From 3222f5676b656cb4eaa28d0f65a890f7dfc35dc0 Mon Sep 17 00:00:00 2001 From: guyotv Date: Wed, 2 Oct 2024 23:18:59 +0200 Subject: [PATCH] =?UTF-8?q?Impl=C3=A9mentation=20d'hyperref=20avec=20corre?= =?UTF-8?q?ctions=20de=20multiples=20adresses=20url?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- =SolutionsOS.tex | 58 +++ Annexe-Exercices/Annexe-Exercices.tex.bak | 5 +- Annexe-MRUA/Annexe-MRUA.tex | 2 +- Annexe-MRUA/Annexe-MRUA.tex.bak | 87 ++++ Annexe-Maree/Annexe-Maree.tex | 2 +- .../Annexe-MesuresDistances.tex | 6 +- .../Annexe-MesuresDistances.tex.bak | 2 +- Annexe-Rotations/Annexe-Rotations.tex | 4 +- Cinematique/Cinematique.tex | 8 +- Cinematique/Cinematique.tex.bak | 12 +- CoursMecaniqueOSDF.mw | 0 CoursMecaniqueOSDF.out | 423 ++++++++++++++++++ CoursMecaniqueOSDF.pdf | Bin 12887069 -> 13393578 bytes CoursMecaniqueOSDF.pdf_ | Bin 0 -> 12895832 bytes Dynamique/Dynamique.tex | 4 +- Dynamique/Dynamique.tex.bak | 4 +- Energie/Energie.tex | 12 +- Introduction/Introduction.tex | 16 +- Introduction/Introduction.tex.bak | 2 +- PhysiqueTheorique/PhysiqueTheorique.tex | 8 +- PhysiqueTheorique/PhysiqueTheorique.tex.bak | 2 +- Preambule.tex | 4 +- Preambule.tex.bak | 10 +- Solutions.tex | 150 +++---- SolutionsOS.tex | 2 +- Thermodynamique/Thermodynamique.tex | 4 +- 26 files changed, 698 insertions(+), 129 deletions(-) create mode 100644 =SolutionsOS.tex create mode 100644 Annexe-MRUA/Annexe-MRUA.tex.bak create mode 100644 CoursMecaniqueOSDF.mw create mode 100644 CoursMecaniqueOSDF.out create mode 100644 CoursMecaniqueOSDF.pdf_ diff --git a/=SolutionsOS.tex b/=SolutionsOS.tex new file mode 100644 index 0000000..09ea4cd --- /dev/null +++ b/=SolutionsOS.tex @@ -0,0 +1,58 @@ + + + + + + +Output from handle ans going to Solutions.tex +File ans already open + +LaTeX Warning: Command \' invalid in math mode on input line 2704. + + +LaTeX Warning: Command \' invalid in math mode on input line 2710. + + +LaTeX Warning: Command \' invalid in math mode on input line 2710. + +Annexe M. + +LaTeX Warning: Command \` invalid in math mode on input line 38. + + +LaTeX Warning: Command \` invalid in math mode on input line 151. + + +LaTeX Warning: Command \` invalid in math mode on input line 212. + + + + + + +LaTeX Warning: Command \' invalid in math mode on input line 660. + + + + +LaTeX Warning: Command \` invalid in math mode on input line 685. + + + + + + +LaTeX Warning: Command \' invalid in math mode on input line 704. + + + + +LaTeX Warning: Command \` invalid in math mode on input line 730. + + + + +LaTeX Warning: Command \' invalid in math mode on input line 746. + + + diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index 679b122..2d82828 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -93,7 +93,7 @@ D&=2\cdot R_{Lune}=d\cdot \alpha\;\Rightarrow\;\alpha=\frac{2\cdot R_{Lune}}{d_{Terre-Lune}}\\ \alpha&=\frac{2\cdot 0,2725\cdot 6,371\cdot 10^6}{3,84\cdot 10^8}=\SI{0,009}{\radian} \end{align*} - Comme \(\SI{180}{\degree}=\SI{\pi}{\radian}\), l'angle considéré est~: + Comme \(\SI{180}{\degree}=\pi\unit{\radian}\), l'angle considéré est~: \[\alpha=0,009\cdot \frac{180}{\pi}=\SI{0,5157}{\degree}\] Comme un degré vaut soixante minutes d'arc, \(\SI{1}{\degree}=\SI{60}{\arcminute}\), on a~: \[\alpha=\SI{0,5157}{\degree}=0,5157\cdot 60=\SI{31}{\arcminute}\] @@ -2875,7 +2875,8 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \begin{figure} \def\svgwidth{7cm} \begin{center} - \input{Annexe-Exercices/Images/cycle2.eps_tex} + %\input{Annexe-Exercices/Images/cycle2.eps_tex} + \includegraphics[scale=0.9]{cycle2.eps} \end{center} \caption{Bilan du cycle\label{exos:cycle1}} \end{figure} diff --git a/Annexe-MRUA/Annexe-MRUA.tex b/Annexe-MRUA/Annexe-MRUA.tex index 3ce2bb3..082a249 100644 --- a/Annexe-MRUA/Annexe-MRUA.tex +++ b/Annexe-MRUA/Annexe-MRUA.tex @@ -36,7 +36,7 @@ C'est ce qu'il fallait démontrer. \begin{figure}[t] \centering -\caption[\emph{Discours concernant deux sciences nouvelles} de Galilée]{\emph{Discours concernant deux sciences nouvelles} de Galilée\label{deuxsciences} \par \scriptsize{Discours dans lesquels Galilée présente ses expériences sur la chute des corps et fonde la mécanique. Il tente aussi d'y créer une science de la résistance des matériaux\endnote{Voir le site de l'encyclopédie Wikipedia~: \url=http://commons.wikimedia.org/wiki/Image:Galileo_Galilei\%2C_Discorsi_e_Dimostrazioni_Matematiche_Intorno_a_Due_Nuove_Scienze\%2C_1638_\%281400x1400\%29.png=.}}} +\caption[\emph{Discours concernant deux sciences nouvelles} de Galilée]{\emph{Discours concernant deux sciences nouvelles} de Galilée\label{deuxsciences} \par \scriptsize{Discours dans lesquels Galilée présente ses expériences sur la chute des corps et fonde la mécanique. Il tente aussi d'y créer une science de la résistance des matériaux\endnote{Voir le site de l'encyclopédie Wikipedia~: \url=http://commons.wikimedia.org/wiki/Image:Galileo\_Galilei\%2C\_Discorsi\_e\_Dimostrazioni\_Matematiche\_Intorno\_a\_Due\_Nuove\_Scienze\%2C\_1638\_\%281400x1400\%29.png=.