c-42e1-b768-ac8ad032b04c"], ". Lembre que o potencial el\[EAcute]trico de um carga puntiforme ", Cell[BoxData[ FormBox["Q", TraditionalForm]],ExpressionUUID-> "c30f4cb1-1f7d-44a0-80ce-b8b042acedac"], " em um ponto a uma dist\[AHat]ncia ", Cell[BoxData[ FormBox["r", TraditionalForm]],ExpressionUUID-> "954150cb-f0d5-43ec-a477-a8bd9725bf6e"], " da carga \[EAcute] igual a ", Cell[BoxData[ FormBox[ RowBox[{"k", " ", RowBox[{"Q", "/", "r"}]}], TraditionalForm]],ExpressionUUID-> "5f488630-4e0a-4239-8474-7832923482e8"], ". 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As linhas irregulares no gr\[AAcute]fico s\[ATilde]o um efeito da baixa \ densidade de pontos, que corrigiremos a seguir." }], "ItemParagraph", CellChangeTimes->{{3.843295759858673*^9, 3.843295850670024*^9}, { 3.8792319217079735`*^9, 3.8792319558688707`*^9}},ExpressionUUID->"80ae9563-1c02-4b66-a481-\ 5e606d8be694"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"0d6bbe14-c226-415e-b1fc-fcf294444606"], Cell[TextData[{ "Se quisermos remover a caixa em torno do gr\[AAcute]fico, utilizamos a op\ \[CCedilla]\[ATilde]o ", StyleBox["Boxed", FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "b220b631-67ce-430a-824f-765271950ea0"], StyleBox["False", FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Para remover os eixos, a op\[CCedilla]\[ATilde]o \[EAcute] ", StyleBox["Axes", FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "e791073a-4171-4103-b36f-fc696cd6c90b"], StyleBox["False", FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Vamos tamb\[EAcute]m ajustar o intervalo no eixo ", Cell[BoxData[ FormBox["z", TraditionalForm]],ExpressionUUID-> "c559c6e4-7ea9-42a9-b732-703350bed3f7"], ", utilizando ", StyleBox["PlotRange", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", e refinar o gr\[AAcute]fico com ", StyleBox["PlotPoints", FontFamily->"Source Code Pro", FontWeight->"Regular"], "." }], "Text", CellChangeTimes->{{3.843295915233144*^9, 3.8432959210819607`*^9}, { 3.843295955496604*^9, 3.843295958139822*^9}, {3.843296011870647*^9, 3.843296086919487*^9}, {3.843297814136956*^9, 3.843297881700449*^9}, { 3.843584988405768*^9, 3.84358498873169*^9}},ExpressionUUID->"bb6c48d3-962e-46a6-9d6a-\ ad3cb2b9b203"], Cell[BoxData[ RowBox[{"grafico", "=", RowBox[{"Plot3D", "[", RowBox[{ RowBox[{ RowBox[{"V", "[", RowBox[{"x", ",", "y"}], "]"}], "/.", "pars"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "15"}], ",", "15"}], "}"}], ",", RowBox[{"{", RowBox[{"y", ",", RowBox[{"-", "15"}], ",", "15"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"Boxed", "\[Rule]", " ", "False"}], ",", RowBox[{"Axes", "\[Rule]", "False"}], ",", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "0.125"}], ",", "0.125"}], "}"}]}], ",", RowBox[{"PlotPoints", "\[Rule]", " ", "80"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.843295993399487*^9, 3.8432960053811626`*^9}, { 3.843296089942525*^9, 3.843296099511011*^9}, {3.843297887958171*^9, 3.8432980374364967`*^9}, {3.843298079822111*^9, 3.8432980985118713`*^9}, { 3.8432983849096003`*^9, 3.843298385962276*^9}, {3.8433020530741663`*^9, 3.843302054450349*^9}, {3.843555680281966*^9, 3.843555680730096*^9}, 3.8435846244812717`*^9},ExpressionUUID->"35e41775-dae6-460d-bcea-\ 1818650dcecd"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"17774e00-e7a4-4e88-b623-1d0211c771fc"], Cell[TextData[{ "\[CapitalEAcute] poss\[IAcute]vel ajustar a posi\[CCedilla]\[ATilde]o do \ observador, atrav\[EAcute]s da op\[CCedilla]\[ATilde]o ", StyleBox["ViewPoint", FontFamily->"Source Code Pro", FontWeight->"Regular"], ", que recebe uma lista contendo as coordenadas ", Cell[BoxData[ FormBox["x", TraditionalForm]],ExpressionUUID-> "5a0c1247-58bc-4419-a276-682f3a73fa7d"], ", ", Cell[BoxData[ FormBox["y", TraditionalForm]],ExpressionUUID-> "f54915fd-63b2-4bff-85a3-8c11a775d13a"], " e ", Cell[BoxData[ FormBox["z", TraditionalForm]],ExpressionUUID-> "12d5da6e-ce27-498a-9be7-016f9bc81ed5"], " do observador, como nos exemplos a seguir. " }], "Text", CellChangeTimes->{{3.84329829620815*^9, 3.843298318549615*^9}, { 3.843298362594396*^9, 3.84329840933783*^9}, {3.843298440613007*^9, 3.843298468809654*^9}},ExpressionUUID->"0748b206-6537-42a5-be25-\ 2fa6fe5f3073"], Cell[BoxData[ RowBox[{ RowBox[{"Show", "[", RowBox[{"grafico", ",", RowBox[{"ViewPoint", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1.77"}], ",", "2.84", ",", "0.51"}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.84329841705902*^9, 3.84329842855435*^9}, { 3.843298513031458*^9, 3.843298520106174*^9}, 3.8435846223536882`*^9},ExpressionUUID->"329aa24c-4d42-4612-8acc-\ 6d4345704a4f"], Cell[BoxData[ RowBox[{ RowBox[{"Show", "[", RowBox[{"grafico", ",", RowBox[{"ViewPoint", "\[Rule]", " ", RowBox[{"{", RowBox[{"0", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}]}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.843298556323887*^9, 3.8432986283554*^9}, { 3.843298847118277*^9, 3.8432988701311913`*^9}, 3.843584621588745*^9},ExpressionUUID->"3c027daa-3ccc-4b0e-baf1-\ a6e130466a9c"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"916aafb6-db56-46e9-abc4-ac7b00e07c2a"], Cell[TextData[{ "Tamb\[EAcute]m \[EAcute] poss\[IAcute]vel ajustar a \ posi\[CCedilla]\[ATilde]o do observador \[OpenCurlyDoubleQuote]girando\ \[CloseCurlyDoubleQuote] a imagem com aux\[IAcute]lio do mouse. Para isso, \ sobre a figura, mantenha o bot\[ATilde]o esquerdo pressionado enquanto \ movimenta o mouse. Ap\[OAcute]s chegar \[AGrave] posi\[CCedilla]\[ATilde]o \ desejada, \[EAcute] poss\[IAcute]vel extrair as coordenadas da \ posi\[CCedilla]\[ATilde]o copiando a imagem (Ctrl-C ou Cmd-C) e colando-a \ (Ctrl-V ou Cmd-V) como argumento da fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Options", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Options.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Options.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", conforme exemplificado abaixo." }], "Text", CellChangeTimes->{{3.843301813765079*^9, 3.843301933487143*^9}, { 3.843306013438015*^9, 3.843306013442376*^9}, {3.8792316830611606`*^9, 3.879231710799524*^9}},ExpressionUUID->"644a62b1-ac31-494c-8dfd-\ 7ed730ae7044"], Cell[BoxData[ RowBox[{ RowBox[{"Show", "[", "grafico", "]"}], " "}]], "Input", CellChangeTimes->{{3.843301940659185*^9, 3.843301942983018*^9}, 3.8435846199233007`*^9},ExpressionUUID->"9a64b630-5e01-460e-8037-\ f58778c3c710"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Options", "[", "]"}], " ", RowBox[{"(*", " ", RowBox[{ "Cole", " ", "a", " ", "imagem", " ", "entre", " ", "os", " ", "colchetes"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.8433019581082563`*^9, 3.843301978397332*^9}, 3.843302027943808*^9, 3.843315501247065*^9, 3.8433155763402853`*^9, { 3.9060425189633703`*^9, 3.906042532571666*^9}},ExpressionUUID->"ac4ba23e-729c-4524-a64c-\ 1c6a721d9c92"], Cell[TextData[{ "Na \[UAcute]ltima sa\[IAcute]da, procure pela op\[CCedilla]\[ATilde]o ", StyleBox["ViewPoint", FontFamily->"Source Code Pro", FontWeight->"Regular"], "." }], "Text", CellChangeTimes->{{3.8792317667620077`*^9, 3.8792317964517727`*^9}},ExpressionUUID->"d2671eb4-91e9-40e9-b8a2-\ ffe45ebda9ba"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"3d840658-e7d8-403c-867c-a584b277195f"], Cell[TextData[{ "Outra forma de fazer gr\[AAcute]ficos de fun\[CCedilla]\[OTilde]es de duas \ vari\[AAcute]veis \[EAcute] utilizar a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["ContourPlot", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/ContourPlot.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/ContourPlot.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", que produz um gr\[AAcute]fico em \[OpenCurlyDoubleQuote]curvas de n\ \[IAcute]vel\[CloseCurlyDoubleQuote]: " }], "Text", CellChangeTimes->{{3.84331583893407*^9, 3.843315839839013*^9}, { 3.843315893265614*^9, 3.843315919509251*^9}, {3.843470235013055*^9, 3.8434702350181427`*^9}, {3.84347029756699*^9, 3.843470320381606*^9}, { 3.8792320136291037`*^9, 3.879232056533741*^9}},ExpressionUUID->"ea9daf24-7658-4d4d-9dc9-\ 9882aa98b880"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"ContourPlot", "[", RowBox[{ RowBox[{ RowBox[{"V", "[", RowBox[{"x", ",", "y"}], "]"}], 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3.843584612679428*^9},ExpressionUUID->"8dc43626-09c3-44c3-a5a8-\ 98058b3a58e8"], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"df1199e5-fe80-43b3-988f-8ac6975ad4f5"], Cell[TextData[{ "O campo vetorial associado ao campo el\[EAcute]trico do dipolo pode ser \ visualizado com a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["VectorPlot", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/VectorPlot.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/VectorPlot.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ". 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"}]], "Input", CellChangeTimes->{{3.843565521072576*^9, 3.8435655426511803`*^9}, { 3.843578680447152*^9, 3.843578753340214*^9}, {3.8435788202057734`*^9, 3.843578825842217*^9}, {3.843578989339519*^9, 3.843579028617633*^9}, { 3.8435790740744143`*^9, 3.8435790797274*^9}, {3.843579945640328*^9, 3.8435799477599792`*^9}, {3.8435801368685303`*^9, 3.843580145152944*^9}, { 3.843580524985551*^9, 3.843580525473777*^9}, {3.843580680126594*^9, 3.843580687284181*^9}, 3.843584602195776*^9},ExpressionUUID->"f997c31c-b74d-4eb8-922e-\ 36bff867fb36"], Cell[TextData[{ "As op\[CCedilla]\[OTilde]es imp\[OTilde]em um tamanho constante para cada \ vetor (VectorScaling", Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]],ExpressionUUID-> "b124d647-e7f6-44ee-83ab-62bd4e8059f1"], "None), definem esse tamanho (VectorSize) e especificam a densidade de \ pontos (VectorPoints)" }], "ItemParagraph", CellChangeTimes->{{3.8435809978843946`*^9, 3.843581108696615*^9}},ExpressionUUID->"ded8b18f-da01-47e7-a5b4-\ 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3.841494862183041*^9}, {3.8414950184999313`*^9, 3.8414950759148483`*^9}, {3.841512344981151*^9, 3.841512400542829*^9}, { 3.84159445169516*^9, 3.8415944912259007`*^9}, {3.842290177474586*^9, 3.842290192109984*^9}, {3.84347507591788*^9, 3.8434750845792*^9}, { 3.843553392048843*^9, 3.843553394736849*^9}},ExpressionUUID->"6026ccad-e8e5-4659-852a-\ 29f271e84752"], Cell[TextData[{ "Uma fun\[CCedilla]\[ATilde]o bastante \[UAcute]til do Mathematica \[EAcute] \ ", StyleBox[ButtonBox["Manipulate", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Manipulate.