From d24295e2c3a846f55ed1a94dbb0c531bd13c1f31 Mon Sep 17 00:00:00 2001 From: vincent Date: Fri, 4 Jan 2019 10:45:52 +0100 Subject: [PATCH] =?UTF-8?q?Deux=20exercices=20suppl=C3=A9mentaires=20avec?= =?UTF-8?q?=20figure=20produite=20avec=20inkscape=20et=20export=20du=20tex?= =?UTF-8?q?te=20de=20la=20figure=20dans=20un=20fichier=20latex.=20Attentio?= =?UTF-8?q?n,=20modifications=20manuelles=20de=20ce=20fichier=20pour=20y?= =?UTF-8?q?=20mettre=20des=20commandes=20latex=20d'=C3=A9quations.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Annexe-Exercices/Annexe-Exercices.aux | 14 +- Annexe-Exercices/Annexe-Exercices.tex | 68 + Annexe-Exercices/Annexe-Exercices.tex.bak | 1484 +++++++++++++++++++ Annexe-Exercices/Images/incline.eps | 152 ++ Annexe-Exercices/Images/incline.eps_tex | 61 + Annexe-Exercices/Images/incline.eps_tex.bak | 61 + Annexe-Exercices/Images/incline.svg | 268 ++++ Annexe-Exercices/Images/incline2.svg | 257 ++++ Annexe-Exercices/Images/inclinePSTR.tex | 575 +++++++ Annexe-Exercices/Images/inclinePSTR.tex.bak | 575 +++++++ Annexe-Exercices/Images/pendante.eps | 121 ++ Annexe-Exercices/Images/pendante.eps_tex | 57 + Annexe-Exercices/Images/pendante.svg | 141 ++ Annexe-Incertitudes/Annexe-Incertitudes.aux | 22 +- Annexe-Maree/Annexe-Maree.tex | 2 +- CoursMecaniqueOSDF.dvi | Bin 1468336 -> 1475996 bytes CoursMecaniqueOSDF.idx | 2 +- CoursMecaniqueOSDF.lof | 1 + CoursMecaniqueOSDF.log | 73 +- CoursMecaniqueOSDF.lot | 4 +- CoursMecaniqueOSDF.pdf | Bin 12245181 -> 12252813 bytes CoursMecaniqueOSDF.ps | 955 ++++++++---- CoursMecaniqueOSDF.tex.bak | 174 +++ CoursMecaniqueOSDF.toc | 10 +- SolutionsOS.tex | 65 + 25 files changed, 4763 insertions(+), 379 deletions(-) create mode 100644 Annexe-Exercices/Annexe-Exercices.tex.bak create mode 100644 Annexe-Exercices/Images/incline.eps create mode 100644 Annexe-Exercices/Images/incline.eps_tex create mode 100644 Annexe-Exercices/Images/incline.eps_tex.bak create mode 100644 Annexe-Exercices/Images/incline.svg create mode 100644 Annexe-Exercices/Images/incline2.svg create mode 100644 Annexe-Exercices/Images/inclinePSTR.tex create mode 100644 Annexe-Exercices/Images/inclinePSTR.tex.bak create mode 100644 Annexe-Exercices/Images/pendante.eps create mode 100644 Annexe-Exercices/Images/pendante.eps_tex create mode 100644 Annexe-Exercices/Images/pendante.svg create mode 100644 CoursMecaniqueOSDF.tex.bak diff --git a/Annexe-Exercices/Annexe-Exercices.aux b/Annexe-Exercices/Annexe-Exercices.aux index 372d8d5..2b7f475 100644 --- a/Annexe-Exercices/Annexe-Exercices.aux +++ b/Annexe-Exercices/Annexe-Exercices.aux @@ -26,8 +26,8 @@ \newlabel{pendante}{{L.3}{212}} \@writefile{toc}{\contentsline {subsection}{\numberline {L.1.9}Relatifs aux forces}{212}} \newlabel{massedelaterre}{{48}{212}} -\@writefile{toc}{\contentsline {subsection}{\numberline {L.1.10}Relatifs \IeC {\`a} l'\IeC {\'e}nergie}{212}} -\newlabel{helico}{{54}{212}} +\@writefile{toc}{\contentsline {subsection}{\numberline {L.1.10}Relatifs \IeC {\`a} l'\IeC {\'e}nergie}{213}} +\newlabel{helico}{{54}{213}} \@writefile{toc}{\contentsline {subsection}{\numberline {L.1.11}Relatifs \IeC {\`a} la conservation de l'\IeC {\'e}nergie}{213}} \@writefile{toc}{\contentsline {subsection}{\numberline {L.1.12}Relatifs \IeC {\`a} l'\IeC {\'e}nergie hydraulique}{213}} \@writefile{toc}{\contentsline {subsection}{\numberline {L.1.13}Relatifs \IeC {\`a} l'\IeC {\'e}nergie \IeC {\'e}olienne}{213}} @@ -36,8 +36,10 @@ \newlabel{ascenseur}{{L.4}{214}} \@writefile{toc}{\contentsline {section}{\numberline {L.2}Solutions}{214}} \@writefile{toc}{\contentsline {section}{\numberline {L.3}Solutions OS}{220}} +\@writefile{lof}{\contentsline {figure}{\numberline {L.5}{\ignorespaces Plan inclin\IeC {\'e}}}{221}} +\newlabel{incline}{{L.5}{221}} \@setckpt{Annexe-Exercices/Annexe-Exercices}{ -\setcounter{page}{221} +\setcounter{page}{222} \setcounter{equation}{0} \setcounter{enumi}{3} \setcounter{enumii}{0} @@ -52,7 +54,7 @@ \setcounter{subsubsection}{0} \setcounter{paragraph}{0} \setcounter{subparagraph}{0} -\setcounter{figure}{4} +\setcounter{figure}{5} \setcounter{table}{0} \setcounter{FancyVerbLine}{0} \setcounter{float@type}{8} @@ -73,7 +75,7 @@ \setcounter{lstnumber}{1} \setcounter{Solution}{26} \setcounter{exc}{71} -\setcounter{Solution OS}{2} -\setcounter{exosc}{2} +\setcounter{Solution OS}{3} +\setcounter{exosc}{3} \setcounter{lstlisting}{0} } diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index d677fcd..efae223 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -892,6 +892,74 @@ On lâche la première à vitesse initiale nulle. Calculez la vitesse de la seco \end{solos} \end{exos} +\begin{exos} + On lâche, à vitesse initiale nulle, une masse m de \unit{3}{\kilo\gram}, sur un plan incliné d'un angle \(\alpha = \unit{20}{\degree}\) sans frottements. Calculez sa vitesse après un temps t de \unit{2}{\second}. Réponse~: \unit{9,72}{\metre\per\second}. + \begin{solos} + Deux forces agissent ici : la réaction du plan, qui lui est normale (c'est-à-dire perpendiculaire), et le poids de la masse. Comme la réaction du plan n'a aucune composante parallèlement au plan, elle ne peut être responsable de l'accélération de la masse le long de celui-ci. + + Il faut donc trouver la composante du poids qui est parallèle au plan incliné. L'angle entre le poids et un plan horizontal est de \unit{90}{\degree}. Quand le plan est incliné, cet angle diminue de la valeur de l'inclinaison. L'angle \(\beta\) entre le plan incliné et le poids et donc \(\beta=90-\alpha\). + + Comme la projection du poids selon l'angle \(\beta\) correspond à sa composante parallèle au plan, dans le triangle rectangle composé du poids comme hypoténuse et de ses composantes parallèle et perpendiculaire au plan, la composante parallèle au plan correspond au côté adjacent. Ainsi, on peut écrire~: + \begin{align*} + P_{//}&=P\cdot\cos(\beta)=P\cdot\cos(90-\alpha)\\ + &=P\cdot\sin(\alpha) + \end{align*} + La seconde loi de Newton s'écrit donc le long du plan incliné~: + \begin{align*} + \sum F^{ext}=P\cdot\sin(\alpha)&=m\cdot a\;\Rightarrow\\ + m\cdot g\cdot\sin(\alpha)&=m\cdot a\;\Rightarrow\\ + a&=g\cdot\sin(\alpha)\;\Rightarrow\\ + a&=9,81\cdot\sin(20)=\unit{3,36}{\metre\per\second\squared} + \end{align*} + Avec une vitesse initiale nulle, pour un MRUA d'accélération calculée ci-dessus, la vitesse au bout d'un temps t=\unit{2}{\second} s'obtient par~: + \[v=a\cdot t+v_0=3,36\cdot 2=\unit{9,72}{\metre\per\second}\] + + \medskip + Une autre manière de résoudre le problème est de procéder avec méthode. Le système qu'on doit choisir est bien évidemment la masse m, puisque c'est de celle-ci qu'on cherche l'accélération pour en trouver la vitesse au bout de \unit{2}{\second}. + \begin{figure}[h] + \centering + \caption[Plan incliné]{Le plan incliné}\label{incline} + \medskip + \def\svgwidth{6cm} + \input{Annexe-Exercices/Images/incline.eps_tex} + \end{figure} + + La figure \ref{incline} présente ensuite le dessin des forces extérieures et le choix du système d'axes. Remarquez que ce dernier l'a été selon l'inclinaison du plan. Il aurait pu ne pas en être ainsi, mais ce choix simplifie les calculs, car la masse étant contrainte à se déplacer le long du plan, son accélération perpendiculairement est nulle. Les équations de la seconde loi de Newton, obtenues par projection des forces extérieures et de l'accélération selon les axes, peuvent alors s'écrire : + \begin{align*} + \sum F^{ext}_x&=P_x=m\cdot a_x &\text{sur l'axe x}\\ + \sum F^{ext}_y&=R-P_y=m\cdot a_y=0 &\text{sur l'axe y} + \end{align*} + car l'accélération perpendiculairement au plan est nulle, comme déjà mentionné. + + \smallskip + Si on considére l'angle \(\alpha\), en s'imaginant le plan incliné horizontal, on comprends qu'il se reporte entre le vecteur poids \(\overleftarrow{P}\) et sa compostante selon y \(P_y\). + + Avec le triangle rectangle formé par le poids et ses composantes et un peu de trigonométrie, on peut en déduire~: + \begin{align*} + P_x &= P\cdot \sin(\alpha)\\ + P_y &= P\cdot \cos(\alpha) + \end{align*} + Comme par ailleurs on sait que \(P=m\cdot g\), on peut réécrire les équations de Newton sur les axes comme~: + \begin{align*} + \sum F^{ext}_x&=m\cdot g\cdot \sin(\alpha)=m\cdot a_x\\ + \sum F^{ext}_y&=R-m\cdot g\cdot \cos(\alpha)=m\cdot a_y=0 + \end{align*} + La première de ces équations permet de trouver l'accélération du bloc selon le plan incliné~: + \begin{align*} + a=a_x&=g\cdot \sin(\alpha)\\ + &=9,81\cdot\sin(20)=\unit{3,36}{\metre\per\second\squared} + \end{align*} + + \smallskip + Mais une information supplémentaire nous est donnée par la seconde équation, c'est la valeur de la réaction R au plan~: + \begin{align*} + R&=m\cdot g\cdot \cos(\alpha)\\ + &=3\cdot 9,81\cdot \cos(20)=\unit{27,67}{\newton} + \end{align*} + + \end{solos} +\end{exos} + \begin{exos} Un exercice de test. \begin{solos} diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak new file mode 100644 index 0000000..e274016 --- /dev/null +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -0,0 +1,1484 @@ +\myclearpage + +\chapter{Exercices}\label{exos} +\lettrine{D}{eux conseils} pour la résolution des exercices : faites un dessin quand cela est possible et expliquez-vous le problème en français. +\Opensolutionfile{ans}[Solutions] % ouvre le fichier Solutions.tex qui va contenir les solutions +\section{Problèmes} + +\subsection{Relatifs à la conversion d'unités et à la notation scientifique} + +\begin{ex}\label{centaure} + Alpha du Centaure\index{Alpha du Centaure} C est l'étoile la plus proche de nous. Elle se situe à \unit{4,238}{AL} de nous. Combien de \kilo\metre cela fait-il ? Exprimez le résultat en notation scientifique. Combien de parsec\index{parsec} (pc) cela fait-il ? Réponse : \unit{1,33}{pc}. + \begin{sol} + Comme le nombre de \kilo\metre est 1000 fois plus petit que le nombre de mètres et que \(\unit{1}{AL}=\unit{9,46\cdot 10^{15}}{\metre}\), on a : + \begin{align*} + \unit{4,238}{AL}&=4,238\cdot 9,46\cdot 10^{15}\\ + &=4\cdot 10^{16}\,m=\unit{4\cdot 10^{13}}{\kilo\metre} + \end{align*} + Comme \(\unit{1}{pc}\approx \unit{3\cdot 10^{16}}{\metre}\), on a : + \[\unit{4\cdot 10^{16}}{\metre}\approx \frac{4\cdot 10^{16}}{3\cdot 10^{16}}=\unit{1,33}{pc}\] + \end{sol} +\end{ex} + +\begin{ex} + Quelle est la distance Terre-Soleil en \kilo\metre, en UA et en AL ? Entre nous et l'étoile (autre que le Soleil) la plus proche, combien de fois peut-on mettre cette distance ? Réponses : \unit{1,496\cdot 