}}} \includegraphics[width=6cm]{Discorsi_e_Dimostrazioni_Matematiche.eps} \end{figure} diff --git a/Annexe-MRUA/Annexe-MRUA.tex.bak b/Annexe-MRUA/Annexe-MRUA.tex.bak new file mode 100644 index 0000000..d105a06 --- /dev/null +++ b/Annexe-MRUA/Annexe-MRUA.tex.bak @@ -0,0 +1,87 @@ +\myclearpage + +\chapter{MRUA développements} + +\section{La position\index{position}} + +\lettrine{P}{our un MRUA}, la position est donnée par \label{demo}: + +\begin{equation} +\fbox{\(\displaystyle x=\frac{1}{2}\cdot a\cdot t^{2}+v_{o}\cdot t+x_{o}\)} +\end{equation} + +Démonstration~: + +Par définition la vitesse moyenne est~: + +\[\overline{v}=\frac{x-x_{o}}{t}\;\Rightarrow\; x=\overline{v}\cdot t+x_{o}\] + +Mais, la vitesse moyenne peut aussi s'exprimer par~: +\[\overline{v}=\frac{v+v_{o}}{2}\] + +Ainsi, on a~: +\[x=\overline{v}\cdot t+v_{o}=\frac{v+v_{o}}{2}\cdot t+x_{o}\] + +Or, par définition de l'accélération (constante)~: +\[\overline{a}=a_{o}=\frac{v-v_{o}}{t}\;\Rightarrow\; v=a_{o}\cdot t+v_{o}\] + +Donc, on a~: + +\begin{align*} +x&=\frac{v+v_{o}}{2}\cdot t+x_{o}=\frac{a_{o}\cdot t+v_{o}+v_{o}}{2}+x_{o}\\ +&=\frac{1}{2}\cdot a\cdot t^{2}+v_{o}\cdot t+x_{o} +\end{align*} + +C'est ce qu'il fallait démontrer. + +\begin{figure}[t] +\centering +\caption[\emph{Discours concernant deux sciences nouvelles} de Galilée]{\emph{Discours concernant deux sciences nouvelles} de Galilée\label{deuxsciences} \par \scriptsize{Discours dans lesquels Galilée présente ses expériences sur la chute des corps et fonde la mécanique. Il tente aussi d'y créer une science de la résistance des matériaux\endnote{Voir le site de l'encyclopédie Wikipedia~: \url=http://commons.wikimedia.org/wiki/Image:Galileo\_Galilei\%2C_Discorsi_e_Dimostrazioni_Matematiche_Intorno_a_Due_Nuove_Scienze\%2C_1638_\%281400x1400\%29.png=.}}} +\includegraphics[width=6cm]{Discorsi_e_Dimostrazioni_Matematiche.eps} +\end{figure} + +\section{Une autre relation bien pratique\label{pratique}} + +\subsection{Cinématique} + +Jusqu'à présent, les relations obtenues (la vitesse et la position) sont fonctions du temps. Il est néanmoins pratique dans bien des cas de disposer d'une relation où le facteur temps n'apparaît pas. Cette relation est facilement obtenue en éliminant le temps des deux équations de la vitesse et de la position. Pour le calcul on part de équations du MRUA suivantes~: + +\begin{align*} +v&=a_{o}\cdot t+v_{o}\\ +x&=\frac{1}{2}\cdot a\cdot t^{2}+v_{o}\cdot t+x_{o} +\end{align*} + +Elles constituent généralement un système de deux équations à deux inconnues, dont le temps t est l'une d'elles. Pour résoudre ce système et éliminer le temps par substitution, on tire t de la première équation~: + +\[t=\frac{v-v_{o}}{a_{o}}\] + +et on le remplace dans la seconde (faire le contraire est aussi réalisable, mais mathématiquement plus complexe)~: + +\begin{align*} +x&=\frac{1}{2}\cdot a_{o}\cdot(\frac{v-v_{o}}{a_{o}})^{2}+v_{o}\cdot\frac{v-v_{o}}{a_{o}}+x_{o}\\ +&=\frac{1}{2}\cdot a_{o}\cdot\frac{(v-v_{o})^{2}}{a_{o}^{2}}+v_{o}\cdot\frac{v-v_{o}}{a_{o}}+x_{o}\\ +&=\frac{1}{2}\cdot\frac{v^{2}-2\cdot v\cdot v_{o}+v_{o}^{2}}{a_{o}}+\frac{v\cdot v_{o}-v_{o}^{2}}{a_{o}}+x_{o}\\ +&=\frac{v^{2}-2\cdot v\cdot v_{o}+v_{o}^{2}+2\cdot v\cdot v_{o}-2\cdot v_{o}^{2}}{2\cdot a_{o}}+x_{o} +\end{align*} + +\[\Rightarrow\; x=\frac{v^{2}-v_{o}^{2}}{2\cdot a_{o}}+x_{o}\;\Rightarrow\] + +\begin{equation}\label{sanst} +\fbox{\(v^{2}=v_{o}^{2}+2\cdot a_{o}\cdot(x-x_{o})\)} +\end{equation} + +Cette relation\label{demo2} est indépendante du temps t. Elle est canoniquement présentée sous cette forme. + +\subsection{Énergie} + +Il faut relever que la relation \ref{sanst} peut aussi être obtenue grâce au théorème de conservation de l'énergie. En effet, imaginons un objet de masse \(m\) à une hauteur \(h\) qu'on lance à une vitesse \(v_o\) vers le bas. Son énergie cinétique initiale est non nulle, de même que son énergie potentielle initiale. Son énergie potentielle finale est nulle. Par contre, son énergie cinétique finale ne l'est pas. Par conservation de l'énergie, on peut écrire~: +\begin{gather*} +E_{cin}^i+E_{pot}^i=E_{cin}^f+E_{pot}^f\;\Rightarrow\\ +\frac{1}{2}\cdot m\cdot v_o^2+m\cdot g\cdot h=\frac{1}{2}\cdot m\cdot v^2\;\Rightarrow\\ +v^2=v_o^2+2\cdot g\cdot h +\end{gather*} +Soit, de manière plus générale, en posant \(g=a\) et \(h=x-x_o\), ce qu'il fallait démontrer~: + +\begin{equation*} +\fbox{\(v^{2}=v_{o}^{2}+2\cdot a_{o}\cdot(x-x_{o})\)} +\end{equation*} \ No newline at end of file diff --git a/Annexe-Maree/Annexe-Maree.tex b/Annexe-Maree/Annexe-Maree.tex index cb2851f..7a5ce05 100644 --- a/Annexe-Maree/Annexe-Maree.tex +++ b/Annexe-Maree/Annexe-Maree.tex @@ -8,7 +8,7 @@ Nous allons ici rappeler la simplicité du phénomène et expliquer pourquoi il \begin{figure}[t] \centering -\caption[Marée]{Marée\label{maree}\endnote{Voir le site de l'encyclopédie Wikipedia~: \url=http://commons.wikimedia.org/wiki/Image:Bay_of_Fundy.jpg= notamment pour le copyright de l'image. Remerciements à son auteur Samuel Wantman.