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Manipulate.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", que permite a manipula\[CCedilla]\[ATilde]o interativa de \ express\[OTilde]es, controlando uma ou mais vari\[AAcute]veis. Vamos nos \ concentrar no uso de ", StyleBox["Manipulate", FontFamily->"Source Code Pro", FontWeight->"Regular"], " com gr\[AAcute]ficos; veja a documenta\[CCedilla]\[ATilde]o para outros \ exemplos." }], "Text", CellChangeTimes->{{3.841495092716528*^9, 3.841495136864716*^9}, { 3.841495220002638*^9, 3.84149522517741*^9}, {3.8415122501164703`*^9, 3.841512255616736*^9}, {3.842336998635304*^9, 3.842337088961656*^9}, { 3.8423371323026047`*^9, 3.842337132306622*^9}, {3.8423576053399067`*^9, 3.842357609130612*^9}, {3.843475452984144*^9, 3.843475547415985*^9}, { 3.843475592697249*^9, 3.843475592701581*^9}, {3.879232376016219*^9, 3.879232382055779*^9}},ExpressionUUID->"2432d87d-accc-41bf-bf07-\ a9da126b9fae"], Cell[TextData[{ "Considere um oscilador harm\[OHat]nico sob amortecimento viscoso, cujo \ movimento \[EAcute] descrito pela equa\[CCedilla]\[ATilde]o diferencial\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["d", "2"], "x"}], RowBox[{"d", " ", SuperscriptBox["t", "2"]}]], "+", RowBox[{"\[Gamma]", FractionBox[ RowBox[{"d", " ", "x"}], RowBox[{"d", " ", "t"}]]}], "+", RowBox[{ SubsuperscriptBox["\[Omega]", "0", "2"], "x"}]}], "=", "0."}], TraditionalForm]],ExpressionUUID->"39b2fb77-8f27-43f9-aa84-32aa140b49b0"], "\nVamos criar um gr\[AAcute]fico interativo que permita, fixos o valor de \ ", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", "0"], TraditionalForm]],ExpressionUUID-> "ae14061a-aaf2-4dd3-8128-76231ee0feb7"], " e as condi\[CCedilla]\[OTilde]es iniciais, estudar os movimentos \ resultantes para diversos valores de ", Cell[BoxData[ FormBox["\[Gamma]", TraditionalForm]],ExpressionUUID-> "6fdb399e-6163-4191-a107-6d32fb6d35c6"], ". Para ajudar a compreender a sintaxe, o primeiro argumento de ", StyleBox["Manipulate", FontFamily->"Source Code Pro", FontWeight->"Regular"], " aparece abaixo em ", StyleBox["laranja", FontColor:>Dynamic[ CurrentValue[{StyleHints, "ColorSet", "Display2"}]]], ". O segundo argumento \[EAcute] uma lista cujo primeiro elemento \[EAcute] \ outra lista, contendo a vari\[AAcute]vel a manipular, seu valor inicial e o r\ \[OAcute]tulo do controle deslizante criado pela fun\[CCedilla]\[ATilde]o; os \ demais argumentos indicam o intervalo permitido de valores e o incremento." }], "Text", CellChangeTimes->{{3.8423371614805603`*^9, 3.842337175856915*^9}, { 3.8423576200879383`*^9, 3.842357620358099*^9}, {3.843481114025661*^9, 3.8434813840592623`*^9}, {3.84348142205219*^9, 3.843481447588128*^9}, { 3.843482569084197*^9, 3.843482725448159*^9}, {3.843483034014436*^9, 3.843483036163526*^9}, {3.8792323188011885`*^9, 3.879232321196108*^9}, { 3.8792323934092107`*^9, 3.8792323961095004`*^9}, 3.906042719589415*^9},ExpressionUUID->"1563b0cc-a456-4ba5-9ae6-\ 1cc66a106766"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Manipulate", "[", RowBox[{ StyleBox[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Evaluate", "[", RowBox[{ RowBox[{"x", "[", "t", "]"}], "/.", RowBox[{"NDSolve", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"x", "''"}], "[", "t", "]"}], "+", RowBox[{"\[Gamma]", "*", RowBox[{ RowBox[{"x", "'"}], "[", "t", "]"}]}], "+", RowBox[{ RowBox[{"\[Omega]0", "^", "2"}], "*", RowBox[{"x", "[", "t", "]"}]}]}], ")"}], "/.", RowBox[{"{", RowBox[{"\[Omega]0", "\[Rule]", " ", "2"}], "}"}]}], ")"}], "\[Equal]", "0"}], ",", RowBox[{ RowBox[{"x", "[", "0", "]"}], "\[Equal]", "1"}], ",", RowBox[{ RowBox[{ RowBox[{"x", "'"}], "[", "0", "]"}], "\[Equal]", "0"}]}], "}"}], ",", "x", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}]}], "]"}]}], "]"}], ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "10"}], "}"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", RowBox[{ "\"\<\!