10^{8}}{\kilo\metre}, \unit{1}{UA}, \unit{16}{\micro AL} et \(268'228\,\times\). + \begin{sol} + La distance Terre-Soleil vaut \unit{1,496\cdot 10^{11}}{\metre}. On a donc : + \[\unit{1,496\cdot 10^{11}}{\metre}=\unit{1,496\cdot 10^{8}}{\kilo\metre}\] + Par ailleurs, par définition de l'unité astronomique (UA), on a : + \[\unit{1,496\cdot 10^{11}}{\metre}=\unit{1}{UA}\] + Finalement, avec $1\,AL=9,46\cdot 10^{15}\,m$, on a : + \begin{align*} + \unit{1,496\cdot 10^{11}}{\metre}&=\frac{1,496\cdot 10^{11}}{9,46\cdot 10^{15}}\\ + &=\unit{1,58\cdot 10^{-5}}{AL}\cong\unit{16}{\micro AL} + \end{align*} + L'exercice \ref{centaure} nous indique que l'étoile la plus proche de nous est à \unit{4,238}{AL}. On a donc : + \[\frac{4,238}{1,58\cdot 10^{-5}}=2,68\cdot 10^5\,\times\] + Alpha du Centaure se trouve donc à \(268'228\,\times\) la distance Terre-Soleil. + \end{sol} +\end{ex} + +\begin{ex} + Combien y a-t-il de fois la distance entre nous et Alpha du Centaure dans le diamètre de notre galaxie la Voie Lactée (\(\phi_{galaxie}=\unit{80'000}{AL}\))? Réponse : \(18'877\,\times\). + \begin{sol} + Le diamètre de notre galaxie est de \unit{80'000}{AL}. L'exercice \ref{centaure} nous indique que la distance à Alpha du Centaure vaut \unit{4,238}{AL}. Ainsi, on a : + \[\frac{80'000}{4,238}=18'877\,\times\] + Le diamètre de la galaxie représente donc \(18'877\,\times\) la distance à l'étoile la plus proche de nous. + \end{sol} +\end{ex} + +\begin{ex} + Sous quel angle (en \degree, en \arcminute et en \arcsecond) voit-on le diamètre de la Lune depuis la Terre ($R_{Lune}=0,2725\cdot R_{Terre}$ ; $d_{Terre-Lune}=\unit{3,84\cdot10^{8}}{\metre}$). Réponses : \unit{0,5157}{\degree}, \unit{31}{\arcminute} et \unit{1860}{\arcsecond}. + \begin{sol} + La distance Terre-Lune est beaucoup plus grande que le diamètre de la Lune. L'angle est donc petit et on peut écrire la relation d'arc donnée par l'équation \ref{relationdarc} : + \begin{align*} + 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}=\unit{0,009}{\rad} + \end{align*} + Comme \(\unit{180}{\degree}=\unit{\pi}{\rad}\), l'angle considéré est : + \[\alpha=0,009\cdot \frac{180}{\pi}=\unit{0,5157}{\degree}\] + Comme un degré vaut soixante minutes d'arc, \(\unit{1}{\degree}=\unit{60}{\arcminute}\), on a : + \[\alpha=\unit{0,5157}{\degree}=0,5157\cdot 60=\unit{31}{\arcminute}\] + et : + \[\alpha=\unit{31}{\arcminute}=31\cdot 60=\unit{1860}{\arcsecond}\] + \end{sol} +\end{ex} + +\begin{ex} + La matière interstellaire est constituée de gaz neutre (hydrogène atomique et moléculaire), de gaz ionisé, de poussières et de particules cosmiques (électrons, protons, ...). La densité d'un nuage de gaz neutre est de \unit{0,1\cdot 10^4}{atomes\per\centi\metre\cubed}. Combien cela fait-il d'atomes par \metre\cubed ? Par \litre ? Réponses : \(10^9\,\text{atomes}\) et \(10^6\,\text{atomes}\). + \begin{sol} + Dans \unit{1}{\metre\cubed}, on trouve \unit{1000}{\deci\metre\cubed} et \unit{10^6}{\centi\metre\cubed}. Ainsi, dans \unit{1}{\metre\cubed}, on trouve un million de fois plus d'atomes que dans \unit{1}{\centi\metre\cubed}. On a donc par \metre\cubed : + \[nb\,atomes=0,1\cdot 10^4\cdot 10^6=10^9\,\text{atomes}\] + Et par litre, c'est-à-dire par \deci\metre\cubed : + \[nb\,atomes=0,1\cdot 10^4\cdot 10^3=10^6\,\text{atomes}\] + \end{sol} +\end{ex} + +\begin{ex} + La première mesure du rayon de la Terre à été faite à Alexandrie\index{Alexandrie} en 235 avant notre ère par Eratosthène\index{Eratosthene@Ératosthène}\footnote{Voir ``Mécanique'', Éric Lindemann, DeBoeck Université, 1999, p. 15,16.}. La méthode qu'il a utilisée est très simple. Il a tout d'abord observé qu'un certain jour de l'année le Soleil éclairait le fond d'un puits à Syène. Il en a déduit qu'à ce moment-là les rayons du Soleil alors parfaitement au zénith\index{zenith@zénith}, pointaient directement vers le centre de la Terre (voir figure \ref{Erathostene}). Par ailleurs, il a mesuré au même moment l'angle fait par ces mêmes rayons au sommet d'un bâton planté 5000 stades plus au nord, à Alexandrie. Comme le montre le schéma \ref{Erathostene}, cet angle est l'angle au centre de la Terre que fait l'arc de cercle déterminé par la partie de méridien entre Syène et Alexandrie. Connaissant cet angle (\(\alpha = \unit{7,5}{\degree}\)) et la longueur de l'arc de cercle (\(L = \unit{5000}{stades}\)), il en déduisit le rayon de la Terre\index{rayon de la Terre} avec une précision extraordinaire pour l'époque : 4 \% d'écart avec la valeur connue aujourd'hui. + + Sachant qu'un stade valait environ \unit{160}{\metre} d'aujourd'hui, calculez le rayon R de la Terre obtenu par Eratosthène. Réponse : \unit{6'111,55}{\kilo\metre}. + +\begin{figure}[ht] + +\caption{Le rayon de la Terre par Eratosthène\label{Erathostene}} +\smallskip{} + +\begin{center} +\input{Annexe-Exercices/Images/Erathostene} +\end{center} +\end{figure} + \begin{sol} + La relation d'arc donnée par l'équation \ref{relationdarc} nous permet d'écrire : + \[L=R\cdot \alpha\,\Rightarrow\,R=\frac{L}{\alpha}\] + Avec la longueur \(L\) en \(m\) et l'angle \(\alpha\) en \rad : + \begin{align*} + L&=5000\cdot 160=\unit{8\cdot 10^5}{\metre}\\ + \alpha&=7,5\cdot \frac{\pi}{180}=\unit{0,13}{\rad} + \end{align*} + Ainsi, le rayon de la Terre d'Eratosthène valait : + \[R=\frac{8\cdot 10^5}{0,13}=\unit{6'111'550}{\metre}=\unit{6'111,55}{\kilo\metre}\] + Sachant que la valeur actuelle du rayon moyen de la Terre vaut : + \[R_{Terre}=\unit{6'371,03}{\kilo\metre}\] + à l'aide de l'équation \ref{defecart}, on peut déterminer l'écart entre les deux valeurs : + \[e=\frac{6'371,03-6'111,55}{6'371,03}\cdot 100=4,1\%\] + \end{sol} +\end{ex} + +\subsection{Relatifs aux notions de déplacement, position et distance parcourue} + +\begin{ex} + Un objet se déplace de la position \(x = \unit{0}{\metre}\) à \(x = \unit{10}{\metre}\), puis à \(x = \unit{11}{\metre}\), puis à \(x = \unit{-5}{\metre}\). Calculez le déplacement total et la distance totale parcourue. Réponses : \unit{-5}{\metre} et \unit{27}{\metre}. + \begin{sol} + Par définition, le déplacement se calcule par : + \[D=\Delta x=x_f-x_i=-5-0=\unit{-5}{\metre}\] + Et la distance parcourue est la distance réellement effectuée : + \[d=10+1+11+5=\unit{27}{\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + Le conducteur d'une automobile qui se déplace à \unit{120}{\kilo\metre\per\hour} est inattentif pendant deux secondes. Quelle distance a-t-il parcourue pendant ce temps ? Réponse : \unit{66,6}{\metre}. + \begin{sol} + Pour passer de \kilo\metre\per\hour en \metre\per\second, il faut diviser par un facteur de 3,6. En effet : + \[\unit{120}{\kilo\metre\hour}=\frac{120\cdot 10^3\,m/h}{3600\,s/h}=\frac{120}{3,6}=\unit{33,3}{\metre\per\second}\] + Ainsi, la distance parcourue en deux secondes est : + \[d=v\cdot t=33,3\cdot 2=\unit{66,6}{\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + La position en mètres d'un objet est donné par l'équation suivante : + + \[x=\begin{cases} + 2\cdot t&si\,0\leq t\leq4s\\ + 8&si\,4\leq t\leq6s\\ + -2\cdot t+20&si\,6\leq t\leq10s + \end{cases}\] + +Calculez le déplacement et la distance parcourue au bout de \unit{10}{\second}. Réponses : \unit{0}{\metre} et \unit{16}{\metre}. + \begin{sol} + Comme la position au bout de \unit{10}{\second} se calcule par : + \[x(10\,s)=-2\cdot 10+20=\unit{0}{\metre}\] + le déplacement est donné par : + \[D=\Delta x=x_f-x_i=0-0=\unit{0}{\metre}\] + Selon l'équation de la position donnée ici, le mouvement de l'objet est le suivant : + \begin{enumerate} + \item à la vitesse constante de \unit{2}{\metre\per\second} l'objet se déplace pendant \unit{4}{\second}, + \item il s'arrête de \unit{4}{} à \unit{6}{\second} et + \item il revient en arrière à la vitesse de \unit{-2}{\metre\per\second} de \unit{6}{} à \unit{10}{\second}. + \end{enumerate} + Ainsi, l'objet parcourt dans un premier temps \(2\cdot 4=\unit{8}{\metre}\) en avant et dans un second temps \unit{8}{\metre} en arrière. La distance parcourue est donc : + \[d=8+8=\unit{16}{\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + Deux trains se dirigent l'un vers l'autre sur une même voie. Ils se déplacent à \unit{100}{\kilo\metre\per\hour} par rapport au sol. Si la distance initiale qui les séparait était de \unit{12}{\kilo\metre}, combien de temps auront-ils roulé quand aura lieu l'accident ? Réponse : \unit{0,06}{\hour}. + \begin{sol} + Deux raisonnements sont possibles : + \begin{itemize} + \item On s'imagine être dans un train qu'on ne voit pas bouger. Sa vitesse par rapport à nous est nulle. L'autre train se trouve au départ à une distance de \unit{12}{\kilo\metre} et se déplace par rapport à nous à une vitesse relative de \unit{200}{\kilo\metre\per\hour} (sa vitesse et notre vitesse sont cumulées). Ainsi, on peut écrire : + \[v=\frac{\Delta x}{t}\;\Rightarrow\;t=\frac{\Delta x}{v}=\frac{12}{200}=\unit{0,06}{\hour}\] + \item On définit le zéro du système d'axes à la position du premier train au moment où ils sont séparés de \unit{12}{\kilo\metre}. L'équation de la position du premier train est alors : + \[x_1=100\cdot t\] + Comme la position initiale du second train vaut \unit{12}{\kilo\metre} et qu'il s'approche, sa vitesse est négative et l'équation de sa position au cours du temps est : + \[x_2=-100\cdot t+12\] + La condition de rencontre s'écrit alors : + \begin{align*} + x_1&=x_2\\ + 100\cdot t&=-100\cdot t+12\;\Rightarrow\\ + 200\cdot t&=12\;\Rightarrow\;t=\frac{12}{200}=\unit{0,06}{\hour} + \end{align*} + \end{itemize} + Le premier raisonnement se fait par rapport à l'un des objets en mouvement. Il est dit relatif. Le second raisonnement se fait par rapport à un référentiel commun : le sol. Il est dit absolu. + + Mais quel que soit le référentiel, le résultat est le même. + \end{sol} +\end{ex} + +\begin{ex} + Une voiture dont la vitesse est de \unit{110}{\kilo\metre\per\hour} se trouve \unit{600}{\metre} derrière un camion dont la vitesse vaut \unit{80}{\kilo\metre\hour}. Combien de temps mettra-t-elle pour rattraper le camion ? Réponse : \unit{0,02}{\hour}. + \begin{sol} + Deux raisonnements sont possibles : + \begin{itemize} + \item On s'imagine être dans le camion qu'on ne voit pas bouger. Sa vitesse par rapport à nous est nulle. La voiture, elle, se trouve au départ à une distance de \unit{0,6}{\kilo\metre} et se déplace par rapport à nous à une vitesse relative de \(110-80=\unit{30}{\kilo\metre\per\hour}\) (sa vitesse est diminuée de notre vitesse, puisqu'on la fuit). Ainsi, on