}} +\caption[Marée]{Marée\label{maree}\endnote{Voir le site de l'encyclopédie Wikipedia~: \url=http://commons.wikimedia.org/wiki/Image:Bay\_of\_Fundy.jpg= notamment pour le copyright de l'image. Remerciements à son auteur Samuel Wantman.}} \includegraphics[width=6cm]{Maree.eps} \end{figure} diff --git a/Annexe-MesuresDistances/Annexe-MesuresDistances.tex b/Annexe-MesuresDistances/Annexe-MesuresDistances.tex index f06417c..2f9d7af 100644 --- a/Annexe-MesuresDistances/Annexe-MesuresDistances.tex +++ b/Annexe-MesuresDistances/Annexe-MesuresDistances.tex @@ -58,7 +58,7 @@ Même de nos jours, si on utilise des images trop petites, l'incertitude sur le \begin{figure}[ht] \centering -\caption[Taille de la lune]{Taille de la lune\label{tailledelalune} \par \scriptsize{Une taille incertaine\endnote{Voir le site de l'encyclopédie Wikipedia~: \url=http://commons.wikimedia.org/wiki/Image:Eclipse_lune.jpg=. notamment pour le copyright de l'image. Remerciements à son auteur Luc Viatour.}}} +\caption[Taille de la lune]{Taille de la lune\label{tailledelalune} \par \scriptsize{Une taille incertaine\endnote{Voir le site de l'encyclopédie Wikipedia~: \url=http://commons.wikimedia.org/wiki/Image:Eclipse\_lune.jpg=. notamment pour le copyright de l'image. Remerciements à son auteur Luc Viatour.}}} \includegraphics[width=6cm]{RayonLune.eps} \end{figure} @@ -93,7 +93,7 @@ en raison du fait que deux droites parallèles sont toujours coupées par une tr L'astronome Cassini, qui détermina pour la première fois la distance Terre-Soleil à partir de la parallaxe de Mars, décrit la mesure ainsi~: \begin{quotation} -``\textit{La meilleure méthode pour chercher la parallaxe de Mars par la correspondance des observations faites à Paris \& en Caïenne auroit été d'observer, par la lunette, la conjonction précise de cette planète avec une étoile fixe. Car si cette conjonction avoit été vue de l'un \& de l'autre lieu au même instant \& précisément de la même manière sans aucune distance, c'eût été une marque qu'il n'y avoit point de parallaxe sensible. S'il y en avoit eu quelque peu, à l'instant que Mars auroit paru toucher par son bord supérieur une Etoile fixe en Caïenne, il auroit paru à Paris un peu éloigné de la même Etoile vers l'Horizon, \& quand il auroit paru à Paris toucher l'Etoile par son bord inférieur, il auroit paru en Caïenne éloigné de la même Etoile vers le Zénit \& cette distance vue d'un lieu \& non pas de l'autre, aurait été attribuée à la parallaxe}'' J. D. Cassini, dans ``Mémoires de l'Académie Royale des Sciences'', volume 8, année 1730.\endnote{\url=http://www.iap.fr/InformationCommunication/ArticlesGrandPublic/Etoiles/Transit/transit_parallaxe_mars_1672.html=} +``\textit{La meilleure méthode pour chercher la parallaxe de Mars par la correspondance des observations faites à Paris \& en Caïenne auroit été d'observer, par la lunette, la conjonction précise de cette planète avec une étoile fixe. Car si cette conjonction avoit été vue de l'un \& de l'autre lieu au même instant \& précisément de la même manière sans aucune distance, c'eût été une marque qu'il n'y avoit point de parallaxe sensible. S'il y en avoit eu quelque peu, à l'instant que Mars auroit paru toucher par son bord supérieur une Etoile fixe en Caïenne, il auroit paru à Paris un peu éloigné de la même Etoile vers l'Horizon, \& quand il auroit paru à Paris toucher l'Etoile par son bord inférieur, il auroit paru en Caïenne éloigné de la même Etoile vers le Zénit \& cette distance vue d'un lieu \& non pas de l'autre, aurait été attribuée à la parallaxe}'' J. D. Cassini, dans ``Mémoires de l'Académie Royale des Sciences'', volume 8, année 1730.\endnote{\url=http://www.iap.fr/InformationCommunication/ArticlesGrandPublic/Etoiles/Transit/transit\_parallaxe\_mars\_1672.html=} \end{quotation} \begin{figure}[ht] @@ -113,7 +113,7 @@ où \(R_T\) est le rayon de la Terre. \smallskip Le résultat est donné par Cassini\index{Cassini} lui-même~: \begin{quotation} -``\textit{Le 5 septembre 1672, trois jours avant l'opposition du Soleil à Mars, nous observâmes à Paris trois Etoiles dans l'Eau Aquarius marquées par Bayerus \(\Psi\), vers lesquelles Mars alloit par son mouvement particulier rétrograde, de sorte que l'on jugeoit qu'il en auroit pu cacher une. Il étoit alors un peu plus septentrional que la plus septentrionale des trois. On prit la hauteur Méridienne de celle-ci qui passoit la première; \& celle de la moyenne vers laquelle le mouvement particulier de Mars s'adressoit. Par le choix des Observations les plus exactes \& les plus conformes entre elles, on fixa à 15" la parallaxe que fait Mars de Paris à Caïenne}'' J. D. Cassini, dans ``Mémoires de l'Académie Royale des Sciences'', volume 8, année 1730.