\(\*StyleBox[\"t\",FontSlant->\"Italic\"]\) (s)\>\"", ",", "\"\<\!\(\*StyleBox[\"x\",FontSlant->\"Italic\"]\) (m)\>\""}], "}"}]}], ",", RowBox[{"PlotRange", "\[Rule]", " ", RowBox[{"{", RowBox[{ RowBox[{"-", "1"}], ",", "1"}], "}"}]}], ",", RowBox[{"Evaluate", "[", "optam", "]"}]}], "]"}], FontColor:>Dynamic[ CurrentValue[{StyleHints, "ColorSet", "Display2"}]]], ",", RowBox[{"{", RowBox[{ StyleBox[ RowBox[{"{", RowBox[{ "\[Gamma]", ",", "0", ",", "\"\\""}], "}"}], FontColor->GrayLevel[0]], ",", "0", ",", "12", ",", "0.1"}], "}"}]}], "]"}], " "}]], "Input", CellChangeTimes->{{3.843482270467662*^9, 3.8434823045375013`*^9}, { 3.843482349902862*^9, 3.8434823543939247`*^9}, {3.843482392872676*^9, 3.843482432699315*^9}, {3.8434824730778017`*^9, 3.8434824843254547`*^9}, { 3.843483432717123*^9, 3.843483439157712*^9}, 3.843584639893716*^9},ExpressionUUID->"715e504e-c7e1-4546-bc30-\ 30f6be60a377"], Cell[TextData[{ "Clicando sobre o s\[IAcute]mbolo ", Cell[BoxData[ FormBox["+", TraditionalForm]],ExpressionUUID-> "9d0ca5bc-a63b-4e96-a126-b2f9d2ae4bf2"], " logo \[AGrave] direita do controle deslizante, surge um mostrador do valor \ de ", Cell[BoxData[ FormBox["\[Gamma]", TraditionalForm]],ExpressionUUID-> "a21fd073-51fe-451e-9200-fcdbe19cf08d"], " e controles para produzir anima\[CCedilla]\[OTilde]es (mais sobre isso em \ instantes). Verifique que se ", Cell[BoxData[ FormBox[ RowBox[{"\[Gamma]", "<", RowBox[{"2", SubscriptBox["\[Omega]", "0"]}]}], TraditionalForm]],ExpressionUUID-> "dbd87500-b855-4506-a082-5c48f16c84bb"], " o oscilador \[EAcute] subamortecido (ainda h\[AAcute] oscila\[CCedilla]\ \[OTilde]es na posi\[CCedilla]\[ATilde]o em fun\[CCedilla]\[ATilde]o do \ tempo), enquanto para ", Cell[BoxData[ FormBox[ RowBox[{"\[Gamma]", ">", RowBox[{"2", SubscriptBox["\[Omega]", "0"]}]}], TraditionalForm]],ExpressionUUID-> "dccea316-4b8a-467e-a772-cefd08ffc186"], " o oscilador \[EAcute] superamortecido (o comportamento da posi\[CCedilla]\ \[ATilde]o em fun\[CCedilla]\[ATilde]o do tempo \[EAcute] monot\[OHat]nico)." }], "ItemParagraph", CellChangeTimes->{{3.843483122857596*^9, 3.843483132621252*^9}, { 3.8434831817330503`*^9, 3.843483186385202*^9}, {3.84348322510807*^9, 3.843483322367715*^9}, {3.843483354352557*^9, 3.843483374523108*^9}, { 3.843483543479166*^9, 3.8434836493737507`*^9}, {3.8792325005645733`*^9, 3.879232500998562*^9}},ExpressionUUID->"1b27ba19-1f7a-4f31-937e-\ cd3a9addc22f"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"bf3623c1-381c-4d74-bccc-d2feab1bf6cc"], Cell[TextData[{ "As anima\[CCedilla]\[OTilde]es podem ser produzidas diretamente atrav\ \[EAcute]s da fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Animate", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Animate.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Animate.