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+\DeclareExerciseCollection{banque} +\xsimsetup{ + step=3, +} + +\pagestyle{empty} + +\begin{document} + +\input{exercises.tex} +\printcollection{banque} + +\end{document} diff --git a/TST_sti2d/06_Exponentielle_complexe/exercises.tex b/TST_sti2d/06_Exponentielle_complexe/exercises.tex index 08bd1e7..0d05fcb 100644 --- a/TST_sti2d/06_Exponentielle_complexe/exercises.tex +++ b/TST_sti2d/06_Exponentielle_complexe/exercises.tex @@ -124,4 +124,100 @@ \item Placer le résultat de ces opérations dans un repère. \end{enumerate} \end{exercise} + +\begin{exercise}[subtitle={Filtres}, step={3}, origin={Création}, topics={Exponentielle complexe}, tags={Complexe}] + Les résistances et les condensateurs sont des composants électroniques utilisés dans le domaine du son pour concevoir des filtres. + + Placé en sortie d'un microphone, un filtre atténue plus ou moins les sons selon leur fréquence $f$, exprimée en Hertz (Hz). + + Pour un filtre donné, l'atténuation d'un son se calcule à l'aide de deux nombres complexes $z_R$. + + Dans tout l'exercice, on suppose que $z_R = 10$ et $z_C = - \dfrac{\np{1000}\sqrt{3}}{f}i$ , où i désigne le nombre complexe de module 1 et d'argument $\dfrac{\pi}{2}$.. + + \noindent + \textbf{Partie A : Effet du filtre sur un son grave} + + On choisit un son grave de fréquence $f = 100$. + + \begin{enumerate} + \item Montrer que $z_C = - 10\sqrt{3} i$. + \item + \begin{enumerate} + \item Déterminer la forme exponentielle de $z_C$. + \item On considère le nombre complexe $Z = z_R + z_C$. On a donc $Z = 10 - 10\sqrt{3} i$. + + Déterminer la forme exponentielle de $Z$. + \item On considère le nombre complexe $z_G$ défini par : $z_G = \dfrac{z_C}{z_R + z_C}$. + + Montrer que $z_G = \dfrac{\sqrt{3}}{2}\text{e}^{- i\frac{\pi}{6}}$. + \item Le module du nombre complexe $z_G$ est appelé gain du filtre. + + Donner la valeur exacte du gain du filtre puis une valeur approchée au centième. + \end{enumerate} + \end{enumerate} + + \noindent + \textbf{Partie B : Effet du filtre sur un son aigu } + + On choisit un son aigu de fréquence $f = \np{1000}\sqrt{3}$. + + \begin{enumerate} + \item Montrer que le nombre complexe $z_G$ défini par $z_G = \dfrac{z_C}{z_R + z_C}$ est égal à $\dfrac{- i}{10 - i}$. + \item Déterminer la forme algébrique de $z_G$ . + \item Calculer la valeur exacte du gain du filtre $\left|z_G\right|$ et en donner une valeur approchée au centième. + \end{enumerate} +\end{exercise} + +\begin{exercise}[subtitle={Bras articulé}, step={3}, origin={Création}, topics={Exponentielle complexe}, tags={Complexe}] + Dans le plan muni d'un repère orthonormé direct \Ouv, le bras articulé d’un robot, fixé au point O, est représenté par deux segments [OA] et [AB], chacun de longueur 2 unités. + + Deux exemples de position du bras articulé sont donnés ci-dessous à titre indicatif. + + \noindent + \begin{minipage}{0.5\linewidth} + \begin{enumerate} + \item + \begin{enumerate} + \item Tracer sur la copie un repère orthonormé \Ouv. + + Placer le point A d'affixe $z_{\text A}= 2i$ puis construire l'extrémité B du bras articulé + lorsque son affixe $z_{\text B}$ a pour argument $\dfrac{\pi}{4}$. + + \item Donner l'affixe du point B sous forme algébrique et sous forme exponentielle. + \end{enumerate} + \end{enumerate} + \end{minipage} + \hfill + \begin{minipage}{0.4\linewidth} + \includegraphics[scale=0.15]{./fig/bras1} + \end{minipage} + + \begin{enumerate} + \setcounter{enumi}{1} + \item L'extrémité B du bras peut-elle atteindre un objet qui se trouve à une distance de $4,5$ + unités du point O? + \end{enumerate} + + \noindent + \begin{minipage}{0.6\linewidth} + \begin{enumerate} + \setcounter{enumi}{2} + \item Pour soulever un objet lourd dont le point d'accroche est le point C (voir figure ci-contre), il faut rigidifier l'articulation en A. On décide alors de bloquer l'angle $\left ( \vec{AO}~,~\vec{AB}\right )$ tel qu'une mesure de cet angle soit constamment égale à $\dfrac{\pi}{2}$ radians. + \begin{enumerate} + \item Déterminer la longueur OB. + \item Le point C a pour affixe $z_{\text C} = 2\sqrt{2}\e^{i\frac{\pi}{12}}$. + + Justifier que l'extrémité B du bras articulé pourra atteindre le point d'accroche C de l'objet. + + \item Lorsque le bras articulé saisit l'objet, les points B et C sont confondus. + + Calculer la mesure de l'angle que forme alors le bras [OA] avec l'axe [O$x$). + \end{enumerate} + \end{enumerate} + \end{minipage} + \hfill + \begin{minipage}{0.35\linewidth} + \includegraphics[scale=0.25]{./fig/bras2} + \end{minipage} +\end{exercise} \collectexercisesstop{banque} diff --git a/TST_sti2d/06_Exponentielle_complexe/fig/bras1.png b/TST_sti2d/06_Exponentielle_complexe/fig/bras1.png new file mode 100644 index 0000000000000000000000000000000000000000..f33d66bd4b56078c23cb3b9002b7606cbf7d816d GIT binary 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