\endnote{\url=http://www.iap.fr/InformationCommunication/ArticlesGrandPublic/Etoiles/Transit/transit_parallaxe_mars_1672.html=} +``\textit{Le 5 septembre 1672, trois jours avant l'opposition du Soleil à Mars, nous observâmes à Paris trois Etoiles dans l'Eau Aquarius marquées par Bayerus \(\Psi\), vers lesquelles Mars alloit par son mouvement particulier rétrograde, de sorte que l'on jugeoit qu'il en auroit pu cacher une. Il étoit alors un peu plus septentrional que la plus septentrionale des trois. On prit la hauteur Méridienne de celle-ci qui passoit la première; \& celle de la moyenne vers laquelle le mouvement particulier de Mars s'adressoit. Par le choix des Observations les plus exactes \& les plus conformes entre elles, on fixa à 15" la parallaxe que fait Mars de Paris à Caïenne}'' J. D. Cassini, dans ``Mémoires de l'Académie Royale des Sciences'', volume 8, année 1730.\endnote{\url=http://www.iap.fr/InformationCommunication/ArticlesGrandPublic/Etoiles/Transit/transit\_parallaxe\_mars\_1672.html=} \end{quotation} Le résultat de la mesure est donc de quinze secondes d'arc. Mais attention, il s'agit de la parallaxe qui est la moitié de l'angle \(\delta\). Celui-ci vaut donc~: \(\delta=0,008332^{\circ}\) ou \SI{1,454e-4}{\radian}. Avec une distance de Paris à Cayenne de \SI{7082,1}{\kilo\metre} cela donne~: \[MO=\frac{7,0821\cdot 10^6}{1,454\cdot 10^{-4}}=\SI{4,87e10}{\metre}\] diff --git a/Annexe-MesuresDistances/Annexe-MesuresDistances.tex.bak b/Annexe-MesuresDistances/Annexe-MesuresDistances.tex.bak index b461632..f06417c 100644 --- a/Annexe-MesuresDistances/Annexe-MesuresDistances.tex.bak +++ b/Annexe-MesuresDistances/Annexe-MesuresDistances.tex.bak @@ -153,4 +153,4 @@ d_{S-T}&=\frac{d_{T-M}}{(1-e)(T_M/T_T)^{2/3}-1}\label{cassiniuaexcentrique}\\ où \(e\) est l'excentricité\index{excentricite@excentricité} de l'orbite de Mars. La valeur obtenue à l'aide de l'équation \ref{cassiniuaexcentrique} ne représente plus alors qu'un écart de 15\%. \section{La distance des étoiles} -On a vu que la parallaxe\index{parallaxe} de Mars est d'environ \angle{;;15} d'arc. Cette valeur est vraiment très petite. Il est donc impossible d'effectuer une mesure de la parallaxe d'une étoile à l'aide de la méthode utilisée pour Mars. Deux observations simultanées en deux endroits différents de la Terre ne permettent pas une telle mesure. Par la méthode de la parallaxe, la seule grandeur qu'il est possible de modifier est la distance entre les deux points d'observation. Comme des distances de l'ordre du rayon de la Terre ne suffisent pas, un effet de parallaxe plus important fut obtenu en effectuant la mesure à six mois d'intervalle. Ainsi, la distance entre les deux ``points de vue'' correspond au diamètre de l'orbite terrestre. La première mesure de la parallaxe d'une étoile (parallaxe stellaire\index{parallaxe!stellaire}) a été faite en 1838 par Friedrich Wilhelm Bessel\index{Bessel} pour la binaire 61 du Cygne. Mais, même pour une telle distance, les parallaxes d'étoiles restent inférieures à la seconde d'arc. Par exemple, pour Proxima du Centaure\index{Proxima du Centaure}, l'étoile la plus proche de nous, la parallaxe vaut 760 millisecondes d'arc. \ No newline at end of file +On a vu que la parallaxe\index{parallaxe} de Mars est d'environ \ang{;;15} d'arc. Cette valeur est vraiment très petite. Il est donc impossible d'effectuer une mesure de la parallaxe d'une étoile à l'aide de la méthode utilisée pour Mars. Deux observations simultanées en deux endroits différents de la Terre ne permettent pas une telle mesure. Par la méthode de la parallaxe, la seule grandeur qu'il est possible de modifier est la distance entre les deux points d'observation. Comme des distances de l'ordre du rayon de la Terre ne suffisent pas, un effet de parallaxe plus important fut obtenu en effectuant la mesure à six mois d'intervalle. Ainsi, la distance entre les deux ``points de vue'' correspond au diamètre de l'orbite terrestre. La première mesure de la parallaxe d'une étoile (parallaxe stellaire\index{parallaxe!stellaire}) a été faite en 1838 par Friedrich Wilhelm Bessel\index{Bessel} pour la binaire 61 du Cygne. Mais, même pour une telle distance, les parallaxes d'étoiles restent inférieures à la seconde d'arc. Par exemple, pour Proxima du Centaure\index{Proxima du Centaure}, l'étoile la plus proche de nous, la parallaxe vaut 760 millisecondes d'arc. \ No newline at end of file diff --git a/Annexe-Rotations/Annexe-Rotations.tex b/Annexe-Rotations/Annexe-Rotations.tex index a7e8b53..33d7984 100644 --- a/Annexe-Rotations/Annexe-Rotations.tex +++ b/Annexe-Rotations/Annexe-Rotations.tex @@ -76,7 +76,7 @@ La figure \ref{milky_way_2005} présente une vue d'artiste de la Voie Lactée te \begin{figure}[th] \centering -\caption[Le Soleil dans la Voie Lactée]{Le soleil dans la Voie Lactée\label{milky_way_2005} \par \scriptsize{Représentation artistique\endnote{Voir le site de l'encyclopédie Wikipedia~: \url=http://commons.wikimedia.org/wiki/Image:Milky_Way_2005.jpg=. Image dans le domaine public. Remerciements à la NASA.}}} +\caption[Le Soleil dans la Voie Lactée]{Le soleil dans la Voie Lactée\label{milky_way_2005} \par \scriptsize{Représentation artistique\endnote{Voir le site de l'encyclopédie Wikipedia~: \url=http://commons.wikimedia.org/wiki/Image:Milky\_Way\_2005.jpg=. Image dans le domaine public. Remerciements à la NASA.