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", que tem uma sintaxe bastante semelhante \[AGrave] da fun\[CCedilla]\ \[ATilde]o ", StyleBox["Manipulate", FontFamily->"Source Code Pro", FontWeight->"Regular"], ". Vamos exemplific\[AAcute]-la com a superposi\[CCedilla]\[ATilde]o de dois \ pulsos gaussianos viajando com mesma velocidade ", Cell[BoxData[ FormBox["v", TraditionalForm]],ExpressionUUID-> "4479bdb5-53cb-4e9f-8589-05b84073138a"], " em sentidos opostos. Suas equa\[CCedilla]\[OTilde]es s\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["y", "1"], "(", RowBox[{"x", ",", "t"}], ")"}], "=", RowBox[{ SubscriptBox["A", "1"], " ", RowBox[{"exp", "(", RowBox[{ RowBox[{"-", SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", RowBox[{"v", " ", "t"}]}], ")"}], "2"]}], "/", SubsuperscriptBox["\[Sigma]", "1", "2"]}], ")"}]}]}], TraditionalForm]],ExpressionUUID->"cbe62e02-6260-41d9-8f8d-f8ca8a347cd5"], " e ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["y", "2"], "(", RowBox[{"x", ",", "t"}], ")"}], "=", RowBox[{ SubscriptBox["A", "2"], " ", RowBox[{"exp", "(", RowBox[{ RowBox[{"-", SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", RowBox[{"v", " ", "t"}]}], ")"}], "2"]}], "/", SubsuperscriptBox["\[Sigma]", "2", "2"]}], ")"}]}]}], TraditionalForm]],ExpressionUUID->"6213a8f3-4d4a-4cb8-8560-6bb09ffbcfca"], ". Esses dois pulsos e sua soma s\[ATilde]o mostrados na anima\[CCedilla]\ \[ATilde]o." }], "Text", CellChangeTimes->{{3.843483796491507*^9, 3.8434838447544928`*^9}, { 3.843483881697599*^9, 3.8434839379266863`*^9}, {3.8434839802275352`*^9, 3.843484007204666*^9}, {3.843484048844255*^9, 3.843484141485447*^9}, { 3.843484201278462*^9, 3.843484253885581*^9}, {3.879232619661908*^9, 3.8792326357778945`*^9}},ExpressionUUID->"381c8ff0-3cd1-4304-bcf6-\ 97989d674a00"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"y1", "[", RowBox[{"x_", ",", "t_"}], "]"}], ":=", RowBox[{"A1", "*", RowBox[{"Exp", "[", RowBox[{ RowBox[{"-", RowBox[{ RowBox[{"(", RowBox[{"x", "-", RowBox[{"v", "*", "t"}]}], ")"}], "^", "2"}]}], "/", RowBox[{"\[Sigma]1", "^", "2"}]}], "]"}]}]}], " "}], "\[IndentingNewLine]", RowBox[{ RowBox[{"y2", "[", RowBox[{"x_", ",", "t_"}], "]"}], ":=", RowBox[{"A2", "*", RowBox[{"Exp", "[", RowBox[{ RowBox[{"-", RowBox[{ RowBox[{"(", RowBox[{"x", "+", RowBox[{"v", "*", "t"}]}], ")"}], "^", "2"}]}], "/", RowBox[{"\[Sigma]2", "^", "2"}]}], "]"}]}]}]}], "Input", CellChangeTimes->{{3.843484143743986*^9, 3.8434841931242037`*^9}, { 3.843484260613014*^9, 3.843484293128703*^9}, 3.843584642255988*^9},ExpressionUUID->"d15e9027-7da7-4533-88d7-\ 3e42572ec680"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"pars", "=", RowBox[{"{", RowBox[{ RowBox[{"v", "\[Rule]", " ", "1"}], ",", RowBox[{"A1", "\[Rule]", " ", "1"}], ",", RowBox[{"A2", "\[Rule]", " ", "1"}], ",", RowBox[{"\[Sigma]1", "\[Rule]", " ", "1"}], ",", RowBox[{"\[Sigma]2", "\[Rule]", " ", "1"}]}], "}"}]}], ";"}], " "}], "\[IndentingNewLine]", RowBox[{"Animate", "[", RowBox[{ StyleBox[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"y1", "[", RowBox[{"x", ",", "t"}], "]"}], "/.", "pars"}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", RowBox[{"(", RowBox[{ RowBox[{"Abs", "[", "A1", "]"}], "+", RowBox[{"Abs", "[", "A2", "]"}]}], ")"}]}], "+", RowBox[{"y2", "[", RowBox[{"x", ",", "t"}], "]"}]}], ")"}], "/.", "pars"}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "2"}], "*", RowBox[{"(", RowBox[{ RowBox[{"Abs", "[", "A1", "]"}], "+", RowBox[{"Abs", "[", "A2", "]"}]}], ")"}]}], "+", RowBox[{"y1", "[", RowBox[{"x", ",", "t"}], "]"}], "+", RowBox[{"y2", "[", RowBox[{"x", ",", "t"}], "]"}]}], ")"}], "/.", "pars"}]}], "}"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "3"}], ",", "3"}], "}"}], ",", RowBox[{"Evaluate", "[", "optam", "]"}], ",", RowBox[{"Axes", "\[Rule]", " ", "False"}], ",", RowBox[{"PlotRange", "\[Rule]", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{ RowBox[{"-", "3"}], RowBox[{"(", RowBox[{ RowBox[{"Abs", "[", "A1", "]"}], "+", RowBox[{"Abs", "[", "A2", "]"}]}], ")"}]}], "/.", "pars"}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Abs", "[", "A1", "]"}], "+", RowBox[{"Abs", "[", "A2", "]"}]}], ")"}], "/.", "pars"}]}], "}"}]}]}], "]"}], FontColor:>Dynamic[ CurrentValue[{StyleHints, "ColorSet", "Display2"}]]], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "3.5"}], ",", "3.5"}], "}"}]}], "]"}]}], "Input", CellChangeTimes->{{3.8434843003189*^9, 3.843484480665618*^9}, { 3.843484513808215*^9, 3.843484595798758*^9}, {3.843484630034522*^9, 3.8434848180380783`*^9}, {3.843484853478908*^9, 3.8434848994188433`*^9}, { 3.843484940291923*^9, 3.843485020857085*^9}, 3.843485054836812*^9, { 3.843485178429537*^9, 3.843485198655505*^9}, 3.8435846464978533`*^9},ExpressionUUID->"7174bdd9-d1ce-4fa4-afcd-\ ab4fde47b21d"], Cell["\<\ Os quatro bot\[OTilde]es no alto da anima\[CCedilla]\[ATilde]o permitem paus\ \[AAcute]-la, controlar sua velocidade e definir se ela corre \ \[OpenCurlyDoubleQuote]para a frente\[CloseCurlyDoubleQuote] ou \ \[OpenCurlyDoubleQuote]para tr\[AAcute]s\[CloseCurlyDoubleQuote]. Tente \ variar os par\[AHat]metros dos pulsos para produzir interfer\[EHat]ncia \ destrutiva.\ \>", "ItemParagraph", CellChangeTimes->{{3.843485478693246*^9, 3.843485524929681*^9}, { 3.843485570537816*^9, 3.8434855998660927`*^9}, {3.843485697192358*^9, 3.8434857218826942`*^9}},ExpressionUUID->"e7bfa070-1cce-490f-88ca-\ 0fa368c9d36f"] }, Closed]], Cell["", "SlideShowNavigationBar", CellTags-> "SlideShowHeader",ExpressionUUID->"dc3647df-f278-4a6d-8cf5-9ad9865172fe"], Cell[TextData[{ "Vamos utilizar o que foi discutido em v\[AAcute]rias aulas diferentes do \ curso e produzir uma anima\[CCedilla]\[ATilde]o do movimento de um \ p\[EHat]ndulo abandonado do repouso, utilizando a fun\[CCedilla]\[ATilde]o ", StyleBox[ButtonBox["Graphics", BaseStyle->"Hyperlink", ButtonData->{ URL["https://reference.wolfram.com/language/ref/Graphics.html"], None}, ButtonNote->"https://reference.wolfram.com/language/ref/Graphics.html"], FontFamily->"Source Code Pro", FontWeight->"Regular"], ", que exemplificaremos no pr\[OAcute]ximo slide." }], "Text", CellChangeTimes->{{3.843487671006527*^9, 3.843487738114689*^9}, { 3.843487905410777*^9, 3.843487921938307*^9}, {3.843488232192889*^9, 3.8434882375010223`*^9}, {3.843498530184815*^9, 3.843498532310059*^9}, { 3.8792326854187193`*^9, 3.8792326870307417`*^9}, {3.9061928549747033`*^9, 3.906192866431754*^9}},ExpressionUUID->"fa53157e-2c66-4c4c-82e8-\ d1668f67f83e"], Cell["\<\ Primeiro, definimos a equa\[CCedilla]\[ATilde]o diferencial a resolver e \ \[OpenCurlyDoubleQuote]preparamos\[CloseCurlyDoubleQuote] a solu\[CCedilla]\ \[ATilde]o:\ \>", "Text", CellChangeTimes->{{3.843487935147121*^9, 3.84348794642972*^9}, { 3.843488066446362*^9, 3.8434880747366962`*^9}, {3.843544250091337*^9, 3.843544250574554*^9}, 3.9061928825304923`*^9},ExpressionUUID->"20f2aa72-30cc-400d-a884-\ 75e02dd999c8"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"eq", "=", RowBox[{ RowBox[{ RowBox[{ RowBox[{"\[Theta]", "''"}], "[", "t", "]"}], "+", 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Abaixo mostramos um desenho correspondente a um instante \ particular. Vamos desenhar tamb\[EAcute]m um conjunto de eixos, para fixar a \ escala.