peut écrire : + \[v=\frac{\Delta x}{t}\;\Rightarrow\;t=\frac{\Delta x}{v}=\frac{0,6}{30}=\unit{0,02}{\hour}\] + \item On définit le zéro du système d'axes à la position de la voiture au moment où la voiture et le camion sont séparés de \unit{0,6}{\kilo\metre}. L'équation de la position de la voiture est alors : + \[x_v=110\cdot t\] + Comme la position initiale du camion vaut \unit{0,6}{\kilo\metre} et qu'il va dans la même direction que la voiture, l'équation de sa position au cours du temps est : + \[x_c=80\cdot t+0,6\] + La condition de rencontre s'écrit alors : + \begin{align*} + x_v&=x_c\\ + 110\cdot t&=80\cdot t+0,6\;\Rightarrow\\ + 30\cdot t&=0,6\;\Rightarrow\;t=\frac{0,6}{30}=\unit{0,02}{\hour} + \end{align*} + \end{itemize} + Le premier raisonnement se fait par rapport à l'un des objets en mouvement. Il est dit relatif. Le second raisonnement se fait par rapport à un référentiel commun : le sol. Il est dit absolu. + + Mais quel que soit le référentiel, le résultat est le même. + \end{sol} +\end{ex} + +\subsection{Relatifs à la notion de vitesse} + +\begin{ex} + Quelle est notre vitesse de rotation approximative à la surface de la Terre ? N'utilisez pas votre machine à calculer et faites tous vos calculs en notation scientifique. Réponse : \unit{1'600}{\kilo\metre\per\hour}. + \begin{sol} + On sait que le rayon de la Terre vaut environ \unit{6'400}{\kilo\metre}. Sa circonférence vaut donc : + \[C=2\cdot \pi\cdot r\approx 2\cdot 3\cdot 6'000=\unit{36'000}{\kilo\metre}\approx \unit{40'000}{\kilo\metre}\] + Sa vitesse se calcule donc ainsi : + \begin{align*} + v&=\frac{C}{t}\approx \frac{40'000}{24}\approx \frac{40'000}{25}\\ + &=40'000\cdot \frac{4}{100}=400\cdot 4=\unit{1'600}{\kilo\metre\per\hour} + \end{align*} + En réalité, on a : + \[v=\frac{2\cdot \pi\cdot r}{t}=\frac{2\cdot \pi\cdot 6'371}{24}=\unit{1668}{\kilo\metre\per\hour}\] + Ce qui correspond à un écart (équation \ref{defecart}) de : + \[e=\frac{1668-1600}{1668}\cdot 100=4\%\] + Ce qui est un bon écart, compte tenu des grosses approximations faites. + \end{sol} +\end{ex} + +\begin{ex} + Un joueur de pétanque tire la boule d'un adversaire. Celle-ci se trouve à \unit{9}{\metre} de lui. En supposant que la boule se déplace en ligne droite, à vitesse constante et que le joueur entende le bruit du carreau sur la boule adverse \unit{1,2}{\second} après l'avoir lancée, trouvez la vitesse de la boule. Le son se propage à une vitesse de \unit{343}{\metre\per\second}. Réponse : \unit{7,666}{\metre\per\second}. + \begin{sol} + Le temps donné \(t_{tot}\) est constitué du temps \(t_{boule}\) mis par la boule pour aller frapper celle de l'adversaire et du temps \(t_{son}\) mis par le son pour revenir se faire entendre par le joueur. On a donc : + \[t_{tot}=1,2=t_{boule}+t_{son}\] + Or, pour avoir la vitesse de la boule sur les \unit{9}{\metre} de son parcours, il nous faut \(t_{boule}\). Pour cela, il faut donc calculer \(t_{son}\), qui est le temps mis pas le son pour parcourir \unit{9}{\metre} à la vitesse de \unit{343}{\metre\per\second} : + \[v=\frac{d}{t}\;\Rightarrow\;t=\frac{d}{v}=\frac{9}{343}=\unit{0,026}{\second}\] + Ainsi, le temps de parcours de la boule est : + \[t_{boule}=1,2-t_{son}=1,2-0,026=\unit{1,174}{\second}\] + Et la vitesse de la boule est finalement : + \[v_{boule}=\frac{9}{1,174}=\unit{7,666}{\metre\per\second}\] + \end{sol} +\end{ex} + +\begin{ex} + Un objet se déplace de \(x_{1} = \unit{3,6}{\centi\metre}\) à \(x_{2} = \unit{-5,2}{\centi\metre}\) dans l'intervalle de temps entre \(t_{1} = \unit{3}{\second}\) et \(t_{2} = \unit{6,8}{\second}\). Déterminez sa vitesse moyenne. Réponse : \unit{-2,32}{\centi\metre\per\second}. + \begin{sol} + Par définition de la vitesse moyenne, on a tout simplement : + \[v=\frac{x_2-x_1}{t_2-t_1}=\frac{-5,2-3,6}{6,8-3}=\unit{-2,32}{\centi\metre\per\second}\] + \end{sol} +\end{ex} + +\begin{ex}\label{vitrotterresoleil} + Quelle est la vitesse de rotation de la Terre autour du Soleil en \unit{}{\kilo\metre\per\second} ? La distance Terre-Soleil est de une unité astronomique (\unit{1}{UA} : le tableau \ref{grandeurs}, page \pageref{grandeurs} en donne la correspondance dans l'unité du système international) et on suppose le mouvement circulaire. Réponse : \unit{29,8}{\kilo\metre\per\second}. + \begin{sol} + Le rayon du cercle parcouru par le Soleil vaut donc en mètres : + \begin{align*} + d_{Terre-Soleil}&=\unit{1}{UA}=\unit{1,496\cdot 10^{11}}{\metre}\\ + &=\unit{1,496\cdot 10^{8}}{\kilo\metre} + \end{align*} + La distance parcourue par le Soleil est donc de : + \[d=2\cdot \pi\cdot d_{Terre-Soleil}=\unit{9,4\cdot 10^8}{\kilo\metre}\] + Comme la période de rotation \(T\), c'est-à-dire le temps mis par la Terre pour faire un tour autour du Soleil, est d'une année, soit \unit{365}{jours}, la vitesse moyenne est : + \[v=\frac{d}{T}=\frac{9,4\cdot 10^8}{365\cdot 24\cdot 3600}=\unit{29,8}{\kilo\metre\per\second}\] + \end{sol} +\end{ex} + +\begin{ex} + Grâce à la modélisation par ordinateur du squelette et des muscles des dinosaures\endnote{Voir l'article complet à l'adresse : \url{http://rspb.royalsocietypublishing.org/content/274/1626/2711} ou un lien à l'adresse \url{http://dinonews.net/rubriq/articles.php5?ref=2007\_sellers\_speed}}, la longueur de la foulée du velociraptor a été établie à \unit{3,058}{\metre} et celle du tyranosaure à \unit{9,559}{\metre}. Le temps d'un cycle de marche a quant à lui été estimé à \unit{0,284}{\second} pour le velociraptor et \unit{1,199}{\second} pour le tyranosaure. Calculez la vitesse de ces deux dinosaures dans les unités du système international. Quelle vitesse cela représente-t-il en \unit{}{\kilo\metre\per\hour} ? Vous pouvez comparer ces vitesses à celle du sprinter de l'exercice \ref{vitsprinter}. Réponses : \(\unit{10,8}{\metre\per\second}=\unit{38,8}{\kilo\metre\per\hour}\) et \(\unit{8}{\metre\per\second}=\unit{28,7}{\kilo\metre\per\hour}\). + \begin{sol} + On a simplement pour le velociraptor : + \[v=\frac{d}{t}=\frac{3,058}{0,284}=\unit{10,8}{\metre\per\second}=\unit{38,8}{\kilo\metre\per\hour}\] + Et pour le tyranosaure : + \[v=\frac{d}{t}=\frac{9,559}{1,199}=\unit{8}{\metre\per\second}=\unit{28,7}{\kilo\metre\per\hour}\] + La comparaison montre que la vitesse d'un sprinter (\(\unit{10}{\metre\per\second}=\unit{36}{\metre\per\second}\) est légèrement inférieure à celle d'un velociraptor. + \end{sol} +\end{ex} + +\subsection{Relatif à la notion d'accélération} + +\begin{ex} + Une voiture roulant à \unit{50}{\kilo\metre\per\hour} freine soudainement pour ne pas heurter un piéton. Si la décélération maximale moyenne que ses pneus peuvent lui permettre est de \unit{-3}{\metre\per\second\squared} (route mouillée), en combien de temps s'arrêtera-t-elle ? Réponse : \unit{4,63}{\second}. + \begin{sol} + Par définition de l'accélération, on a : + \begin{align*} + a&=\frac{v_f-v_i}{t}\;\Rightarrow\\ + t&=\frac{v_f-v_i}{a}=\frac{0-50/3,6}{-3}=\unit{4,63}{\second} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex}\label{vitsprinter} + Calculez l'accélération moyenne d'un sprinter\footnote{{}The physics of sports, A. Armenti, New York, 1992, p. 112.} qui parvient à une vitesse de \unit{10}{\metre\per\second} en \unit{9,9}{\second}. Réponse : \unit{1,01}{\metre\per\second\squared}. + \begin{sol} + On a simplement : + \[a=\frac{v_f-v_i}{t}=\frac{10-0}{9,9}=\unit{1,01}{\metre\per\second\squared}\] + \end{sol} +\end{ex} + +\begin{ex} + Déterminez l'accélération moyenne dans les cas suivants : + +\begin{itemize} +\item Un avion DC 10 qui part du repos atteint, en \unit{50}{\second}, une vitesse de \unit{350}{\kilo\metre\per\hour} au moment du décollage. +\item Un avion s'approche d'un porte-avions à \unit{190}{\kilo\metre\per\hour}. Il se pose et est arrêté par un câble de retenue en \unit{5}{\second}. +\item Une capsule spatiale passe d'une vitesse nulle à \unit{1450}{\kilo\metre\per\hour} en \unit{3}{\second}. +\end{itemize} + +Réponses : \unit{1,94}{\metre\per\second\squared}, \unit{-10,56}{\metre\per\second\squared} et \unit{134}{\metre\per\second\squared}. + \begin{sol}On a successivement : + \begin{itemize} + \item pour le DC 10 : + \[a=\frac{350/3,6-0}{50}=\unit{1,94}{\metre\per\second\squared}\] + \item pour l'avion sur le porte-avions : + \[a=\frac{0-190/3,6}{5}=\unit{-10,56}{\metre\per\second\squared}\] + C'est une décélération et l'accélération est donc négative. + \item pour la capsule spatiale : + \[a=\frac{1450/3,6-0}{3}=\unit{134}{\metre\per\second\squared}\] + \end{itemize} + \end{sol} +\end{ex} + +\begin{ex} + A l'instant \(t = \unit{3}{\second}\), une particule se trouve en \(x = \unit{7}{\metre}\) à la vitesse de \unit{4}{\metre\per\second}. A \(t = \unit{7}{\second}\), elle est en \(x = \unit{-5}{\metre}\) à la vitesse de \unit{-2}{\metre\per\second}. Calculez sa vitesse et son accélération moyennes. Réponses : \unit{-3}{\metre\per\second} et \unit{-1,5}{\metre\per\second\squared}. + \begin{sol} + Par définition de la vitesse moyenne, on a : + \[v=\frac{x_f-x_i}{t_f-t_i}=\frac{-5-7}{7-3}=\unit{-3}{\metre\per\second}\] + Par définition de l'accélération moyenne, on a : + \[a=\frac{v_f-v_i}{t_f-t_i}=\frac{-2-4}{7-3}=\unit{-1,5}{\metre\per\second\squared}\] + \end{sol} +\end{ex} + +\subsection{Relatif au MRU} + +\begin{ex} + Une voiture se déplace à la vitesse constante de \unit{50}{\kilo\metre\per\hour} pendant \unit{5}{\minute}. Esquissez les graphes horaires de la position, de la vitesse et de l'accélération pendant cette période. Quelle est la distance totale parcourue ? Par quelle grandeur cette distance est-elle représentée sur le graphe de la vitesse en fonction du temps ? Réponse : \unit{4'167}{\metre}. + \begin{sol} + Les graphes sont présentés à la figure \ref{graphesmru}. + \begin{figure}[!t] + \centering + \caption{Graphes horaires du MRU.