}}} \includegraphics[width=6cm]{Milky_Way_2005_soleil.eps} \end{figure} @@ -108,7 +108,7 @@ v&=\frac{d}{t}=\frac{2\cdot \pi\cdot R_{S\rightarrow G}}{T_s}\\ \end{align*} Cette vitesse est incroyable. Nous ne la ressentons à nouveau pas ou peu toujours à cause de l'inertie\index{inertie}. -Notons que cette vitesse est la même pour toutes les étoiles proches du Soleil qui participent au mouvement de rotation autour du centre de la galaxie. Mais le Soleil a aussi un mouvement propre, c'est-à-dire qu'une partie de sa vitesse ne correspond pas à sa vitesse de rotation autour du centre de la galaxie. Cette composante vaut environ \SI{20}{\kilo\metre\per\second}.\endnote{Voir le site \url=http://www.dil.univ-mrs.fr/~gispert/enseignement/astronomie/5eme_partie/voieLactee.php=} +Notons que cette vitesse est la même pour toutes les étoiles proches du Soleil qui participent au mouvement de rotation autour du centre de la galaxie. Mais le Soleil a aussi un mouvement propre, c'est-à-dire qu'une partie de sa vitesse ne correspond pas à sa vitesse de rotation autour du centre de la galaxie. Cette composante vaut environ \SI{20}{\kilo\metre\per\second}.\endnote{Voir le site \url=http://www.dil.univ-mrs.fr/~gispert/enseignement/astronomie/5eme\_partie/voieLactee.php=} Relevons enfin une règle bien pratique pour la transformation d'unité entre les \si{\metre\per\second} et les \si{\kilo\metre\per\hour}. On a en effet~: \[\SI{1}{\kilo\metre\per\hour}=\frac{1\,km}{1\,h}=\frac{1000\,m}{3600\,s}=1 / 3,6\unit{\metre\per\second}\] diff --git a/Cinematique/Cinematique.tex b/Cinematique/Cinematique.tex index 0fa125c..0939bc1 100644 --- a/Cinematique/Cinematique.tex +++ b/Cinematique/Cinematique.tex @@ -278,7 +278,7 @@ Contrairement à la plupart des galaxies\index{galaxie@galaxie} qui s'éloignent \begin{figure} \centering -\caption[Collision Andromède-Voie Lactée]{Collision Andromède-Voie Lactée\label{andromede_collision} \par \scriptsize{Rencontre, vue depuis la Terre, d'Andromède et de la Voie Lactée. Vue d'artiste\endnote{Voir le site de Wikicommon (notamment pour le copyright~: sans copyright)~: \url=http://commons.wikimedia.org/wiki/Image:Andromeda_collision.jpg=}.}} +\caption[Collision Andromède-Voie Lactée]{Collision Andromède-Voie Lactée\label{andromede_collision} \par \scriptsize{Rencontre, vue depuis la Terre, d'Andromède et de la Voie Lactée. Vue d'artiste\endnote{Voir le site de Wikicommon (notamment pour le copyright~: sans copyright)~: \url=http://commons.wikimedia.org/wiki/Image:Andromeda\_collision.jpg=}.}} \includegraphics[width=7cm]{Andromede_collision.eps} \end{figure} @@ -555,7 +555,7 @@ sublunaire\index{sublunaire@sublunaire}, ceux qui se déplacent à la surface de Pour les physiciens actuels, la même physique doit être valable dans tout l'Univers\index{univers@univers}. Ainsi, la Lune\index{Lune@Lune}, comme un autre objet à la surface de la Terre, est soumise à son poids\index{poids@poids}, c'est-à-dire à l'attraction\index{attraction@attraction} de la Terre. Elle devrait donc tomber sur celle-ci. Or, manifestement, elle ne le fait pas. Cela signifie-t-il alors que son poids est nul ? Si on considère que la Lune est un objet comme un autre, cela ne peut être le cas. Comment donc expliquer que la Lune ne tombe pas ? -On pourrait répondre à cette question en admettant, même si cela paraît paradoxal, qu'en réalité elle tombe. L'idée est la suivante~: supposons qu'on lance un objet horizontalement un peu au-dessus de la surface de la Terre. Appelons \(v\) la vitesse horizontale initiale. Si \(v\) est nulle, l'objet tombe en chute libre\index{chute@chute!libre}. Si \(v\) est non nulle, mais petite, l'objet est en mouvement balistique\index{balistique@balistique} et il tombe sur la terre quelques mètres plus loin. Plus \(v\) est grande, plus la distance qu'il parcourt à la surface de la Terre est grande. Si la vitesse est assez grande, l'objet semble suivre la courbure\index{courbure@courbure} de la Terre, tout en tombant petit à petit jusqu'à sa surface. A la limite, pour une vitesse donnée, l'objet tombe \og en même temps\fg{} que la courbure\index{courbure@courbure} de la Terre voit \og descendre\fg{} sa surface (cf. figure \ref{chutelune}). C'est alors comme s'il la ratait en permanence. Ainsi, il peut tomber sur la Terre tout en tournant autour d'elle. La figure \ref{chutelunenewton} présente l'illustration\endnote{Mes plus vifs remerciements à Emilio Segrè Visual Archives pour son autorisation de reproduire ici ce document. Le lien vers l'image est~: \url|http://photos.aip.org/quickSearch.jsp?qsearch=Newton+isaac+H5&group=10&Submit=GO|} utilisée par Newton dans les principia de 1728 pour expliquer la chute de la lune\footnote{Avec mes plus vifs remerciements à Emilio Segrè Visual Archives pour son autorisation de reproduire ici cette magnifique illustration de Newton tirée des Principia de 1728 présentant la relation entre chute et satellisation. On la trouve aussi dans \cite[p. 164]{EL99}.