From 6b2bff430ba554ad68ce1ad056a30050078546d6 Mon Sep 17 00:00:00 2001 From: Nicolas Pettiaux Date: Thu, 2 Jun 2022 08:03:02 +0200 Subject: [PATCH] premier ajout --- COURS-2-Energie-OHEXERCRESOL.tex | 250 ++++++++++++++++++ .../COURS2EnergieOHEXERCRESOL-img001.png | Bin 0 -> 29747 bytes .../COURS2EnergieOHEXERCRESOL-img002.png | Bin 0 -> 182401 bytes .../COURS2EnergieOHEXERCRESOL-img003.png | Bin 0 -> 13464 bytes .../COURS2EnergieOHEXERCRESOL-img004.png | Bin 0 -> 158161 bytes .../COURS2EnergieOHEXERCRESOL-img005.png | Bin 0 -> 119085 bytes .../COURS2EnergieOHEXERCRESOL-img006.png | Bin 0 -> 41473 bytes .../COURS2EnergieOHEXERCRESOL-img007.png | Bin 0 -> 323560 bytes .../COURS2EnergieOHEXERCRESOL-img008.png | Bin 0 -> 198890 bytes .../COURS2EnergieOHEXERCRESOL-img009.png | Bin 0 -> 188159 bytes .../COURS2EnergieOHEXERCRESOL-img010.png | Bin 0 -> 245510 bytes .../COURS2EnergieOHEXERCRESOL-img011.png | Bin 0 -> 171517 bytes .../COURS2EnergieOHEXERCRESOL-img012.png | Bin 0 -> 191643 bytes .../COURS2EnergieOHEXERCRESOL-img013.png | Bin 0 -> 236920 bytes .../COURS2EnergieOHEXERCRESOL-img014.png | Bin 0 -> 144026 bytes 15 files changed, 250 insertions(+) create mode 100644 COURS-2-Energie-OHEXERCRESOL.tex create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img001.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img002.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img003.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img004.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img005.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img006.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img007.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img008.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img009.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img010.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img011.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img012.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img013.png create mode 100644 COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img014.png diff --git a/COURS-2-Energie-OHEXERCRESOL.tex b/COURS-2-Energie-OHEXERCRESOL.tex new file mode 100644 index 0000000..56c1918 --- /dev/null +++ b/COURS-2-Energie-OHEXERCRESOL.tex @@ -0,0 +1,250 @@ +% This file was converted to LaTeX by Writer2LaTeX ver. 1.6.1 +% see http://writer2latex.sourceforge.net for more info +\documentclass[11pt]{article} +\usepackage{amsmath,amssymb,amsfonts} + +%\usepackage[a4paper, total={17cm, 26cm}]{geometry} +\usepackage[a4paper,margin=1.5cm]{geometry} +\usepackage{color} +\usepackage{hyperref} % Pour liens internets cliquables +\hypersetup{ + colorlinks=true, + linkcolor=blue, + filecolor=magenta, + urlcolor=blue, %cyan + pdftitle={Cours de physique - DLPP - 2022}, + pdfpagemode=FullScreen, + } + +\usepackage{hyperref} +\usepackage[ + type={CC}, + modifier={by-nc-sa}, + version={3.0}, +]{doclicense} + +\usepackage{multicol} +\setlength{\columnsep}{0.5cm} +\def\columnseprulecolor{\color{black}} + +% pour les pieds de page et entêtes +\usepackage{fancyhdr} + +\pagestyle{fancy} +\fancyhf{} +\lhead{Cours de physique} +\rhead{Oscillateur harmonique} +\lfoot{En cours de rédaction et correction - ne pas distribuer - \ccbyncsa} +\rfoot{Page \thepage} + +\usepackage{fontspec} +\usepackage{xunicode} +\usepackage{xltxtra} +\usepackage{array} +\usepackage{hhline} +\usepackage{graphicx} +\usepackage{polyglossia} +\setdefaultlanguage{french} + +\newcounter{Text} +\renewcommand\theText{\arabic{Text}} +\title{Cours de physique de $6^e$ secondaire - 2021-2022 \\ +En cours de rédaction et correction - ne pas distribuer \\ +tout commentaire bienvenu par email à \\ + manueldephysique@educode.be} +\author{Alexandra David - Corinne Leyssen - Nicolas Pettiaux - Matteo Poncé} + +\begin{document} +\maketitle +\doclicenseThis + +\setcounter{tocdepth}{10} +\renewcommand\contentsname{Table des matières} +\tableofcontents + +\hrulefill + +\begin{multicols}{2} + +\section{Énergie de l’oscillateur harmonique} + +\subsection{Vidéos à regarder} +\begin{enumerate} +\item \href{https://videos.domainepublic.net/w/k4SYtXTppaRqy5fQQV3foV}{Bilan énergétique de l'oscillateur horizontal} +\item \href{https://videos.domainepublic.net/w/k19sGJLazDaDXk2Xvz2HpX}{Énergie d'un oscillateur masse-ressort horizontal} +\end{enumerate} + +\subsection{Différentes formes d’énergie d’un oscillateur harmonique} +\begin{enumerate} +\item Energie cinétique~(due à la vitesse) : $E= \frac{1}{2} mv^2$ +\item Energie potentielle gravifique~(due à la hauteur) : $E=mgh$ +\item Energie potentielle élastique~(due à la compression ou dilatation d’un ressort) $E=\frac{1}{2} ky^2$ +\end{enumerate} + +\subsection{Energie totale d’un oscillateur harmonique} +L’énergie totale mécanique d’un oscillateur harmonique est la somme des énergies cinétique et potentielle (gravifique +pour un pendule simple et élastique pour un ressort horizontal). + +Dans le cas où les frottements sont négligés, l’énergie totale reste constante (principe de conservation d’énergie). + +Exprimons mathématiquement ce principe en répondant à la question : + +En toute généralité, quelle est l’énergie totale d’un oscillateur harmonique~ (que ce soit un pendule simple ou un +pendule élastique) ? + +Lorsqu’un oscillateur harmonique est à une position extrême (+A ou -A), l’énergie cinétique est nulle et l’énergie +potentielle maximale (énergie potentielle gravifique pour un pendule simple et énergie potentielle élastique pour un +ressort horizontal). + +De même, pour un oscillateur harmonique (quel qu’il soit), lorsque la vitesse est maximale, l’énergie potentielle est +nulle (énergie potentielle gravifique pour un pendule simple et énergie potentielle élastique pour un ressort +horizontal). L’énergie totale de l’OH ($E_T$) est donc égale à $E=\frac{1}{2}mv_{\text{max}}^2$ + +Or nous savons que : $E_{\text{T}}=\frac{1}{2}m v_{\text{max}}^2$ avec $v_{\text{max}}=A\omega $. Donc +$E_{\text{T}}=\frac{1}{2}mv_{\text{max}}^2=\frac{1}{2}mA^2\omega ^2$ + +Or $T$ et $\omega $ ne varient pas au cours de l’oscillation, elles sont constantes. + +Notons $k=m\omega ^2$ où k est une constante. On trouve $E_{\text{totale}}=\frac{1}{2}kA^2$ +qui est donc l’énergie totale d’un oscillateur harmonique. + + +\subsection{Que représente k ? } +L’énergie totale d’un oscillateur harmonique est~ $E_{\text{T}}=\frac 1 2kA^2$ : + +Que représente physiquement cette constante $k=m\omega ^2$? + +Pour un pendule élastique (un ressort) + +k est la constante de raideur du ressort $F=kx$(loi de Hooke) où $x$~étant l’allongement du ressort à l’équilibre +lorsque ce dernier est soumis à une force de traction (ou de compression) F. + +Pour un pendule simple $k=m\omega ^2$ $\omega =2\frac{\pi } T$ et $T=2\pi \sqrt{\frac L g}$. Don $\omega +^2=\frac{4\pi ^2}{T^2}=4\pi ^2\frac 1{4\pi ^2}\frac g L=\frac g L$ +et $\omega=\sqrt{\frac{g}{L}}$ où $L$ est longueur du pendule et $m$, sa masse. + +\subsection[Évolution au cours du temps des énergies cinétique, potentielle et totale. ]{Évolution au cours du temps des +énergies cinétique, potentielle et totale. } +\begin{center} +\begin{minipage}{8.89cm} + + +\includegraphics[width=8cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img001.png} +On remarque que lorsque l’énergie cinétique est maximale alors l’énergie potentielle +est nulle et vice versa. Il y a constamment conversion de l’énergie cinétique en potentielle et vice versa, de telle +sorte que l’énergie totale reste constante. +\end{minipage} +\end{center} + +Variation de l’énergie cinétique $E_c(t)=\frac{1}{2}mv(t)=\frac{1}{2}m\omega ^2A^2\text{cos}^2(\omega t+\phi )$ + +Variation de l’énergie potentielle $E_p(t)=\frac{1}{2}\mathit{ky}^2=\frac{1}{2}kA^2\text{sin}^2(\omega t+\phi )$ + +L’énergie totale reste constante. Elle est égale à la somme des énergies cinétique et potentielle. + +\begin{equation*} +E_c(t)+E_p(t)=E_t=\text{constante} +\end{equation*} + + +\subsection{Énergie d’un oscillateur harmonique - exercices} + +\begin{center} +\begin{minipage}{5.992cm} + \includegraphics[width=5.457cm,height=4.239cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img002.png} +\end{minipage} +\end{center} +\subsubsection{Exercice 