\label{graphesmru}} + \includegraphics{GraphesMRU.eps} + \end{figure} + + La distance totale parcourue se calcule simplement : + \[d=v\cdot t=\frac{50}{3,6}\cdot 5\cdot 60=\unit{4'167}{\metre}\] + Sur le graphe de la vitesse en fonction du temps, la distance parcourue apparaît simplement être l'aire sous le graphe. En effet, la base \(t=5\cdot 60=\unit{300}{\second}\) multipliée par la hauteur \(v=50/3,6=\unit{13,9}{\metre\per\second}\) donne bien la distance parcourue. + \end{sol} +\end{ex} + +\begin{ex} + Une voiture de sport se déplaçant à la vitesse constante de \unit{160}{\kilo\metre\per\hour} est prise en chasse par une voiture de police alors que celle-ci à \unit{1}{\kilo\metre} de retard. Pour rattraper la voiture de sport, la voiture de police prend très rapidement une vitesse de \unit{180}{\kilo\metre\per\hour}. Au bout de combien de temps et de quelle distance la police rattrapera la voiture de sport ? Réponses : \unit{0,05}{\hour} et \unit{9}{\kilo\metre}. + \begin{sol} + On va décrire mathématiquement le mouvement des voitures de sport et de police. + + Les deux mouvements sont des MRU. On peut donc écrire, dans un système d'axes dont l'origine est sur la voiture de police au moment où elle entame sa poursuite : + \begin{align*} + v_{police}&=180\cdot t\\ + v_{sport}&=160\cdot t+1 + \end{align*} + La condition de rencontre s'écrit alors : + \begin{align*} + v_{police}=v_{sport}\;\Rightarrow\,180\cdot t&=160\cdot t+1\;\Rightarrow\\ + 20\cdot t&=1\\ + t&=\frac{1}{20}=\unit{0,05}{\hour} + \end{align*} + La voiture de police se trouve alors à une distance de l'origine du système d'axes de : + \[x_{police}=180\cdot 0,05=\unit{9}{\kilo\metre}\] + Alors que la voiture de sport est à la même place : + \[x_{sport}=160\cdot 0,05+1=\unit{9}{\kilo\metre}\] + Ce qu'il fallait montrer. + \end{sol} +\end{ex} + +\subsection{Relatif au MRUA} + +\begin{ex} + Une voiture de police à l'arrêt démarre la poursuite d'un chauffard lorsque celui-ci la dépasse à une vitesse de \unit{144}{\kilo\metre\per\hour}. Son accélération est alors de \unit{5}{\metre\per\second\squared}. En combien de temps et sur quelle distance rattrape-t-elle le chauffard ? Quelle est alors sa vitesse ? Réponses : \unit{16}{\second}, \unit{640}{\metre} et \unit{288}{\kilo\metre\per\hour}. + \begin{sol} + Ici, aucune symétrie n'est présente. Comme les deux voitures ne sont pas à vitesse constante, on ne peut calculer de vitesse relative pour résoudre le problème. Il faut donc procéder en décrivant les deux mouvements par rapport au sol. Ainsi, avec \(\unit{144}{\kilo\metre\per\hour}=\unit{40}{\metre\per\second}\), on peut écrire : + \begin{align*} + MRU\;&\Rightarrow\;x_{chauffard}=40\cdot t\\ + MRUA\;&\Rightarrow\;x_{police}=\frac{1}{2}\cdot 5\cdot t^2 + \end{align*} + La condition de rencontre permet alors de trouver le temps cherché : + \begin{align*} + x_{chauffard}&=x_{police}\;\Rightarrow\\ + 40\cdot t&=\frac{1}{2}\cdot 5\cdot t^2\;\Rightarrow\\ + 40&=2,5\cdot t\;\Rightarrow\;t=\unit{16}{\second} + \end{align*} + car la solution \(t=\unit{0}{\second}\) est à rejeter. En effet, elle correspond au début de la poursuite. + + La position à laquelle se trouvent les deux voitures, qui est en même temps la distance qu'elles ont parcourues, est alors : + \begin{align*} + x_{chauffard}&=40\cdot 16=640\,m\\ + x_{police}&=\frac{1}{2}\cdot 5\cdot 16^2=\unit{640}{\metre} + \end{align*} + Les deux positions sont bien évidemment les mêmes. + + Quant aux vitesse lors de la rencontre, elles sont : + \begin{align*} + v_{chauffard}&=40\,m/s=144\,km/h\\ + v_{police}&=5\cdot 16=\unit{80}{\metre\per\second}=\unit{288}{\kilo\metre\per\hour} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Une voiture entre en collision frontalement avec un arbre. Sa vitesse juste avant le choc était de \unit{50}{\kilo\metre\per\hour}. Elle est stoppée net sur une distance de \unit{1,5}{\metre} (le moteur est complètement écrasé). + + Calculez la valeur de l'accélération (ici une décélération), exprimez-la comme un multiple de l'accélération terrestre \(g\) (\(g = \unit{9,81}{\metre\per\second\squared}\)) et calculez le temps que dure la collision. Réponses : \(\unit{-64,3}{\metre\per\second\squared}=-6,55\cdot g\) et \unit{0,216}{\second}. + \begin{sol} + On fait l'hypothèse d'un MRUA. La voiture est stoppée sur une distance de \unit{1,5}{\metre}. On peut donc écrire : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot d\;\Rightarrow\\ + 0^2&=(\frac{50}{3,6})^2+2\cdot a\cdot 1,5\;\Rightarrow\\ + a&=-\frac{13,89^2}{3}=\unit{-64,3}{\metre\per\second\squared}=-6,55\cdot g + \end{align*} + Le temps de collision est donc de : + \begin{align*} + a&=\frac{v_f-v_i}{t}\;\Rightarrow\\ + t&=\frac{v_f-v_i}{a}=\frac{0-50/3,6}{-64,3}=\unit{0,216}{\second} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Le chauffeur d'une voiture roulant à \unit{40}{\metre\per\second} aperçoit soudain un kangourou \unit{70}{\metre} devant lui. Quel sera l'avenir de l'animal si le temps de réaction du chauffeur est de \unit{0,8}{\second} et sa décélération maximale de \unit{8}{\metre\per\second\squared} ? Réponse : bien sombre. + \begin{sol} + Oublions la position du kangourou et calculons la distance totale d'arrêt \(d_t\). Elle se compose de la distance de réaction \(d_r\) et la distance de freinage \(d_f\) : + \[d_t=d_r+d_f\] + Pour la distance de réaction, on a : + \[d_r=v\cdot t=40\cdot 0,8=\unit{32}{\metre}\] + Pour la distance de freinage, il faut faire l'hypothèse d'un MRUA. Comme le mouvement est une décélération, c'est-à-dire un freinage, l'accélération est négative, et on a : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot d_f\;\Rightarrow\\ + 0^2&=40^2+2\cdot (-8)\cdot d_f\;\Rightarrow\\ + d_f&=\frac{40^2}{16}=\unit{100}{\metre} + \end{align*} + Ainsi, la distance totale d'arrêt vaut : + \[d_t=32+100=\unit{132}{\metre}\] + Comme le kangourou se trouve à \unit{70}{\metre}, son avenir serait bien sombre s'il n'avait pas cette prodigieuse capacité à rebondir. + \end{sol} +\end{ex} + +\begin{ex} + Un plongeur capable de sauter sur place très haut peut s'élever de \unit{50}{\centi\metre}. Quelle doit être sa vitesse initiale (supposez que le mouvement est un MRUA d'accélération \(g = \unit{9,81}{\metre\per\second\squared}\))? Un dauphin peut s'élever lui \unit{6}{\metre} au-dessus de l'eau. Quelle est sa vitesse verticale initiale ? Réponses : \unit{11,3}{\kilo\metre\per\hour} et \unit{39}{\kilo\metre\per\hour}. + \begin{sol} + Un objet qui n'est soumis qu'à son poids est en chute libre, même s'il monte. Ainsi, l'accélération du plongeur, comme du dauphin, dans sa phase d'ascension vaut \unit{-9,81}{\metre\per\second\squared}. En effet, on a une décélération. Comme celle-ci est constante et que la vitesse au sommet est nulle, on peut écrire pour le plongeur : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot h\;\Rightarrow\\ + 0^2&=v_o^2+2\cdot (-9,81)\cdot 0,5\;\Rightarrow\\ + v_o&=\sqrt{2\cdot 9,81\cdot 0,5}=\unit{3,132}{\metre\per\second}=\unit{11,3}{\kilo\metre\per\hour} + \end{align*} + Et de la même manière, pour le dauphin : + \[v_o=\sqrt{2\cdot 9,81\cdot 6}=\unit{10,85}{\metre\per\second}=\unit{39}{\kilo\metre\per\hour}\] + \end{sol} +\end{ex} + +\begin{ex} + Un plongeur s'élance du haut de la plate-forme des dix mètres d'un plongeoir. + \begin{enumerate} + \item Combien de temps met-il pour arriver dans l'eau s'il se laisse tomber ?\label{caspremier} + \item Combien de temps met-il pour arriver dans l'eau si sa vitesse initiale est horizontale et vaut \unit{1}{\metre\per\second} ? + \item Combien de temps met-il pour arriver dans l'eau si sa vitesse initiale est verticale vers le haut et vaut \unit{1}{\metre\per\second} ?\label{castroisieme} + \end{enumerate} + Dans les cas \ref{caspremier} et \ref{castroisieme}, à quelle vitesse arrive-t-il dans l'eau ? Réponses : \unit{1,43}{\second}, \unit{1,43}{\second} et \unit{1,53}{\second}. + \begin{sol} + Le plongeur est en chute libre. Son accélération vaut donc \(g=\unit{9,81}{\metre\per\second\squared}\). On fixe un axe vertical dont l'origine se situe à \unit{10}{\metre} et qui pointe vers le bas. On peut alors écrire : + \begin{align*} + x&=\frac{1}{2}\cdot g\cdot t^2+v_{o\,verticale}\cdot t\;\Rightarrow\\ + 10&=\frac{1}{2}\cdot 9,81\cdot t^2+v_{o\,verticale}\cdot t + \end{align*} + Avec dans le premier cas, comme dans le second, une vitesse initiale verticale nulle, on peut écrire : + \begin{align*} + 10&=\frac{1}{2}\cdot 9,81\cdot t^2\;\Rightarrow\\ + t&=\sqrt{\frac{2\cdot 10}{9,81}}=\unit{1,43}{\second} + \end{align*} + Ce qui donne une vitesse juste avant d'entrer dans l'eau de : + \[v=a\cdot t=9,81\cdot 1,43=\unit{14}{\metre\per\second}=\unit{50,5}{\kilo\metre\per\hour}\] + Le premier et le second cas ne sont pas différents du point de vue du temps de chute. Néanmoins, la distance parcourue par le plongeur qui se déplace horizontalement est plus grande que celle du plongeur qui se laisse tomber. Mais sa vitesse totale (horizontale et verticale) est aussi plus grande. On peut ainsi comprendre qu'ils arriveraient en bas simultanément s'ils partaient en même temps. + + Le troisième cas est plus complexe puisqu'il faut tenir compte d'une vitesse \(v_o=\unit{-1}{\metre\per\second}\) dans le sens contraire de l'axe : + \begin{align*} + 10&=\frac{1}{2}\cdot 9,81\cdot t^2-1\cdot t\;\Rightarrow\\ + 0&=4,905\cdot t^2-t-10 + \end{align*} + Ce qui constitue une équation à une inconnue (\(t\)), mais du second degré. Sa solution est donnée par : + \begin{align*} + t&=\frac{1\pm \sqrt{1^2-4\cdot 4,905\cdot (-10)}}{2\cdot 4,905}\\ + &=\frac{1\pm 14}{9,81}=\begin{cases}\unit{1,53}{\second}\\\unit{-1,33}{\second}\end{cases} + \end{align*} + Évidemment, la solution négative est à rejeter et la solution positive est supérieure au temps de chute calculé précédemment puisque le plongeur parcourt une certaine distance vers le haut avant de tomber. + + En ce qui concerne la vitesse, dans le troisième cas on peut simplement déterminer la vitesse par : + \[v=a\cdot t=9,81\cdot 1,53-1=\unit{14}{\metre\per\second}=\unit{50,5}{\kilo\metre\per\hour}\] + Ce qui donne la même valeur que précedemment en raison de la faible vitesse verticale initiale et de l'arrondi. Celle-ci doit cependant être comptée et doit l'être négativement (\(v_o=\unit{-1}{\metre\per\second}\)), car elle est vers le haut alors que l'axe est vers le bas. + \end{sol} +\end{ex} + +\begin{ex} + Un objet est lâché à la surface de la terre d'une hauteur h. Déterminez la vitesse à laquelle il arrive au sol en fonction de h et de g. Réponse : \(\sqrt{2\cdot g\cdot h}\). + \begin{sol} + Cet objet est en chute libre. Son accélération vaut donc \(g\). On peut écrire pour un MRUA : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot h\;\Rightarrow\\ + v&=\sqrt{2\cdot g\cdot h} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Une voiture de course tente d'arriver en \unit{4}{\second} à \unit{100}{\kilo\metre\per\hour} départ arrêté. Déterminez quelle accélération (supposée constante) il lui faut pour cela et la distance qu'elle va parcourir. Réponses : \unit{6,94}{\metre\per\second\squared} et \unit{55,56}{\metre}. + \begin{sol} + Par