} (voir aussi page \pageref{chuteprincipia}). +On pourrait répondre à cette question en admettant, même si cela paraît paradoxal, qu'en réalité elle tombe. L'idée est la suivante~: supposons qu'on lance un objet horizontalement un peu au-dessus de la surface de la Terre. Appelons \(v\) la vitesse horizontale initiale. Si \(v\) est nulle, l'objet tombe en chute libre\index{chute@chute!libre}. Si \(v\) est non nulle, mais petite, l'objet est en mouvement balistique\index{balistique@balistique} et il tombe sur la terre quelques mètres plus loin. Plus \(v\) est grande, plus la distance qu'il parcourt à la surface de la Terre est grande. Si la vitesse est assez grande, l'objet semble suivre la courbure\index{courbure@courbure} de la Terre, tout en tombant petit à petit jusqu'à sa surface. A la limite, pour une vitesse donnée, l'objet tombe \og en même temps\fg{} que la courbure\index{courbure@courbure} de la Terre voit \og descendre\fg{} sa surface (cf. figure \ref{chutelune}). C'est alors comme s'il la ratait en permanence. Ainsi, il peut tomber sur la Terre tout en tournant autour d'elle. La figure \ref{chutelunenewton} présente l'illustration\endnote{Mes plus vifs remerciements à Emilio Segrè Visual Archives pour son autorisation de reproduire ici ce document. Le lien vers l'image est~: \url|http://photos.aip.org/quickSearch.jsp?qsearch=Newton+isaac+H5\&group=10\&Submit=GO|} utilisée par Newton dans les principia de 1728 pour expliquer la chute de la lune\footnote{Avec mes plus vifs remerciements à Emilio Segrè Visual Archives pour son autorisation de reproduire ici cette magnifique illustration de Newton tirée des Principia de 1728 présentant la relation entre chute et satellisation. On la trouve aussi dans \cite[p. 164]{EL99}.} (voir aussi page \pageref{chuteprincipia}). \begin{figure}[ht] \centering @@ -687,7 +687,7 @@ Associé au très grand astronome Tycho Brahé\index{Brahe@Brahé!Tycho}, Johane Avec les observations de Galilée\index{Galilee@Galilée} (cratères sur la Lune\index{cratere@cratère!lunaire}, taches solaires\index{tache@tache!solaire}, phases de Vénus\index{phase@phase!de vénus} et satellites de Jupiter\index{satellite@satellite!de jupiter}), celles de Tycho Brahé sur le mouvement des comètes\index{comete@comète} à travers les sphères cristallines\index{sphere@sphère!cristalline} censées \og porter\fg{} les planètes, les calculs de Kepler vont non seulement permettre l'abandon de l'idée de fixité de la Terre, mais plus tard trouver une place importante dans la nouvelle physique élaborée par Newton. \smallskip -\raggedleft{\footnotesize{Portrait de Kepler tiré de Wikipedia\endnote{Voir \url=http://fr.wikipedia.org/wiki/Image:Johannes_Kepler_1610.jpg=}}} +\raggedleft{\footnotesize{Portrait de Kepler tiré de Wikipedia\endnote{Voir \url=http://fr.wikipedia.org/wiki/Image:Johannes\_Kepler\_1610.jpg=}}} \end{minipage} \hfill \begin{minipage}[b]{6.5cm} @@ -700,7 +700,7 @@ Avec les observations de Galilée\index{Galilee@Galilée} (cratères sur la Lune \begin{figure}[t] \centering -\caption[Astronomica pars Optica]{Astronomica pars Optica\label{oeil}\endnote{Voir le site de wikipedia~: \url=http://fr.wikipedia.org/wiki/Image:Kepler_Optica.jpg= notamment pour le copyright de l'image.}} +\caption[Astronomica pars Optica]{Astronomica pars Optica\label{oeil}\endnote{Voir le site de wikipedia~: \url=http://fr.wikipedia.org/wiki/Image:Kepler\_Optica.jpg= notamment pour le copyright de l'image.}} \includegraphics[width=6cm]{Kepler_Optica.eps} \end{figure} diff --git a/Cinematique/Cinematique.tex.bak b/Cinematique/Cinematique.tex.bak index d0f6dca..516985f 100644 --- a/Cinematique/Cinematique.tex.bak +++ b/Cinematique/Cinematique.tex.bak @@ -130,7 +130,7 @@ Un objet accélère de 0 à \SI{100}{\kilo\metre\per\hour} en \SI{10}{\second}. Réponse~: attention, il faut que les unités du dénominateur (\si{\second}) correspondent à celles du numérateur (\si{\kilo\metre\per\hour}). On doit donc soit transformer des \si{\kilo\metre\per\hour{}} en \si{\kilo\metre\per\second}, soit des secondes en heures~: \begin{itemize} -\item \(\SI{100}{\kilo\metre\per\hour}=\SI{100/3600}{\kilo\metre\per\second}=\SI{0,028}{\kilo\metre\per\second}\) +\item \(\SI{100}{\kilo\metre\per\hour}=100/3600\si{\kilo\metre\per\second}=\SI{0,028}{\kilo\metre\per\second}\) Ainsi, l'accélération vaut alors~: @@ -278,7 +278,7 @@ Contrairement à la plupart des galaxies\index{galaxie@galaxie} qui s'éloignent \begin{figure} \centering -\caption[Collision Andromède-Voie Lactée]{Collision Andromède-Voie Lactée\label{andromede_collision} \par \scriptsize{Rencontre, vue depuis la Terre, d'Andromède et de la Voie Lactée. Vue d'artiste\endnote{Voir le site de Wikicommon (notamment pour le copyright~: sans copyright)~: \url=http://commons.wikimedia.org/wiki/Image:Andromeda_collision.jpg=}.}} +\caption[Collision Andromède-Voie Lactée]{Collision Andromède-Voie Lactée\label{andromede_collision} \par \scriptsize{Rencontre, vue depuis la Terre, d'Andromède et de la Voie Lactée. Vue d'artiste\endnote{Voir le site de Wikicommon (notamment pour le copyright~: sans copyright)~: \url=http://commons.wikimedia.org/wiki/Image:Andromeda\_collision.jpg=}.