1} +Un pendule simple de longueur égale à 40 cm et d’une masse de 50 g est lâché lorsqu’il fait un angle de 10° avec la +verticale. + +\begin{enumerate} +\item Calculez son énergie potentielle maximale. +\item Calculez sa vitesse maximale. +\item Calculez sa vitesse à mi-hauteur. +\item Quelle est son énergie totale ? +\end{enumerate} +\subsubsection[Exercice 2]{Exercice 2} +\begin{center} +\begin{minipage}{3.81cm} + \includegraphics[width=3cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img003.png} +\end{minipage} +\end{center} +Pour lancer une boule (masse 50 g) de « flipper », on comprime de 10 cm un ressort d’une constante de raideur égale à +200 N/m. Quelle sera la vitesse de la boule lorsqu’elle aborde le virage au bout d’une course rectiligne de 1,5 m après +qu’elle ait quitté le ressort. Négligez tout frottement ! + +\begin{enumerate} +\item si le flipper est horizontal ? +\item s’il fait un angle de 5° avec l’horizontale ? +\end{enumerate} +\subsubsection[Exercice 3]{Exercice 3} +\begin{center} +\begin{minipage}{6.276cm} + \includegraphics[width=5.75cm,height=5.539cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img004.png} +\end{minipage} +\end{center} +Une balle de 500g est lancée verticalement vers le haut sur un ressort de constante de raideur égale à 32 N/m et de +masse négligeable. La vitesse de lancer de 2 m/s. + +Le ressort se comprime de 12 cm lorsque la bille atteint sa hauteur maximale. + +Quelle est la hauteur atteinte par la bille ? + +\subsubsection{Exercice 4} +Un fusil de fléchettes comprend un ressort de raideur k = 250 N/m, de longueur à vide l0 = 12 cm et qui, comprimé par la +fléchette de masse 25 g, ne mesure plus que l = 4,0 cm. + +\begin{enumerate} +\item Avec quelle vitesse la fléchette sort-elle du fusil dans le cas d’un tir horizontal. Faire le calcul sans tenir +compte du frottement entre fléchette et fusil. +\item Quelle altitude maximale peut-elle atteindre dans le cas d’un tir vertical ? Faire le calcul sans tenir compte du +frottement entre fléchette et fusil ni de la résistance de l’air. + +\end{enumerate} +\subsubsection[Exercice 5]{Exercice 5} +\begin{center} +\begin{minipage}{8.876cm} + \includegraphics[width=8cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img005.png} +\end{minipage} +\end{center} +La masse de 2 kg de la figure ci-contre est suspendue au plafond avec un ressort de masse négligeable et dont la +constante de raideur vaut 200 N/m. Au départ, le ressort n’est pas étiré ni comprimé. On laisse alors tomber la masse +sans la pousser. On aura alors un mouvement d’oscillation de la masse. + +\begin{enumerate} +\item Quelle sera la distance parcourue par le ressort avant qu’il n’entame sa remontée verticale ? +\item Quelle sera la vitesse maximale du ressort ? +\end{enumerate} + + +\begin{center} +\begin{minipage}{3.817cm} + \includegraphics[width=3cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img006.png} +\end{minipage} +\end{center} +\subsubsection{Exercice 6} +Le pendule de la figure ci-contre est en mouvement harmonique et a une vitesse de 5 m/s quand il passe par sa position +d’équilibre. Quelle est la vitesse du pendule lorsqu’il fait un angle de 10° par rapport à la verticale ? + +\end{multicols} + + +\section{Résolutions } + + \includegraphics[width=15cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img007.png} + + \includegraphics[width=15cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img008.png} + + \includegraphics[width=15cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img009.png} + + \includegraphics[width=15cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img010.png} + + \includegraphics[width=15cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img011.png} + + \includegraphics[width=15cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img012.png} + + \includegraphics[width=15cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img013.png} + + \includegraphics[width=15cm]{COURS2EnergieOHEXERCRESOL-img/COURS2EnergieOHEXERCRESOL-img014.png} +\end{document} diff --git 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