définition de l'accélération, on a : + \[a=\frac{v-v_o}{t}=\frac{100/3,6-0}{4}=\unit{6,94}{\metre\per\second\squared}\] + La distance parcourue est alors : + \[d=\frac{1}{2}\cdot 6,94\cdot 4^2=\unit{55,56}{\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + Une moto passe de \unit{50}{\kilo\metre\per\hour} à \unit{100}{\kilo\metre\per\hour} sur une distance de \unit{59}{\metre}. A combien de \(g\) son conducteur est-il soumis ? Réponse : un demi \(g\). + \begin{sol} + Le temps n'est pas donné. On doit donc écrire : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot d\;\Rightarrow\\ + (100/3,6)^2&=(50/3,6)^2+2\cdot a\cdot 59\;\Rightarrow\\ + a&=\frac{(100/3,6)^2-(50/3,6)^2}{2\cdot 59}\\ + &=\unit{4,905}{\metre\per\second\squared} + \end{align*} + Ce qui représente une accélération d'un demi \(g\). + \end{sol} +\end{ex} + +\begin{ex} + Un plongeur s'élance de la plate-forme des cinq mètres avec une vitesse horizontale de \unit{2}{\metre\per\second}. + \begin{enumerate} + \item Combien de temps mettra-t-il pour arriver dans l'eau ? + \item Si on suppose sa vitesse horizontale constante, à quelle distance du pied du bord de la plate-forme arrivera-t-il dans l'eau ? + \end{enumerate} + + Réponses : \unit{1,01}{\second} et \unit{2,02}{\metre}. + \begin{sol} + Le temps de chute est le même que celui d'un objet tombant verticalement. En effet, seul le poids est présent et l'objet est en chute libre. Ainsi, on peut écrire : + \begin{align*} + h&=\frac{1}{2}\cdot g\cdot t^2\;\Rightarrow\;5=\frac{1}{2}\cdot 9,81\cdot t^2\;\Rightarrow\\ + t&=\sqrt{\frac{2\cdot 5}{9,81}}=\unit{1,01}{\second} + \end{align*} + Si la vitesse horizontale est constante, on peut encore écrire pour la distance horizontale \(d\) parcourue : + \[d=v_{horiz}\cdot t=2\cdot 1,01=\unit{2,02}{\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + Une petite fusée d'enfant a pour deux minutes de carburant. Elle s'élève dans les airs à une vitesse moyenne de \unit{3}{\metre\per\second} pendant la phase de poussée. A la fin de cette phase, sa vitesse atteint \unit{4}{\metre\per\second} (son accélération n'est pas constante durant cette première phase). Calculez à quelle hauteur elle s'est élevée et combien de temps à duré son vol. Réponses : \unit{360,8155}{\metre} et \unit{128,98}{\second}. + \begin{sol} + Commençons par calculer la distance qu'elle a parcouru pendant la phase de poussée. Pour cela, on a la vitesse moyenne \(\overline{v}\) et le temps que dure le mouvement. Ainsi, on peut poser : + \[d_{pouss\acute ee}=\overline{v}\cdot t=3\cdot 2\cdot 60=\unit{360}{\metre}\] + A ce moment-là, au bout de deux minutes et à \unit{360}{\metre} d'altitude, la poussée cesse (le moteur s'arrête). Si on imagine un axe \(y\) orienté vers le haut et dont l'origine se situe au sol, on a donc une fusée qui se situe en \(y_o=\unit{360}{\metre}\) avec une vitesse \(v_o=\unit{4}{\metre\per\second}\) et qui n'est plus soumise qu'à son poids. Elle est donc en chute libre, même si elle monte, et son accélération dirigée vers le bas, dans le sens contraire du mouvement, est une décélération qui vaut : \(g=\unit{-9,81}{\metre\per\second\squared}\). Pendant quelques instants, elle va donc encore monter jusqu'à s'arrêter. Pour calculer la distance sur laquelle elle s'arrête, comme on ne dispose pas du temps qu'elle met pour le faire, on doit écrire : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot d\;\Rightarrow\\ + 0^2&=4^2+2\cdot (-9,81)\cdot d\;\Rightarrow\\ + d&=\frac{16}{2\cdot 9,81}=\unit{0,8155}{\metre}=\unit{81,55}{\centi\metre} + \end{align*} + Ainsi la hauteur totale à laquelle est parvenue la fusée vaut : + \[h_{tot}=360+0,8155=\unit{360,8155}{\metre}\] + Pour déterminer le temps de vol, on dispose en premier lieu du temps de poussée qui est de deux minutes. Il faut encore calculer le temps de chute de la fusée entre le moment ou la poussée cesse et celui ou elle arrive au sol. Pour cela, il faut écrire : + \begin{align*} + y&=\frac{1}{2}\cdot a\cdot t^2+v_o\cdot t+y_o\;\Rightarrow\\ + y&=\frac{1}{2}\cdot (-9,81)\cdot t^2+4\cdot t+360\;\Rightarrow\\ + 0&=-4,905\cdot t^2+4\cdot t+360 + \end{align*} + car on cherche le temps mis pour arriver au sol, soit à \(y=0\). C'est une équation du second degré à une inconnue, dont la solution est : + \begin{align*} + t&=\frac{-4\pm \sqrt{4^2-4\cdot (-4,905)\cdot 360}}{2\cdot (-4,905)}\\ + &=\frac{-4\pm 84,14}{-9,81}=\begin{cases}\unit{-8,17}{\second}\\\unit{8,98}{\second}\end{cases} + \end{align*} + La solution \(t=\unit{8,98}{\second}\) est évidemment la bonne. + + Le temps total est donc finalement de : + \[t_{tot}=2\cdot 60+8,98=\unit{128,98}{\second}\simeq \unit{2}{\minute}\unit{9}{\second}\] + \end{sol} +\end{ex} + +\subsection{Relatifs à la physique aristotélicienne} + +\begin{ex} + Un scooter se déplaçant à la vitesse de \unit{5}{\kilo\metre\per\hour} sur un lac gelé horizontal tire verticalement un petit obus éclairant. Il monte et retombe en \unit{2}{\second}. Déterminez, selon la théorie aristotélicienne du mouvement son lieu d'atterrissage sur le lac. A ce moment-là, déterminez sa distance au scooter. Réponse : \unit{2,76}{\metre}. + \begin{sol} + Pour Aristote, au moment où l'obus est sorti du canon, plus aucune action horizontale vers l'avant ne s'exerce sur lui. Il cessera donc de se déplacer horizontalement et retombera exactement là où il a quitté le canon. + + Pendant l'élévation et la chute de l'obus, le scooter avance à vitesse constante. La distance dont il s'est déplacé par rapport à l'obus (qui n'a selon Aristote pas bougé horizontalement) est donc de : + \[d=v\cdot t=\frac{5}{3,6}\cdot 2=\unit{2,76}{\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + La vigie d'un trois-mâts laisse tomber son couteau. Il met \unit{0,8}{\second} pour arriver sur le pont. Le bateau se déplace à \unit{8}{\kilo\metre\per\hour}. Selon la théorie d'Aristote, déterminez à quelle distance du pied du mât il va se planter. Réponse : \unit{1,78}{\metre}. + \begin{sol} + Rappelons que selon Aristote, dès le moment où on a lâché le couteau, plus aucune force horizontale ne s'exerce sur lui et il va tomber parfaitement verticalement. Or, le bateau avance pendant ce temps. La distance au pied du mât à laquelle tombe le couteau est donc de : + \[d=v\cdot t=\frac{8}{3,6}\cdot 0,8=\unit{1,78}{\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + Un touriste visitant Paris laisse tomber une pièce de cinq francs du haut du premier étage de la Tour Eiffel. En tenant compte de la vitesse de Paris autour le l'axe de rotation de la Terre et selon la cinématique d'Aristote, calculez la distance au pied du point de chute à laquelle la pièce va arriver au sol. Le temps de chute est de \unit{2,1}{\second} et la latitude de Paris vaut \(\beta=\unit{48}{\degree}\unit{48}{\arcminute}\). On ne tiendra compte que du déplacement de Paris dû à la rotation de la Terre sur elle-même. Réponse : \unit{640,5}{\metre}. + \begin{sol} + Pour connaître la vitesse de rotation de la Terre à Paris, il faut connaître la distance parcourue en \unit{24}{\hour}. Pour cela, il faut connaître sa distance \(r\) à l'axe de rotation de la Terre (voir figure \ref{eiffelmecanique}). + \begin{figure}[!t] + \centering + \caption{Chute aristotélicienne de la tour Eiffel.\label{eiffelmecanique}} + \includegraphics{EiffelMecanique.eps} + \end{figure} + On a d'après la figure \ref{eiffelmecanique} que : + \[r=R\cdot \cos(\beta)=6'371\cdot \cos(48,8^{\circ})=\unit{4'197}{\kilo\metre}\] + car : + \begin{align*} + R&=R_{Terre}=\unit{6'371}{\kilo\metre}\\ + \beta&=\unit{48}{\degree}\unit{48}{\arcminute}=\unit{48,8}{\degree} + \end{align*} + Ainsi, la vitesse de la tour Eiffel est : + \begin{align*} + v&=\frac{2\cdot \pi\cdot r}{t}=\frac{2\cdot \pi\cdot 4'197}{24}\\ + &=\unit{1099}{\kilo\metre\per\hour}=\unit{305}{\metre\per\second} + \end{align*} + et, selon Aristote, pendant la chute de la pièce, la Tour Eiffel devrait s'être déplacée de : + \[d=v\cdot t=305\cdot 2,1=\unit{640,5}{\metre}\] + Ce n'est évidemment pas le cas. L'inertie de la pièce l'en empêche. + \end{sol} +\end{ex} + +\subsection{Relatifs à la physique newtonienne} + +Pour résoudre un problème de mécanique newtonienne, aucune méthode n'est prescrite. Cependant, dans la mesure où le problème est bien posé (c'est là une étape particulièrement difficile à réaliser et pour laquelle il faut rester ouvert à toute bonne idée), une suite d'opérations assez bien définie permet de résoudre clairement le problème. + +Attention, cette ``procédure\index{procédure}'', qui présente beaucoup d'avantages, est aussi assez stricte pour éliminer des idées originales parfaitement valables et efficaces. Il faut donc l'utiliser sans dogmatisme. Elle consiste en ce qui suit : +\bigskip +\begin{boiteepaisseavecuntitre} +{ Méthode de résolution des problèmes } +\begin{description} +\item[Choisir le système.] C'est l'étape la plus importante. Elle est fondamentale, car c'est du choix du système que dépend la possibilité d'exprimer les grandeurs nécessaires à la résolution du problème. En particulier, il faut choisir le système de manière à éliminer, dans la mesure du possible, les grandeurs qui sont inconnues et non nécessaires. + +\item[Choisir un système d'axes.] Simple et, si possible, orienté dans le sens supposé de l'accélération. Cela permet d'éliminer un certain nombre de fautes de signe. + +\item[Faire un dessin avec forces extérieures.] Il ne faut en oublier aucune. La détermination du caractère extérieur des forces en jeu est évidemment aussi une étape difficile. Mais elle est fondamentale. + +\item[Spécifier les contraintes sur le système.] Par exemple : se déplace uniquement horizontalement. + +\item[Écrire les équations du mouvement.] Il s'agit d'écrire la seconde loi de Newton sur chaque axe. Il faut donc décomposer les vecteurs forces et accélération sur ceux-ci. + +\item[Résoudre le problème.] Vérifier que le nombre d'équations est égal au nombre d'inconnues. A l'aide des équations décrivant les contraintes et des équations du mouvement, résoudre alors le problème. + +\item[Vérifier la solution sur des cas simples.] Par exemple, si on a obtenu l'accélération d'un objet sur un plan incliné, il faut vérifier que si celui-ci est vertical, alors l'accélération est celle de la chute libre. +\end{description} +\end{boiteepaisseavecuntitre} +\bigskip + +\begin{ex}\label{exfusee} + Une fusée d'une masse de \unit{60}{tonnes} décolle verticalement. Au bout d'une minute, elle atteint la vitesse de \unit{1000}{\kilo\metre\per\hour}. En supposant l'accélération constante, calculez la force totale des moteurs. Réponse : \unit{866'400}{\newton}. + \begin{sol} + Le schéma de la situation est donné dans la figure \ref{fusee}. + \begin{figure}[!t] + \centering + \caption{Une fusée.