}} \includegraphics[width=7cm]{Andromede_collision.eps} \end{figure} @@ -292,7 +292,7 @@ Vitesse de la lumière~: & \SI{300000}{\kilo\metre\per\second} \end{tabular} \end{center} -Il existe plusieurs manières de résoudre ce problème. En voici une. On commence par déterminer la vitesse d'Andromède en AL/an. Pour cela, on commence par l'exprimer en \si{\kilo\metre\per} an~: +Il existe plusieurs manières de résoudre ce problème. En voici une. On commence par déterminer la vitesse d'Andromède en AL/an. Pour cela, on commence par l'exprimer en \si{\kilo\metre\per an}~: \begin{align*} &\SI{500000}{\kilo\metre\per\hour}=5\cdot10^{5}\cdot 24\cdot 365\\ &=\SI{4,38e9}{\kilo\metre\per an} @@ -555,7 +555,7 @@ sublunaire\index{sublunaire@sublunaire}, ceux qui se déplacent à la surface de Pour les physiciens actuels, la même physique doit être valable dans tout l'Univers\index{univers@univers}. Ainsi, la Lune\index{Lune@Lune}, comme un autre objet à la surface de la Terre, est soumise à son poids\index{poids@poids}, c'est-à-dire à l'attraction\index{attraction@attraction} de la Terre. Elle devrait donc tomber sur celle-ci. Or, manifestement, elle ne le fait pas. Cela signifie-t-il alors que son poids est nul ? Si on considère que la Lune est un objet comme un autre, cela ne peut être le cas. Comment donc expliquer que la Lune ne tombe pas ? -On pourrait répondre à cette question en admettant, même si cela paraît paradoxal, qu'en réalité elle tombe. L'idée est la suivante~: supposons qu'on lance un objet horizontalement un peu au-dessus de la surface de la Terre. Appelons \(v\) la vitesse horizontale initiale. Si \(v\) est nulle, l'objet tombe en chute libre\index{chute@chute!libre}. Si \(v\) est non nulle, mais petite, l'objet est en mouvement balistique\index{balistique@balistique} et il tombe sur la terre quelques mètres plus loin. Plus \(v\) est grande, plus la distance qu'il parcourt à la surface de la Terre est grande. Si la vitesse est assez grande, l'objet semble suivre la courbure\index{courbure@courbure} de la Terre, tout en tombant petit à petit jusqu'à sa surface. A la limite, pour une vitesse donnée, l'objet tombe \og en même temps\fg{} que la courbure\index{courbure@courbure} de la Terre voit \og descendre\fg{} sa surface (cf. figure \ref{chutelune}). C'est alors comme s'il la ratait en permanence. Ainsi, il peut tomber sur la Terre tout en tournant autour d'elle. La figure \ref{chutelunenewton} présente l'illustration\endnote{Mes plus vifs remerciements à Emilio Segrè Visual Archives pour son autorisation de reproduire ici ce document. Le lien vers l'image est~: \url|http://photos.aip.org/quickSearch.jsp?qsearch=Newton+isaac+H5&group=10&Submit=GO|} utilisée par Newton dans les principia de 1728 pour expliquer la chute de la lune\footnote{Avec mes plus vifs remerciements à Emilio Segrè Visual Archives pour son autorisation de reproduire ici cette magnifique illustration de Newton tirée des Principia de 1728 présentant la relation entre chute et satellisation. On la trouve aussi dans \cite[p. 164]{EL99}.} (voir aussi page \pageref{chuteprincipia}). +On pourrait répondre à cette question en admettant, même si cela paraît paradoxal, qu'en réalité elle tombe. L'idée est la suivante~: supposons qu'on lance un objet horizontalement un peu au-dessus de la surface de la Terre. Appelons \(v\) la vitesse horizontale initiale. Si \(v\) est nulle, l'objet tombe en chute libre\index{chute@chute!libre}. Si \(v\) est non nulle, mais petite, l'objet est en mouvement balistique\index{balistique@balistique} et il tombe sur la terre quelques mètres plus loin. Plus \(v\) est grande, plus la distance qu'il parcourt à la surface de la Terre est grande. Si la vitesse est assez grande, l'objet semble suivre la courbure\index{courbure@courbure} de la Terre, tout en tombant petit à petit jusqu'à sa surface. A la limite, pour une vitesse donnée, l'objet tombe \og en même temps\fg{} que la courbure\index{courbure@courbure} de la Terre voit \og descendre\fg{} sa surface (cf. figure \ref{chutelune}). C'est alors comme s'il la ratait en permanence. Ainsi, il peut tomber sur la Terre tout en tournant autour d'elle. La figure \ref{chutelunenewton} présente l'illustration\endnote{Mes plus vifs remerciements à Emilio Segrè Visual Archives pour son autorisation de reproduire ici ce document. Le lien vers l'image est~: \url|http://photos.aip.org/quickSearch.jsp?qsearch=Newton+isaac+H5\&group=10\&Submit=GO|} utilisée par Newton dans les principia de 1728 pour expliquer la chute de la lune\footnote{Avec mes plus vifs remerciements à Emilio Segrè Visual Archives pour son autorisation de reproduire ici cette magnifique illustration de Newton tirée des Principia de 1728 présentant la relation entre chute et satellisation. On la trouve aussi dans \cite[p. 164]{EL99}.