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Elle f\ \[UHat]t enlev\[EAcute]e par P\[AHat]ris qui \[EAcute]tait follement amoureux \ d\[CloseCurlyQuote]elle, ce qui provoqua la guerre de Troie. Il n'y a pas eu \ de guerre \[AGrave] propos de la cyclo\[IDoubleDot]de, mais son histoire est \ riche d'\[EAcute]v\[EAcute]nements et elle a exerc\[EAcute] un attrait \ manifeste sur les grands penseurs du XVI", Cell[BoxData[ \(TextForm\`I\^e\)]], " si\[EGrave]cle.\nD'un point de vue cin\[EAcute]matique, la cyclo\ \[IDoubleDot]de ou \[LeftGuillemet] roulette \[RightGuillemet] comme \ l'appelait Pascal, est la courbe d\[EAcute]crite par un point d'une roue qui \ roule sur un plan sans glisser. L'\[EAcute]tude de cette courbe aux propri\ \[EAcute]t\[EAcute]s remarquables remonte au paradoxe d'Aristote : pourquoi \ deux cercles concentriques, l'un ayant un diam\[EGrave]tre inf\[EAcute]rieur \ \[AGrave] l'autre, parcourent-ils une distance \[EAcute]gale s'ils sont tourn\ \[EAcute]s selon un cercle, et pourquoi une fois s\[EAcute]par\[EAcute]s, \ parcourent-ils des distances proportionnelles \[AGrave] leurs \ diam\[EGrave]tres ? En 1634, Gille Personne (Roberval) parvint \[AGrave] d\ \[EAcute]terminer la forme de cette courbe, et entre 1634 et 1637, il trouva \ la quadrature et le volume engendr\[EAcute]s par la rotation d'un arc de la \ courbe autour de sa base. Il expliqua en outre la fa\[CCedilla]on de la \ construire par points. En 1638, Descartes et Fermat donn\[EGrave]rent leur \ propres solutions \[AGrave] la quadrature d'une arche de la \ cyclo\[IDoubleDot]de et ils trouv\[EGrave]rent une m\[EAcute]thode alg\ \[EAcute]brique pour d\[EAcute]terminer la tangente \[AGrave] la roulette. En \ 1658, Pascal, dans une lettre circulaire anonyme, lance un d\[EAcute]fi aux \ math\[EAcute]maticiens :" }], "Text", CellMargins->{{2, 0}, {Inherited, Inherited}}], Cell[TextData[{ "\[LeftGuillemet] ", StyleBox["Nous \[EAcute]tant occup\[EAcute] il y a quelques mois de \ diverses questions touchant la cyclo\[IDoubleDot]de et son centre de gravit\ \[EAcute], plusieurs probl\[EGrave]mes vinrent se pr\[EAcute]senter \[AGrave] \ notre esprit. Nous en demandons instamment la solution aux g\[EAcute]om\ \[EGrave]tres les plus illustres de l\[CloseCurlyQuote]univers", FontSlant->"Italic"], " \[RightGuillemet]" }], "Text", CellMargins->{{11.3125, 11.3125}, {Inherited, Inherited}}], Cell["\<\ En d\[EAcute]cembre de la m\[EHat]me ann\[EAcute]e, il publie un \ recueil contenant ses m\[EAcute]thodes et ses r\[EAcute]sultats au sujet de \ la d\[EAcute]termination des centres de gravit\[EAcute], des surfaces et des \ volumes li\[EAcute]s \[AGrave] la cyclo\[IDoubleDot]de. En 1659, Huygens d\[EAcute]couvre les propri\[EAcute]t\[EAcute]s isochrones \ du pendule cyclo\[IDoubleDot]dal et en 1697, Jacques et Jean Bernoulli \ montrent que la courbe de descente la plus rapide pour un point pesant, dite \ courbe brachystochrone, est une cyclo\[IDoubleDot]de.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Objectifs", "Section"], Cell["\<\ \[Bullet] prise de vue sur l'histoire de la cyclo\[IDoubleDot]de; \[Bullet] \[EHat]tre capable de construire \[LeftGuillemet] \ cin\[EAcute]matiquement \[RightGuillemet] une cyclo\[IDoubleDot]de \[Bullet] savoir r\[EAcute]aliser quelques animations pertinentes \[AGrave] \ propos de la cyclo\[IDoubleDot]de.