\label{fusee}} + \includegraphics{Fusee.eps} + \end{figure} + La seconde équation de Newton s'écrit : + \[\overrightarrow F+\overrightarrow P=m\cdot \overrightarrow a\] + En projection sur l'axe et en raison de la définition du poids, on a : + \[F-P=m\cdot a\;\Rightarrow\;F=m\cdot a+m\cdot g\] + Or, comme l'accélération vaut : + \[a=\frac{v_f-v_i}{t}=\frac{1000/3,6-0}{60}=\unit{4,63}{\metre\per\second\squared}\] + on a : + \[F=60\cdot 10^3\cdot 4,63+60\cdot 10^3\cdot 9,81=\unit{866'400}{\newton}\] + \end{sol} +\end{ex} + +\begin{ex} + Une voiture tire une remorque à vitesse constante. Si la force de frottement qui s'exerce sur la remorque vaut \unit{500}{\newton}, quelle est la force exercée par la voiture sur la remorque ? Quelle est la somme des forces qui s'exercent sur la remorque ? Répondez aussi à ces deux questions si la voiture a une accélération de \unit{5}{\metre\per\second\squared} et la remorque une masse de \unit{500}{\kilo\gram}. Réponses : \unit{500}{\newton}, nulle, \unit{3000}{\newton} et \unit{2500}{\newton}. + \begin{sol} + Le schéma de la situation est donné par la figure \ref{remorque}. + \begin{figure}[!t] + \centering + \caption{Une remorque\label{remorque}} + \includegraphics[width=6cm]{Remorque.eps} + \end{figure} + + La remorque avance à vitesse constante. La première loi de Newton nous indique alors que la somme des forces qui s'exercent sur elle est nulle. On peut considérer successivement le cas des forces verticales et celui des forces horizontales. + + Verticalement, la position de la voiture ne change pas. Elle est verticalement à l'arrêt. La réaction du sol \(\overrightarrow R\) est donc égale en grandeur, mais opposée, au poids \(\overrightarrow P\), comme présenté dans la figure \ref{remorque}. + + Horizontalement par contre, la remorque se déplace. Mais elle le fait à vitesse constante et donc, là encore, la somme des forces horizontales qui s'exercent sur elle est nulle. La force de frottement \(\overrightarrow F_{fr}\) est égale en grandeur et opposée à la force \(\overrightarrow F\) exercée par la voiture pour tirer la remorque. Ainsi : + \[F=F_{fr}=\unit{500}{\newton}\] + + Si la voiture a une accélération, la situation des forces verticales ne change pas. On a toujours : \(\overrightarrow R=-\overrightarrow P\). Par contre, la sommes des forces horizontales n'est plus nulle. On a, selon l'axe de la figure \ref{remorque} : + \begin{align*} + F-F_{fr}&=m\cdot a\;\Rightarrow\\ + F&=m\cdot a+F_{fr}=500\cdot 5+500=\unit{3000}{\newton} + \end{align*} + et la somme des forces qui s'exercent sur la remorque est : + \[F-F_{fr}=m\cdot a=500\cdot 5=\unit{2500}{\newton}\] + \end{sol} +\end{ex} + +\begin{ex} + Un ascenseur de masse \(m = \unit{200}{\kilo\gram}\), dans lequel une personne de \unit{60}{\kilo\gram} se trouve, monte avec une accélération de \unit{4}{\metre\per\second\squared}. Quelle est la force exercée par le câble sur l'ascenseur ? Réponse : \unit{3590,6}{\newton}. + \begin{sol} + Ce problème est identique au problème \ref{exfusee} de la fusée. Il suffit de remplacer la fusée par l'ascenseur et de considérer la force de propulsion de la fusée comme la force de traction du câble. Considérons donc la figure \ref{fusee}. Selon l'axe considéré, on a : + \begin{align*} + F-P&=m\cdot a\,\Rightarrow\\ + F&=m\cdot a+m\cdot g=260\cdot 4+260\cdot 9,81\\ + &=\unit{3590,6}{\newton} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Une voiture de deux tonnes roulant à \unit{50}{\kilo\metre\per\hour} freine brusquement pour s'arrêter sur une distance de \unit{40}{\metre}. Calculez la force de frottement des pneus et expliquez précisément d'où elle vient. Réponse : \unit{-4'822,5}{\newton}. + \begin{sol} + Pour que la voiture ralentisse, il faut que la force qui s'exerce sur elle soit vers l'arrière. C'est la force de frottement du sol sur les roues qui la freine. En effet, sur la glace elle ne s'exerce pas et la voiture ne peut freiner. + + Tant la force que l'accélération sont donc dirigées vers l'arrière. La décélération se calculant par : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot d\;\Rightarrow\\ + a&=\frac{0^2-(50/3,6)^2}{2\cdot 40}=\unit{-2,4}{\metre\per\second\squared} + \end{align*} + on trouve aisément la force de frottement de la route sur les pneus qui ralentit la voiture : + \[F_{fr}=m\cdot a=2000\cdot (-2,4)=\unit{-4'822,5}{\newton}\] + Elle est négative, donc bien dirigée vers l'arrière. + \end{sol} +\end{ex} + +\begin{ex} + Un train de \unit{300}{tonnes} (locomotive : \unit{50}{tonnes}, wagons : \unit{250}{tonnes} accélère de \unit{0}{} à \unit{10}{\kilo\metre\per\hour} en une minute. On néglige les frottements. + + \begin{enumerate} + \item Calculez la force \(F_{tot}\) nécessaire pour réaliser cette augmentation de vitesse. + \item Quelle force \(F\) la locomotive exerce-t-elle sur les wagons ? + \item Combien de temps durerait le démarrage si la locomotive n'avait pas de wagons (elle exerce la même force qu'au premier point) ? + \end{enumerate} + + Réponses : \unit{13'889}{\newton}, \unit{11'500}{\newton} et \unit{10}{\second}. + \begin{sol} + Ce problème illustre bien l'importance du choix du système. Comme tout se déroule horizontalement, on ne va considérer que les forces horizontales. Les verticales existent, mais n'interviennent pas. + \begin{enumerate} + \item Comme on cherche la force nécessaire à l'augmentation de vitesse du train dans son entier, considérons pour système le train entier. Sa masse totale est \(M=\unit{300\cdot 10^3}{\kilo\gram}\). Son accélération est : + \[a=\frac{v_f-v_i}{t}=\frac{10/3,6-0}{60}=\unit{0,046}{\metre\per\second\squared}\] + On peut donc écrire : + \[F_{tot}=M\cdot a=300\cdot 10^3\cdot 0,046=\unit{13'889}{\newton}\] + \item Cette fois-ci, on cherche la force exercée sur une partie du train : les wagons. On ne va donc considérer comme système que les wagons. Une seule force \(F\) les tire vers l'avant avec la même accélération que celle du train dans son ensemble. Leur masse est \(m=\unit{250\cdot 10^3}{\kilo\gram}\). On a donc : + \[F=m\cdot a=250\cdot 10^3\cdot 0,046=\unit{11'500}{\newton}\] + Valeur inférieure à celle du point précédent, car il ne faut pas tirer la locomotive. + \item Ici, le système est évidemment la locomotive seule de masse \(m'=\unit{50\cdot 10^3}{\kilo\gram}\). On peut donc calculer l'accélération : + \begin{align*} + F_{tot}&=m'\cdot a\;\Rightarrow\\ + a&=\frac{F_{tot}}{m'}=\frac{13'889}{50\cdot 10^3}=\unit{0,28}{\metre\per\second\squared} + \end{align*} + et finalement le temps : + \[a=\frac{v_f-v_i}{t}\;\Rightarrow\;t=\frac{10/3,6-0}{0,28}=\unit{10}{\second}\] + \end{enumerate} + \end{sol} +\end{ex} + +\begin{ex} + On lance une balle de masse \(m=\unit{100}{\gram}\) vers le haut avec une vitesse \(v_o=\unit{2}{\metre\per\second}\). Si elle est soumise à une force de frottement \(F_{fr}=\unit{0,05}{\newton}\) pendant tout son vol, jusqu'à quelle hauteur monte-t-elle ? Réponse : \unit{19,4}{\centi\metre}. + \begin{sol} + Pendant la montée, la balle subit deux forces extérieures vers le bas : son poids \(P=m\cdot g\) et la force de frottement \(F_{fr}\). En prenant un axe vertical dirigé vers le haut, on peut donc écrire : + \begin{align*} + \sum F^{ext}=-m\cdot g-F_{fr}&=m\cdot a\;\Rightarrow\\ + -0,1\cdot 9,81-0,05&=0,1\cdot a\;\Rightarrow\\ + a&=\unit{-10,31}{\metre\per\second\squared} + \end{align*} + La balle a donc une décélération constante. Ainsi il s'agit d'un MRUA et, connaissant les vitesses initiale et finale et l'accélération, on peut écrire : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot h\;\Rightarrow\\ + 0&=2^2-2\cdot 10,31\cdot h\;\Rightarrow\\ + h&=\unit{0,194}{\metre}=\unit{19,4}{\centi\metre} + \end{align*} + Évidemment, elle monte moins haut que s'il n'y avait pas de frottements. + \end{sol} +\end{ex} + +\begin{ex} + Un vaisseau spatial d'une masse \(m=\unit{9}{tonnes}\) se déplace à une vitesse \(v_o=\unit{10'000}{\kilo\metre\per\hour}\) pendant cinq minutes. Puis il enclenche ses moteurs. Ils lui fournissent une poussée \(P\) de \unit{1000}{\newton} pendant \unit{30}{\second}. Quelle est la distance parcourue jusqu'au moment où il coupe ses moteurs ? Réponse : \unit{916,7}{\kilo\metre}. + \begin{sol} + La première phase du mouvement se déroule à vitesse constante. La distance parcourue pendant cinq minutes, c'est-à-dire \(1/12\,h\), est donc donnée par : + \[d_{MRU}=v_o\cdot t=10'000\cdot \frac{1}{12}=\unit{833,3}{\kilo\metre}\] + La seconde phase du mouvement se déroule à accélération constante. En effet, la force de poussée et la masse étant constantes, on peut écrire : + \begin{align*} + \sum F^{ext}=P&=m\cdot a\;\Rightarrow\\ + 1000&=9'000\cdot a\;\Rightarrow\\ + a&=\unit{0,111}{\metre\per\second\squared} + \end{align*} + Connaissant l'accélération, on peut ensuite calculer la distance parcourue en MRUA grâce au temps de poussée : + \begin{align*} + d_{MRUA}&=\frac{1}{2}\cdot a\cdot t^2+v_o\cdot t\\ + &=\frac{1}{2}\cdot 0,111\cdot 30^2+\frac{10'000}{3,6}\cdot 30\\ + &=83'383,3\,m=\unit{83,4}{\kilo\metre} + \end{align*} + Au total, la distance parcourue est donc de : + \[d_{tot}=d_{MRU}+d_{MRUA}=833,3+83,4=\unit{916,7}{\kilo\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + Un électron initialement au repos est accéléré jusqu'à une vitesse de 1\% de la vitesse de la lumière sur une distance de \unit{30}{\metre}. Quelle est la force totale qui s'exerce sur lui ? Réponse : \unit{1,37\cdot 10^{-19}}{\newton}. + \begin{sol} + Son accélération, supposée constante, est donnée par : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot d\;\Rightarrow\\ + (0,01\cdot 3\cdot 10^8)^2&=0^2+2\cdot a\cdot 30\;\Rightarrow\\ + a&=\unit{1,5\cdot 10^{11}}{\metre\per\second\squared} + \end{align*} + La force totale qui s'exerce sur lui est alors donnée par : + \[F_{tot}=m\cdot a=9,1\cdot 10^{-31}\cdot 1,5\cdot 10^{11}=\unit{1,37\cdot 10^{-19}}{\newton}\] + \end{sol} +\end{ex} + +\begin{ex} + Une araignée de masse \(m=\unit{5}{\gram}\) est suspendue par son fil au plafond. Elle ne bouge pas. Quelle force le fil exerce-t-il sur elle ? Elle se met à descendre en freinant sa chute à l'aide de son fil avec une force \(F=\unit{0,01}{\newton}\). Si