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\usepackage{array} \usepackage{comment} \usepackage{enumerate} +\usepackage{siunitx} % pour la gestion des unités +\sisetup{ + per-mode = symbol-or-fraction, + exponent-product = \cdot, + output-decimal-marker = {,} +} \begin{document} @@ -517,6 +523,71 @@ Bien d'autres exemples existent et font l'objet d'études approfondies\endnote{\ \end{overlayarea} \end{frame} +\begin{frame}{Transformation des unités} +Le calcul précédent est intéressant car il montre que les changements d'unités sont omniprésents en physique. + +\smallskip +Revenons sur la toute dernière partie du calcul : transformer des heures en années. En y réfléchissant, on comprend vite qu'une clé du calcul sera le chiffre \[r= 24\cdot 365=\SI{8760}{\hour\per an}\] + +\smallskip +Voici trois manières de savoir si ce chiffre \(r\) doit diviser ou multiplier le nombre d'heures qu'on veut transformer en années. +\end{frame} + +\begin{frame} +\begin{columns} +\begin{column}{0.3\textwidth} +\begin{block}{Rapport de grandeurs} +\scriptsize \textbf{Question}~: dans une année, le nombre d'heures est plus grand que 1 ou plus petit ? + +\smallskip +\textbf{Réponse}~: plus grand. + +Cela implique que si deux chiffres représentent une durée identique, celui en heures est plus grand que celui en années ou celui en années est plus petit que celui en heures. + +\smallskip +\textbf{Conséquence}~: il faut diviser le nombre d'heures par le chiffres rapport r. +\end{block} +\end{column} +\begin{column}{0.3\textwidth} +\begin{block}{Règle de trois} +\scriptsize +\begin{align*} +1 an &\rightarrow \SI{8760}{\hour}\\ +? &\leftarrow \SI{4,8e13}{\hour} +\end{align*} +Par produits croisés : +\begin{align*} +? \cdot 8760 &= 1\cdot 4,8\cdot 10^{13}\\ +? &= \frac{4,8\cdot 10^{13}}{8760}\\ +&=\SI{5,5e9}{ans} +\end{align*} +On a donc bien une division par r. +\end{block} +\end{column} +\begin{column}{0.3\textwidth} +\begin{block}{Unités} +\scriptsize +Les crochets [\dots] signifient~: \og les unités de \fg{} +\begin{align*} +t_h&=\SI{4,8e13}{\hour}\,[t_h]=\si{\hour}\\ +r&=\SI{8760}{\hour\per an}\;\;[r]=\si{\hour\per an} +\end{align*} +La question est~: +\(t_{an}=t_h\cdot r\;\text{ou}\;t_{an}=t_h/r\) + +\smallskip +Un calcul d'unité donne~: +\begin{align*} +[t_{an}]&=[t_h]\cdot [r]=\si{\hour\cdot \hour\per an}\\ +&=\si{\hour\squared\per an}\\ +[t_{an}]&=[t_h]/[r]=\si{\hour\per(\hour\per an)}\\ +&=\si{\hour\cdot (an\per\hour)}=\si{an} +\end{align*} +Pour avoir des années, il faut donc diviser par r. +\end{block} +\end{column} +\end{columns} +\end{frame} \subsubsection{Le système solaire} \begin{frame}{Mouvement du soleil} diff --git a/MecaPres.toc b/MecaPres.toc index 68aabe0..4f0a05b 100644 --- a/MecaPres.toc +++ b/MecaPres.toc @@ -1,4 +1,4 @@ -\select@language {french} +\babel@toc {french}{} \beamer@sectionintoc {1}{Structures}{2}{0}{1} \beamer@subsectionintoc {1}{1}{L'infiniment grand}{2}{0}{1} \beamer@subsubsectionintoc {1}{1}{1}{L'univers}{3}{0}{1} @@ -14,10 +14,10 @@ \beamer@sectionintoc {2}{Mouvements}{32}{0}{2} \beamer@subsectionintoc {2}{1}{L'infiniment grand}{32}{0}{2} \beamer@subsubsectionintoc {2}{1}{1}{Les galaxies}{33}{0}{2} -\beamer@subsubsectionintoc {2}{1}{2}{Le syst\IeC {\`e}me solaire}{38}{0}{2} -\beamer@subsubsectionintoc {2}{1}{3}{La terre}{40}{0}{2} -\beamer@subsubsectionintoc {2}{1}{4}{La lune}{43}{0}{2} -\beamer@subsectionintoc {2}{2}{L'infiniment petit}{47}{0}{2} -\beamer@subsubsectionintoc {2}{2}{1}{Probabilit\IeC {\'e}s}{48}{0}{2} -\beamer@subsubsectionintoc {2}{2}{2}{Principe d'incertitude}{50}{0}{2} -\beamer@subsubsectionintoc {2}{2}{3}{\IeC {\`A} la limite}{51}{0}{2} +\beamer@subsubsectionintoc {2}{1}{2}{Le syst\IeC {\`e}me solaire}{40}{0}{2} +\beamer@subsubsectionintoc {2}{1}{3}{La terre}{42}{0}{2} +\beamer@subsubsectionintoc {2}{1}{4}{La lune}{45}{0}{2} +\beamer@subsectionintoc {2}{2}{L'infiniment petit}{49}{0}{2} +\beamer@subsubsectionintoc {2}{2}{1}{Probabilit\IeC {\'e}s}{50}{0}{2} +\beamer@subsubsectionintoc {2}{2}{2}{Principe d'incertitude}{52}{0}{2} +\beamer@subsubsectionintoc {2}{2}{3}{\IeC {\`A} la limite}{53}{0}{2}