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Activit\[EAcute]s", "Subsubtitle", FormatType->TextForm], Cell[TextData[{ StyleBox["1.", FontWeight->"Bold"], " ", StyleBox["Le paradoxe d'Aristote", FontWeight->"Bold"], "\na) Dessinez deux cercles concentriques, l'un ayant un diam\[EGrave]tre \ inf\[EAcute]rieur \[AGrave] l'autre.\nb) Faites rouler ces cercles :\n\ \[Bullet] sur un plan horizontal tangent au plus grand cercle;\n\[Bullet] sur \ un plan horizontal tangent au plus petit cercle.\n\[CapitalEAcute]noncez, \ \[AGrave] partir de cette animation, le paradoxe d'Aristote.\n\n", StyleBox["2.", FontWeight->"Bold"], " ", StyleBox["Horaires", FontWeight->"Bold"], "\nRappel : un horaire donne la position d'un mobile en fonction du temps.\n\ a) Donnez l'horaire ", Cell[BoxData[ FormBox[ OverscriptBox[ FormBox[ SubscriptBox[ StyleBox["r", FontSlant->"Italic"], StyleBox["c", FontSlant->"Italic"]], "TextForm"], FormBox["\[RightArrow]", "TextForm"]], TextForm]]], "(", StyleBox["t", FontSlant->"Italic"], ") du centre ", StyleBox["C", FontSlant->"Italic"], " d'un cercle de rayon ", StyleBox["r", FontSlant->"Italic"], " qui roule sur un plan horizontal \[AGrave] vitesse ", Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["v", FontSlant->"Italic"], FormBox["\[RightArrow]", "TextForm"]], TextForm]]], ".\nb) Donnez l'horaire ", Cell[BoxData[ FormBox[ OverscriptBox[ FormBox[ SubscriptBox[ StyleBox["r", FontSlant->"Italic"], StyleBox["p", FontSlant->"Italic"]], "TextForm"], FormBox["\[RightArrow]", "TextForm"]], TextForm]]], "(", StyleBox["t", FontSlant->"Italic"], ") d'un point ", StyleBox["P", FontSlant->"Italic"], " d\[EAcute]crivant une trajectoire circulaire de rayon ", StyleBox["r", FontSlant->"Italic"], " \[AGrave] vitesse \nangulaire \[Omega] constante.\nc) Donnez l'horaire ", Cell[BoxData[ FormBox[ OverscriptBox[ FormBox[ StyleBox["r", FontSlant->"Italic"], "TextForm"], FormBox["\[RightArrow]", "TextForm"]], TextForm]]], "(", StyleBox["t", FontSlant->"Italic"], ") d'un point ", StyleBox["P", FontSlant->"Italic"], " situ\[EAcute] \[AGrave] une distance ", StyleBox["d", FontSlant->"Italic"], " du centre d'un cercle de rayon ", StyleBox["r", FontSlant->"Italic"], " roulant sans glisser sur un plan horizontal.\n\n", StyleBox["3. Cyclo\[IDoubleDot]des", FontWeight->"Bold"], "\nConstruisez une animation permettant de faire rouler sans glissement une \ roue de rayon ", StyleBox["r", FontSlant->"Italic"], " sur un plan et d'obtenir la trajectoire d'un point solidaire de la roue \ et situ\[EAcute] \[AGrave] une distance ", StyleBox["d", FontSlant->"Italic"], " du centre." }], "Text", CellMargins->{{Inherited, 1.375}, {Inherited, Inherited}}], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\ \(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\ \[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\)\)], \ "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Pour en savoir plus", "Subsubtitle", FormatType->TextForm], Cell[TextData[{ StyleBox["\[Bullet]", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["\[NonBreakingSpace]", FontColor->GrayLevel[0], FontVariations->{"CompatibilityType"->0}], StyleBox["Jean-Luc Verley. ", FontColor->GrayLevel[0]], StyleBox["Autour de la cyclo\[IDoubleDot]de (\[AGrave] rechercher sur le \ web).