elle descend d'une hauteur \(d=\unit{2}{\metre}\), combien de temps met-elle pour le faire et à quelle vitesse arrive-t-elle en bas ? Réponses : \unit{0,716}{\second} et \unit{5,59}{\metre\per\second}. + \begin{sol} + Tant que l'araignée ne bouge pas, la force exercée par le fil est égale à son poids, c'est-à-dire : + \[F=m\cdot g=0,005\cdot 9,81=\unit{0,049}{\newton}\] + Pendant sa chute, la seconde loi de Newton permet d'écrire, en utilisant un axe dans le sens de la chute : + \begin{align*} + \sum F^{ext}=m\cdot g-F&=m\cdot a\;\Rightarrow\\ + 0,005\cdot 9,81-0,01&=0.005\cdot a\;\Rightarrow\\ + a&=\unit{7,81}{\metre\per\second\squared} + \end{align*} + Avec une vitesse initiale nulle, le temps de chute sur une hauteur de \unit{2}{\metre} est : + \begin{align*} + h&=\frac{1}{2}\cdot a\cdot t^2\;\Rightarrow\\ + t&=\sqrt{\frac{2\cdot h}{a}}=\sqrt{\frac{2\cdot 2}{7,81}}=\unit{0,716}{\second} + \end{align*} + Et la vitesse finale est alors : + \[v=a\cdot t=7,81\cdot 0,716=\unit{5,59}{\metre\per\second}\] + \end{sol} +\end{ex} + +\begin{ex} + Une corde aux extrémités de laquelle sont suspendues deux masses \(m_1=\unit{2}{\kilo\gram}\) et \(m_2=\unit{3}{\kilo\gram}\) est suspendue à une poulie (voir figure \ref{poulie}). On lâche les masses (il s'agit de ce qu'on appelle une machine de Atwood\index{Atwood}). Quelle est leur accélération ? Réponse : \unit{1,962}{\metre\per\second\squared}. + \begin{figure}[ht] + \caption[La poulie]{La poulie\label{poulie} \par \scriptsize{ou machine de Atwood\index{Atwood}}} + % ligne pour placement correct de l'input + \smallskip + \begin{center} + \input{Annexe-Exercices/Images/Poulie} + \end{center} + \medskip + \end{figure} + + \begin{sol} + L'ensemble se comporte comme un système en une dimension dont l'accélération se fait dans le sens de la masse la plus grande et qui est freiné par la masse la plus faible. Si on définit le sens positif de l'axe dans le sens du mouvement, on peut écrire : + \begin{align*} + \sum F^{ext}=m_2\cdot g-m_1\cdot g&=(m_1+m_2)\cdot a\;\Rightarrow\\ + 3\cdot 9,81-2\cdot 9,81&=(2+3)\cdot a\;\Rightarrow\\ + a&=\unit{1,962}{\metre\per\second\squared} + \end{align*} + \end{sol} +\end{ex} + +\Closesolutionfile{ans} + +\optv{OS}{ +\Opensolutionfile{ansos}[SolutionsOS] + +\begin{exos} + Soient deux masses, la première, de valeur M=\unit{5}{\kilo\gram}, posée sur un plan horizontal sans frottements et la seconde, m=\unit{2}{\kilo\gram}, reliée à la première par une corde sans masse et pendant dans le vide, comme présenté sur la figure \ref{pendante}. + \begin{figure}[h] + \caption[Masse pendante]{La masse pendante}\label{pendante} + \smallskip + \begin{center} + \def\svgwidth{6cm} + \input{Annexe-Exercices/Images/pendante.eps_tex} + \end{center} + \end{figure} + +On lâche la première à vitesse initiale nulle. Calculez la vitesse de la seconde alors que la première à parcouru une distance d=\unit{50}{\centi\metre}. Réponse~: \unit{2,8}{\metre\per\second}. + \begin{solos} + On commence par choisir le système. Pour éviter de devoir calculer la tension dans la corde, on le choisit comme constitué de la corde et des deux masses M et m. + + Les forces extérieures sont alors au nombre de trois. Le plan horizontal exerce sur la masse M une force de soutient verticale égale et opposée à son poids (mais qui ne sont pas l'action et la réaction l'une de l'autre), puisque la masse se déplace horizontalement. Ces deux forces s'annulent donc. Reste le poids de la masse m, seule force extérieure à agir pour accélérer le système. + + On peut donc écrire, selon la seconde loi de Newton~: + \[\sum F^{ext}=m\cdot g=(M+m)\cdot a\] + où M+m est la masse du système. Ainsi, finalement, on peut calculer la valeur de l'accélération~: + \begin{align*} + &m\cdot g=(M+m)\cdot a\;\Rightarrow\\ + &a=\frac{m}{M+m}\cdot g=\frac{2}{5}\cdot 9,81=\unit{3,924}{\metre\per\second\squared} + \end{align*} + \medskip + À partir de l'accélération, on peut ensuite calculer la vitesse au bout d'un mètre, grâce à l'équation du MRUA~: + \begin{align*} + v^2&=v_0^2+2\cdot a\cdot d\;\Rightarrow\\ + v^2&=0+2\cdot 3,924\cdot 1\;\Rightarrow\\ + v&=\sqrt{2\cdot 3,924}=\unit{2,8}{\metre\per\second} + \end{align*} + \end{solos} +\end{exos} + +\begin{exos} + On lâche, à vitesse initiale nulle, une masse m de \unit{3}{\kilo\gram}, sur un plan incliné d'un angle \(\alpha = \unit{20}{\degree}\) sans frottements. Calculez sa vitesse après un temps t de \unit{2}{\second}. Réponse~: \unit{9,72}{\metre\per\second}. + \begin{solos} + Deux forces agissent ici : la réaction du plan, qui lui est normale (c'est-à-dire perpendiculaire), et le poids de la masse. Comme la réaction du plan n'a aucune composante parallèlement au plan, elle ne peut être responsable de l'accélération de la masse le long de celui-ci. + + Il faut donc trouver la composante du poids qui est parallèle au plan incliné. L'angle entre le poids et un plan horizontal est de \unit{90}{\degree}. Quand le plan est incliné, cet angle diminue de la valeur de l'inclinaison. L'angle \(\beta\) entre le plan incliné et le poids et donc \(\beta=90-\alpha\). + + Comme la projection du poids selon l'angle \(\beta\) correspond à sa composante parallèle au plan, dans le triangle rectangle composé du poids comme hypoténuse et de ses composantes parallèle et perpendiculaire au plan, la composante parallèle au plan correspond au côté adjacent. Ainsi, on peut écrire~: + \begin{align*} + P_{//}&=P\cdot\cos(\beta)=P\cdot\cos(90-\alpha)\\ + &=P\cdot\sin(\alpha) + \end{align*} + La seconde loi de Newton s'écrit donc le long du plan incliné~: + \begin{align*} + \sum F^{ext}=P\cdot\sin(\alpha)&=m\cdot a\;\Rightarrow\\ + m\cdot g\cdot\sin(\alpha)&=m\cdot a\;\Rightarrow\\ + a&=g\cdot\sin(\alpha)\;\Rightarrow\\ + a&=9,81\cdot\sin(20)=\unit{3,36}{\metre\per\second\squared} + \end{align*} + Avec une vitesse initiale nulle, pour un MRUA d'accélération calculée ci-dessus, la vitesse au bout d'un temps t=\unit{2}{\second} s'obtient par~: + \[v=a\cdot t+v_0=3,36\cdot 2=\unit{9,72}{\metre\per\second}\] + + \medskip + Une autre manière de résoudre le problème est de procéder avec méthode. Le système qu'on doit choisir est bien évidemment la masse m, puisque c'est de celle-ci qu'on cherche l'accélération pour en trouver la vitesse au bout de \unit{2}{\second}. + \begin{figure}[h] + \centering + \caption[Plan incliné]{Le plan incliné}\label{incline} + \medskip + \def\svgwidth{6cm} + \input{Annexe-Exercices/Images/incline.eps_tex} + \end{figure} + + La figure \ref{incline} présente ensuite le dessin des forces extérieures et le choix du système d'axes. Remarquez que ce dernier l'a été selon l'inclinaison du plan. Il aurait pu ne pas en être ainsi, mais ce choix simplifie les calculs, car la masse étant contrainte à se déplacer le long du plan, son accélération perpendiculairement est nulle. Les équations de la seconde loi de Newton, obtenues par projection des forces extérieures et de l'accélération selon les axes, peuvent alors s'écrire : + \begin{align*} + \sum F^{ext}_x&=P_x=m\cdot a_x &\text{sur l'axe x}\\ + \sum F^{ext}_y&=R-P_y=m\cdot a_y=0 &\text{sur l'axe y} + \end{align*} + car l'accélération perpendiculairement au plan est nulle, comme déjà mentionné. + + \smallskip + Si on considére l'angle \(\alpha\), en s'imaginant le plan incliné horizontal, on comprends qu'il se reporte entre le vecteur poids \(\overleftarrow{P}\) et sa compostante selon y \(P_y\). + + Avec le triangle rectangle formé par le poids et ses composantes et un peu de trigonométrie, on peut en déduire : + \begin{align*} + P_x &= P\cdot \sin(\alpha)\\ + P_y &= P\cdot \cos(\alpha) + \end{align*} + Comme par ailleurs on sait que \(P=m\cdot g\), on peut réécrire les équations de Newton sur les axes comme : + \begin{align*} + \sum F^{ext}_x&=m\cdot g\cdot \sin(\alpha)=m\cdot a_x\\ + \sum F^{ext}_y&=R-m\cdot g\cdot \cos(\alpha)=m\cdot a_y=0} + \end{align*} + La première de ces équations permet de trouver l'accélération du bloc selon le plan incliné : + \begin{align*} + a=a_x&=g\cdot \sin(\alpha)\\ + &=9,81\cdot\sin(20)=\unit{3,36}{\metre\per\second\squared} + \end{align*} + + \smallskip + Mais une information supplémentaire nous est donnée par la seconde équation, c'est la valeur de la réaction R au plan : + \begin{align*} + R&=m\cdot g\cdot \cos(\alpha)\\ + &=3\cdot 9,81\cdot \cos(20)=\unit{27,67}{\newton} + \end{align*} + + \end{solos} +\end{exos} + +\begin{exos} + Un exercice de test. + \begin{solos} + Un corrigé de test. + \end{solos} +\end{exos} + +\Closesolutionfile{ansos} +} + +\Opensolutionfile{ans}[Solutions] + +\subsection{Relatifs aux forces} + +\begin{ex} + Une personne de masse \(m = \unit{70}{\kilo\gram}\) prend l'ascenseur. Elle se place sur une balance. Durant la première phase de montée, l'accélération vaut \unit{2}{\metre\per\second\squared}. Puis suit une phase à vitesse constante et enfin l'ascenseur décélère à \unit{3}{\metre\per\second\squared}. Trouvez dans chaque cas la force exercée par la balance sur la personne (en divisant par l'accélération terrestre, on obtient la valeur que marquerait une balance sous les pieds de la personne). Réponses : \unit{84,3}{\kilo\gram}, \unit{70}{\kilo\gram} et \unit{48,6}{\kilo\gram}. + \begin{sol} + La figure \ref{ascenseur} présente la situation. + \begin{figure}[!t] + \centering + \caption{Un ascenseur\label{ascenseur}} + \includegraphics{Ascenseur.eps} + \end{figure} + + On choisit comme système la personne. Les forces extérieures sont alors : + \begin{itemize} + \item son poids \(\overrightarrow P\) et + \item la force \(\overrightarrow R\) exercée par la balance sur la personne. Sa réaction, la force exercée par la personne sur la balance, permet à cette dernière de donner une indication du poids de la personne, indiqué malheureusement en kilogrammes (et non en newtons, comme cela devrait être le cas) par la balance. + \end{itemize} + + Selon l'axe donné, on peut écrire : + \[R-P=m\cdot a\;\Rightarrow\;R=m\cdot a+m\cdot g\] + A partir de là, on peut considérer les différents cas : + \begin{enumerate} + \item Quand l'ascenseur prend de la vitesse, l'accélération vaut \unit{2}{\metre\per\second\squared}. La réaction de la balance est alors : + \[R=70\cdot 2+70\cdot 9,81=\unit{826,7}{\newton}\] + et la balance indiquerait : + \[m=\frac{826,7}{9,81}=\unit{84,3}{\kilo\gram}\] + La personne a un poids plus important qu'à l'arrêt et donc l'indication donnée en terme de masse par la balance pourrait laisser croire à~\cdot