} (voir aussi page \pageref{chuteprincipia}). \begin{figure}[ht] \centering @@ -591,7 +591,7 @@ a&=\frac{32\cdot d_{t-l}}{T^{2}}=\frac{32\cdot3,84404\cdot10^{8}}{\left(30\cdot2 Cette valeur est du bon ordre de grandeur puisqu'en réalité l'accélération\index{acceleration@accélération} de la Lune vers la Terre vaut~: \[a=\SI{2,7e-3}{\metre\per\second\squared}\] -Bien entendu, ces deux valeurs sont néanmoins assez différentes. Cependant, étant donné les hypothèses très grossières qui ont été faites, obtenir un bon ordre de grandeur est un résultat remarquable. En effet, on a considéré que le mouvement était balistique\index{balistique@balistique}, c'est-à-dire qu'à tout instant la direction dans laquelle la Lune tombe est la même. Or, étant donné la courbure\index{courbure@courbure} de la Terre, sur un quart de période, cette direction varie de \(90^{\circ}\). En réalité donc, le mouvement n'est pas balistique\index{balistique@balistique}, mais central\index{mouvement@mouvement!central}, c'est-à-dire que la chute de la Lune se fait toujours vers un même point. Ainsi, le mouvement réel de la Lune ne se fait pas sur une parabole\index{parabole@parabole}, comme dans le cas d'un mouvement balistique\index{balistique@balistique}, mais sur une ellipse\index{ellipse@ellipse} (très proche d'un cercle). L'annexe \ref{chutelunecirculaire} présente un calcul plus correct de l'accélération de la Lune basé sur une orbite circulaire\index{orbite@orbite!circulaire}. Il permet aussi d'en déduire la forme de la force de gravitation\index{force@force!de gravitation} donnée par la loi de la gravitation universelle\index{loi@loi!de la gravitation universelle} (voir paragraphe \ref{gravitationuniverselle}) qui est inversément proportionnelle au carré de la distance à la Lune. Ce calcul est néanmoins encore une approximation puisque l'orbite n'est pas circulaire mais elliptique. +Bien entendu, ces deux valeurs sont néanmoins assez différentes. Cependant, étant donné les hypothèses très grossières qui ont été faites, obtenir un bon ordre de grandeur est un résultat remarquable. En effet, on a considéré que le mouvement était balistique\index{balistique@balistique}, c'est-à-dire qu'à tout instant la direction dans laquelle la Lune tombe est la même. Or, étant donné la courbure\index{courbure@courbure} de la Terre, sur un quart de période, cette direction varie de \(90^{\circ}\). En réalité donc, le mouvement n'est pas balistique\index{balistique@balistique}, mais central\index{mouvement@mouvement!central}, c'est-à-dire que la chute de la Lune se fait toujours vers un même point. Ainsi, le mouvement réel de la Lune ne se fait pas sur une parabole\index{parabole@parabole}, comme dans le cas d'un mouvement balistique\index{balistique@balistique}, mais sur une ellipse\index{ellipse@ellipse} (très proche d'un cercle). L'annexe \ref{chutelunecirculaire} présente un calcul plus correct de l'accélération de la Lune basé sur une orbite circulaire\index{orbite@orbite!circulaire}. Il permet aussi d'en déduire la forme de la force de gravitation\index{force@force!de gravitation} donnée par la loi de la gravitation universelle\index{loi@loi!de la gravitation universelle} (voir paragraphe \ref{gravitationuniverselle}) qui est inversement proportionnelle au carré de la distance à la Lune. Ce calcul est néanmoins encore une approximation puisque l'orbite n'est pas circulaire mais elliptique. Quoi qu'il en soit, l'idée que la Lune tombe en permanence sur la Terre peut parfaitement expliquer qu'elle tienne apparemment en apesanteur\index{apesanteur@apesanteur} au-dessus de notre tête, pour autant qu'elle soit animée d'une vitesse\index{vitesse@vitesse} non nulle parallèle à la surface de la Terre, que celle-ci ait une valeur bien précise et, bien entendu, qu'elle ne soit soumise à aucun frottement\index{frottement@frottement}. @@ -713,7 +713,7 @@ Un exemple très intéressant de cinématique du mouvement, au sens d'une descri L'orbite des planètes est une ellipse\index{orbite@orbite!elliptique} dont le Soleil est à l'un des foyers. \item[Deuxième loi\index{Kepler@Kepler!seconde loi}] -En des temps égaux, la surface balayée par la distance entre le Toleil et la planète est toujours la même. +En des temps égaux, la surface balayée par la distance entre le Soleil et la planète est toujours la même. \item[Troisième loi\index{Kepler@Kepler!troisième loi}] Le rapport du cube du demi-grand axe\index{axe@axe!demi-grand} d'une planète au carré de sa période de révolution\index{periode@période!de révolution} est constant pour tous les corps tournant autour du Soleil. diff --git a/CoursMecaniqueOSDF.mw b/CoursMecaniqueOSDF.mw new file mode 100644 index 0000000..e69de29 diff --git a/CoursMecaniqueOSDF.out b/CoursMecaniqueOSDF.out new file mode 100644 index 0000000..885fc52 --- /dev/null +++ b/CoursMecaniqueOSDF.out @@ -0,0 +1,423 @@ +\BOOKMARK [0][-]{chapter.1}{\376\377\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n}{}% 1 +\BOOKMARK [1][-]{section.1.1}{\376\377\000D\000u\000\040\000t\000e\000m\000p\000s}{chapter.1}% 2 +\BOOKMARK [1][-]{section.1.2}{\376\377\000D\000e\000\040\000l\000'\000i\000n\000f\000i\000n\000i\000m\000e\000n\000t\000\040\000g\000r\000a\000n\000d\000\040\000\340\000\040\000l\000'\000i\000n\000f\000i\000n\000i\000m\000e\000n\000t\000\040\000p\000e\000t\000i\000t}{chapter.1}% 3 +\BOOKMARK 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