\n\[Bullet]", FontColor->GrayLevel[0]], StyleBox[" http://p7app.geneve.ch:8080/webMathematica/Bindex.html", FontColor->GrayLevel[0]] }], "Text", CellFrame->True, CellMargins->{{Inherited, 1}, {Inherited, Inherited}}, FontFamily->"Lucida Grande", FontColor->GrayLevel[0.900008], Background->GrayLevel[0.833326]] }, Open ]], Cell[CellGroupData[{ Cell["Corrig\[EAcute]", "Subtitle"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox[" ", FontFamily->"Lucida Grande"], StyleBox["Le paradoxe d'Aristote", FontFamily->"Lucida Grande", FontWeight->"Bold"] }], "Subsubsection", FormatType->TextForm], Cell["Dessinons les deux cercles concentriques :", "Text"], Cell[BoxData[{ RowBox[{ StyleBox[\(x0 = 0;\), FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[\( (*\ coordonn\[EAcute]es\ du\ centre\ des\ cercles\ *) \), FontColor->RGBColor[0, 0, 1]]}], "\[IndentingNewLine]", StyleBox[\(y0 = 0;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(r1 = 1;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(r2 = 0.8;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", \(Show[ Graphics[{Circle[{x0, y0}, \ r1], Circle[{x0, y0}, r2]}], AspectRatio \[Rule] Automatic]\)}], "Input"], Cell["Faisons rouler les cercles sur un plan horizontal :", "Text", FormatType->TextForm], Cell[BoxData[{ StyleBox[\(x0 = 0; 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\ (*\ coordonn\[EAcute]es\ du\ centre\ des\ cercles\ *) \[IndentingNewLine]\ \(y0 = 0;\)\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(\[CapitalDelta]x = \ 0.25;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(r1 = 1;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(r2 = 0.6;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", RowBox[{ StyleBox[\(xmax = 2 \[Pi]*r1;\), FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[\( (*\ longueur\ de\ la\ piste\ *) \), FontColor->RGBColor[0, 0, 1]]}], "\[IndentingNewLine]", \(Table[ Show[Graphics[{Circle[{x, y0}, \ r1], Line[{{\(-r1\), \(-r1\)}, {xmax + r1, \(-r1\)}}], \[IndentingNewLine]Hue[1], Line[{{x, y0}, r1 {x + Cos[\(-\[Pi]\)/2 - x/r1], y0 + Sin[\(-\[Pi]\)/2 + x/r1]}}], PointSize[0.02], Point[{x, y0}]}, PlotRange \[Rule] {{\(-r1\), xmax + r1}, Automatic}], AspectRatio \[Rule] Automatic], {x, x0, xmax, \[CapitalDelta]x}]\)}], "Input"], Cell[BoxData[{ StyleBox[\(x0 = 0; \ (*\ coordonn\[EAcute]es\ du\ centre\ des\ cercles\ \ *) \[IndentingNewLine]y0 = 0;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(\[CapitalDelta]x = \ 0.25;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(r1 = 1;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(r2 = 0.6;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", RowBox[{ StyleBox[\(xmax = 2 \[Pi]*r2\), FontColor->RGBColor[0, 0, 1]], StyleBox[";", FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[\( (*\ longueur\ de\ la\ piste\ *) \), FontColor->RGBColor[0, 0, 1]], StyleBox["\[IndentingNewLine]", FontColor->RGBColor[0, 0, 1]], \(Table[ Show[Graphics[{Circle[{x, y0}, \ r2], Line[{{\(-r1\), \(-r2\)}, {xmax + r1, \(-r2\)}}], \[IndentingNewLine]Hue[1], PointSize[0.02], Point[{x, y0}], Line[{{x, y0}, {x + r2*Cos[\(-\[Pi]\)/2 - x/r2], r2*Sin[\(-\[Pi]\)/2 + x/r2]}}]}, PlotRange \[Rule] {{\(-r1\), xmax + r1}, Automatic}], AspectRatio \[Rule] Automatic], {x, x0, xmax, \[CapitalDelta]x}]\)}]}], "Input"], Cell[BoxData[{ StyleBox[\(r = 1;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(d = 1;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(v = 2 \[Pi];\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(omega = v/r;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(phi = \(-\[Pi]\)/2;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(tmin = 0;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(tmax = 2;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(\[CapitalDelta]t = 0.1;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", StyleBox[\(ps = 0.02;\), FontColor->RGBColor[0, 0, 1]], "\[IndentingNewLine]", \(h1[t_] := {v*t, r}\), "\[IndentingNewLine]", \(h2[t_] := d {Cos[\(-omega\)*t + phi], Sin[\(-omega\)*t + phi]}\), "\[IndentingNewLine]", \(h[t_] := h1[t] + h2[t]\), "\[IndentingNewLine]", \(ParametricPlot[ h[t], {t, tmin, tmax}, AspectRatio \[Rule] Automatic];\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(coord = Table[h[t], {t, tmin, tmax, \[CapitalDelta]t}]\)], "Input"], Cell[BoxData[ \({{0, 0}, {0.040533278425485486`, 0.19098300562505255`}, {0.3055805451407637`, 0.6909830056250524`}, {0.9338990758587224`, 1.3090169943749472`}, {1.9254888705793611`, 1.8090169943749475`}, {3.141592653589793`, 2.`}, {4.357696436600225`, 1.8090169943749475`}, {5.349286231320864`, 1.3090169943749477`}, {5.977604762038823`, 0.6909830056250528`}, {6.242652028754101`, 0.19098300562505277`}, {6.283185307179586`, 0.`}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(pts = Map[Point, coord]\)], "Input"], Cell[BoxData[ \({Point[{0, 0}], Point[{0.040533278425485486`, 0.19098300562505255`}], Point[{0.3055805451407637`, 0.6909830056250524`}], Point[{0.9338990758587224`, 1.3090169943749472`}], Point[{1.9254888705793611`, 1.8090169943749475`}], Point[{3.141592653589793`, 2.`}], Point[{4.357696436600225`, 1.8090169943749475`}], Point[{5.349286231320864`, 1.3090169943749477`}], Point[{5.977604762038823`, 0.6909830056250528`}], Point[{6.242652028754101`, 0.19098300562505277`}], Point[{6.283185307179586`, 0.`}]}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(centres = Table[h1[t], {t, tmin, tmax, \[CapitalDelta]t}]\)], "Input"], Cell[BoxData[ \({{0, 1}, {0.6283185307179586`, 1}, {1.2566370614359172`, 1}, {1.884955592153876`, 1}, {2.5132741228718345`, 1}, {3.141592653589793`, 1}, {3.769911184307752`, 1}, {4.39822971502571`, 1}, {5.026548245743669`, 1}, {5.654866776461628`, 1}, {6.283185307179586`, 1}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(rayons = Table[r, {\((tmax - tmin)\)/\[CapitalDelta]t + 1}]\)], "Input"], Cell[BoxData[ \({1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(cercles = MapThread[Circle, {centres, rayons}, 1]\)], "Input"], Cell[BoxData[ \({Circle[{0, 1}, 1], Circle[{0.6283185307179586`, 1}, 1], Circle[{1.2566370614359172`, 1}, 1], Circle[{1.884955592153876`, 1}, 1], Circle[{2.5132741228718345`, 1}, 1], Circle[{3.141592653589793`, 1}, 1], Circle[{3.769911184307752`, 1}, 1], Circle[{4.39822971502571`, 1}, 1], Circle[{5.026548245743669`, 1}, 1], Circle[{5.654866776461628`, 1}, 1], Circle[{6.283185307179586`, 1}, 1]}\)], "Output"] }, Open ]], Cell[BoxData[{ RowBox[{ StyleBox[\(r = 1;\), FontColor->RGBColor[0, 0, 1]], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox[\( (*\ rayon\ de\ la\ roue\ *) \), FontColor->RGBColor[0, 0, 1]]}], "\[IndentingNewLine]", StyleBox[\(d = 5; 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