\ \>", "Text", CellChangeTimes->{{3.9061933460978365`*^9, 3.906193357577413*^9}, { 3.906193451731741*^9, 3.9061934864293413`*^9}, {3.9061937739172297`*^9, 3.9061937803456526`*^9}, 3.9061939102729044`*^9, {3.9061939874510975`*^9, 3.90619399082495*^9}, {3.906194565627941*^9, 3.9061945683920016`*^9}, { 3.906195158331247*^9, 3.9061952315500746`*^9}, {3.906195959651882*^9, 3.906195960586592*^9}},ExpressionUUID->"b7ceb810-a897-4658-90ed-\ 29403b550968"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"L", "=", "1"}], ";"}], " ", RowBox[{"(*", " ", RowBox[{"Comprimento", " ", "da", " ", "haste"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"raio", "=", RowBox[{"0.05", "*", "L"}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{"Raio", " ", "do", " ", "disco"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"xc", "=", RowBox[{"L", "*", RowBox[{"Sin", "[", RowBox[{"angulo", 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", StyleBox["AnimationRunning", FontFamily->"Source Code Pro", FontWeight->"Regular"], Cell[BoxData[ FormBox["\[Rule]", TraditionalForm]], FontFamily->"Source Code Pro", FontWeight->"Regular",ExpressionUUID-> "c44bcf89-4e58-44c0-8e54-ecf47564495a"], StyleBox["False", FontFamily->"Source Code Pro", FontWeight->"Regular"], " indica que a anima\[CCedilla]\[ATilde]o n\[ATilde]o deve se iniciar \ automaticamente. 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As gotas caem simultaneamente nas posi\ \[CCedilla]\[OTilde]es ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "1"], ",", "0"}], ")"}], TraditionalForm]], ExpressionUUID->"1c6f7a76-4c47-412e-95ae-820cc1a2f559"], " e ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ SubscriptBox["x", "2"], ",", "0"}], ")"}], TraditionalForm]], ExpressionUUID->"e1625c22-0e94-43e0-88cd-6bd8a24d3147"], ", e vamos supor que as alturas das torneiras produzem ondas com amplitudes \ parametrizadas por ", Cell[BoxData[ FormBox[ SubscriptBox["A", "1"], TraditionalForm]],ExpressionUUID-> "5b3f046e-378c-4532-9725-908d1f43d36c"], " e ", Cell[BoxData[ FormBox[ SubscriptBox["A", "2"], TraditionalForm]],ExpressionUUID-> "4807de74-eebd-4682-86a9-b9ad9c83c9a8"], ". Por conserva\[CCedilla]\[ATilde]o da energia, a amplitude da onda \ produzida por cada gota decai com o inverso da dist\[AHat]ncia at\[EAcute] a \ \[OpenCurlyDoubleQuote]fonte\[CloseCurlyDoubleQuote] correspondente. A onda \ resultante em uma posi\[CCedilla]\[ATilde]o ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"x", ",", "y"}], ")"}], TraditionalForm]],ExpressionUUID-> "3f51c449-821d-4367-bd98-0bfd506cd512"], " do tanque \[EAcute] a soma das ondas produzidas por cada gota." }], "Text", CellChangeTimes->{{3.843553497203477*^9, 3.843553793114109*^9}},ExpressionUUID->"5ff24c5e-458c-431e-8c5f-\ ddfd87b4942c"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"r1", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "-", "x1"}], ")"}], "^", "2"}], "+", RowBox[{"y", "^", "2"}]}], "]"}]}], " ", RowBox[{"(*", " ", RowBox[{ "Dist\[AHat]ncia", " ", "at\[EAcute]", " ", "a", " ", "primeira", " ", "fonte"}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"r2", "[", RowBox[{"x_", ",", "y_"}], "]"}], ":=", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "-", "x2"}], ")"}], "^", "2"}], "+", RowBox[{"y", "^", "2"}]}], "]"}]}], " 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