de l'embonpoint. + \item Pendant la phase à vitesse constante, l'accélération vaut \unit{0}{\metre\per\second\squared}. La réaction de la balance est alors : + \[R=70\cdot 0+70\cdot 9,81=\unit{686,7}{\newton}\] + et la balance indiquerait : + \[m=\frac{686,7}{9,81}=\unit{70}{\kilo\gram}\] + La personne a le même poids qu'à l'arrêt et donc l'indication donnée en terme de masse par la balance est juste. + \item Quand l'ascenseur perd de la vitesse, l'accélération vaut \unit{-3}{\metre\per\second\squared}. La réaction de la balance est alors : + \[R=70\cdot (-3)+70\cdot 9,81=\unit{476,7}{\newton}\] + et la balance indiquerait : + \[m=\frac{476,7}{9,81}=\unit{48,6}{\kilo\gram}\] + La personne a un poids moins important qu'à l'arrêt et donc l'indication donnée en terme de masse par la balance pourrait laisser croire à~\cdot une maladie. + \end{enumerate} + On voit qu'une balance indique quelque chose de relatif à l'état de mouvement de la personne. Elle n'indique donc pas une quantité de matière, c'est-à-dire une masse. Une balance devrait donc être graduée en newtons. Elle l'est en \kilo\gram, car on a l'habitude de se peser à l'arrêt (bien que~\cdot la Terre tournant~\cdot) et à la surface de la Terre. Dans ce cas bien précis, il suffit en effet de diviser son indication (ou de reporter face à la graduation une indication de masse) par \gram pour obtenir la masse. + \end{sol} +\end{ex} + +\begin{ex} + Calculez la grandeur de la force exercée par deux boules de pétanque de masse \(m = \unit{1}{\kilo\gram}\) l'une sur l'autre. Elles sont distantes de \unit{0,5}{\metre}. Réponse : \unit{2,668\cdot 10^{-10}}{\newton}. + \begin{sol} + La loi de la gravitation universelle donne : + \begin{align*} + F&=G\cdot \frac{M\cdot m}{d^2}\\ + &=6,67\cdot 10^{-11}\cdot \frac{1\cdot 1}{0,5^2}=\unit{2,668\cdot 10^{-10}}{\newton} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex}\label{massedelaterre} + Calculez la masse de la Terre. On donne \(g=\unit{9,81}{\metre\per\second\squared}\) et le rayon terrestre moyen \(R_{T}=\unit{6'370}{\kilo\metre}\). Réponse : \unit{5,97\cdot 10^{24}}{\kilo\gram}. + \begin{sol} + Le poids à la surface de la Terre peut être calculé de deux manières. A l'aide de la loi de la gravitation universelle et à l'aide de sa définition \(P=m\cdot g\). Il s'agit de la même force. On peut donc écrire : + \begin{align*} + m\cdot g&=G\cdot \frac{M_{Terre}\cdot m}{R_{Terre}^2}\;\Rightarrow\\ + g&=G\cdot \frac{M_{Terre}}{R_{Terre}^2}\;\Rightarrow\\ + M_{Terre}&=\frac{g\cdot R_{Terre}^2}{G}\\ + &=\frac{9,81\cdot (6'372\cdot 10^3)^2}{6,67\cdot 10^{-11}}=\unit{5,97\cdot 10^{24}}{\kilo\gram} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Calculez l'accélération de la pesanteur de la Lune avec les données des tables. Réponse : \unit{1,62}{\metre\per\second\squared}. + \begin{sol} + Comme au problème \ref{massedelaterre}, on a : + \[g=G\cdot \frac{M}{R^2}\] + Mais ici il s'agit de la Lune. Ainsi : + \begin{align*} + g_{Lune}&=G\cdot \frac{M_{Lune}}{R_{Lune}^2}\\ + &=6,67\cdot 10^{-11}\cdot \frac{7,35\cdot 10^{22}}{(1,738\cdot 10^6)^2}\\ + &=\unit{1,62}{\metre\per\second\squared} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Quelle force faut-il exercer sur un ressort de constante \(k = \unit{800}{\newton\per\metre}\) pour le déformer de \unit{10}{\centi\metre} ? Réponse : \unit{80}{\newton}. + \begin{sol} + On a simplement : + \[F=k\cdot x=800\cdot 0,1=\unit{80}{\newton}\] + \end{sol} +\end{ex} + +\begin{ex} + Quelle est la constante d'un ressort qui s'allonge de \unit{12}{\centi\metre} lorsqu'on lui suspend une masse de \unit{100}{\gram} ? Réponse : \unit{8,175}{\newton\per\metre}. + \begin{sol} + Une masse de \unit{100}{\gram} exerce une force + \[F=m\cdot g=0,1\cdot 9,81=\unit{0,981}{\newton}\] + sur le ressort. On a donc : + \[F=k\cdot x\;\Rightarrow\;k=\frac{F}{x}=\frac{0,981}{0,12}=\unit{8,175}{\newton\per\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + Une voiture de deux tonnes roulant à \unit{50}{\kilo\metre\per\hour} freine brusquement pour s'arrêter sur une distance horizontale de \unit{40}{\metre}. Calculez le c\oe fficient de frottement des pneus sur la route. Réponse : \unit{0,25}{}. + \begin{sol} + L'accélération de la voiture est (hypothèse MRUA) : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot d\;\Rightarrow\\ + a&=\frac{v^2-v_o^2}{2\cdot d}=\frac{13,8^2}{2\cdot 40}\\ + &=\unit{2,41}{\metre\per\second\squared} + \end{align*} + La force exercée par la route sur les pneus est donc de : + \[F_{fr}=m\cdot a=2000\cdot 2,41=\unit{4822,5}{\newton}\] + Or, par définition de la force de frottement sec, on a : + \begin{align*} + F_{fr}&=\mu\cdot R=\mu\cdot m\cdot g\;\Rightarrow\\ + \mu&=\frac{F_{fr}}{m\cdot g}=\frac{4822,5}{2000\cdot 9,81}=\unit{0,25}{} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Une voiture a une masse \(m=\unit{1000}{\kilo\gram}\) dont \unit{600}{} s'exercent sur les roues avant et \unit{400}{} sur les roues arrière. Calculez l'accélération maximale de cette automobile si le c\oe fficient de frottement pneu-route vaut \(\mu_{o}=\unit{0,5}{}\). La route est horizontale. + + \begin{enumerate} + \item La traction se fait par les roues avant uniquement, + \item La traction est arrière, + \item C'est une traction quatre roues. + \end{enumerate} + + Réponses : \unit{2,943}{\metre\per\second\squared}, \unit{1,962}{\metre\per\second\squared} et \unit{4,905}{\metre\per\second\squared}. + \begin{sol} + La force maximale entre les pneus et la route est donnée par : + \[F_{fr}=\mu_{o}\cdot R=\mu_{o}\cdot m\cdot g\] + On a alors : + \begin{enumerate} + \item Avec les roues avant uniquement, la force maximale est : + \[F_{fr}=0,5\cdot 600\cdot 9,81=\unit{2943}{\newton}\] + Et l'accélération : + \[a=\frac{F_{fr}}{m}=\frac{2943}{1000}=\unit{2,943}{\metre\per\second\squared}\] + \item Avec les roues arrière uniquement, la force maximale est : + \[F_{fr}=0,5\cdot 400\cdot 9,81=\unit{1962}{\newton}\] + Et l'accélération : + \[a=\frac{F_{fr}}{m}=\frac{1962}{1000}=\unit{1,962}{\metre\per\second\squared}\] + \item Avec les roues avant et arrière ensemble, la force maximale est : + \[F_{fr}=0,5\cdot 1000\cdot 9,81=\unit{4905}{\newton}\] + Et l'accélération : + \[a=\frac{F_{fr}}{m}=\frac{4905}{1000}=\unit{4,905}{\metre\per\second\squared}\] + \end{enumerate} + \end{sol} +\end{ex} + +\subsection{Relatifs à l'énergie} + +\begin{ex}\label{helico} + Un hélicoptère monte une masse de \unit{50}{\kilo\gram} avec une accélération de \unit{4}{\metre\per\second\squared} sur une distance de \unit{100}{\metre}. Calculez le travail de la force de traction qui s'exerce sur la masse. Réponse : \unit{69'050}{\joule}. + \begin{sol} + La seconde loi de Newton permet de calculer la valeur de la force de traction : + \begin{align*} + F-m\cdot g&=m\cdot a\;\Rightarrow\\ + F&=m\cdot a+m\cdot g\\ + &=50\cdot 4+50\cdot 9,81=\unit{690,5}{\newton} + \end{align*} + La force de traction et le déplacement étant parallèles et de même sens, on a : + \[A=\overrightarrow F\cdot \overrightarrow d=F\cdot d=690,5\cdot 100=\unit{69'050}{\joule}\] + \end{sol} +\end{ex} + +\begin{ex} + Quelle est l'énergie potentielle de la masse du problème \ref{helico} à une hauteur de \unit{100}{\metre} ? Quelle est son énergie cinétique à la même place si la vitesse initiale est nulle ? Réponses : \unit{49'050}{\joule} et \unit{20'022,25}{\joule}. + \begin{sol} + Par définition de l'énergie potentielle, on a : + \[E_{pot}=m\cdot g\cdot h=50\cdot 9,81\cdot 100=\unit{49'050}{\joule}\] + Pour le calcul de l'énergie cinétique, si on fait l'hypothèse d'un MRUA, la vitesse de la masse au bout de \unit{100}{\metre} est : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot d\;\Rightarrow\\ + v&=\sqrt{2\cdot a\cdot d}=\sqrt{2\cdot 4\cdot 100}=\unit{28,3}{\metre\per\second} + \end{align*} + et l'énergie cinétique est alors : + \[E_{cin}=\frac{1}{2}\cdot m\cdot v^2=\frac{1}{2}\cdot 50\cdot 28,3^2=\unit{20'022,25}{\joule}\] + \end{sol} +\end{ex} + +\begin{ex} + Estimez l'énergie cinétique + + \begin{itemize} + \item d'un coureur de \unit{100}{\metre} de masse \unit{80}{\kilo\gram}, + \item d'une voiture de masse \unit{800}{\kilo\gram} roulant à \unit{10}{\metre\per\second} et + \item d'une balle de fusil de \unit{10}{\gram} se déplaçant à \unit{800}{\metre\per\second}. + \end{itemize} + + Réponses : \unit{4000}{\joule}, \unit{40'000}{\joule} et \unit{3200}{\joule}. + \begin{sol} + Dans chacun des cas, l'énergie cinétique se calcule par : + \[E_{cin}=\frac{1}{2}\cdot m\cdot v^2\] + \begin{itemize} + \item Un bon coureur met environ \unit{10}{\second} pour parcourir \unit{100}{\metre}. Sa vitesse moyenne est donc de : + \[v=\frac{d}{t}=\frac{100}{10}=\unit{10}{\metre\per\second}\] + Son énergie cinétique est donc de : + \[E_{cin}=\frac{1}{2}\cdot 80\cdot 10^2=\unit{4000}{\joule}\] + \item Simplement, on calcule l'énergie cinétique : + \[E_{cin}=\frac{1}{2}\cdot 800\cdot 10^2=\unit{40'000}{\joule}\] + \item Il faut mettre la masse en \kilo\gram pour calculer l'énergie cinétique : + \[E_{cin}=\frac{1}{2}\cdot 0,01\cdot 800^2=\unit{3200}{\joule}\] + \end{itemize} + \end{sol} +\end{ex} + +\begin{ex} + Calculez le travail du poids d'un objet de masse \(m = \unit{3}{\kilo\gram}\) quand on le monte à vitesse constante de \unit{4}{\metre}, le déplace de \unit{5}{\metre} horizontalement, le redescend de \unit{4}{\metre} et le ramène à sa position initiale. Calculez le travail pour chaque étapes et le travail total. Réponses : \unit{-117,72}{\joule}, 0, \unit{117,72}{\joule}, 0 et 0. + \begin{sol} + Procédons par étapes : + \begin{itemize} + \item si on monte de \unit{4}{\metre}, le poids et le déplacement sont opposés et le travail du poids est : + \begin{align*} + A&=m\cdot g\cdot d\cdot \cos(\alpha)\\ + &=3\cdot 9,81\cdot 4\cdot \cos(180)=\unit{-117,72}{\joule} + \end{align*} + \item lors d'un déplacement horizontal, le poids et le déplacement sont perpendiculaires et le travail est nul (car \(\cos(90^{\circ})=0\)), + \item si on descend de \unit{4}{\metre}, le poids et le déplacement sont dans le même sens et le travail du poids est : + \begin{align*} + A&=m\cdot g\cdot d\cdot \cos(\alpha)\\ + &=3\cdot 9,81\cdot 4\cdot \cos(0)=\unit{117,72}{\joule} + \end{align*} + \item lors d'un déplacement horizontal, le poids et le déplacement sont perpendiculaires et le travail est nul (car \(\cos(90^{\circ})=0\)). + \end{itemize} + + Ainsi, le travail total est la somme des travaux pour chaque déplacement : + \[A_{tot}=-117,72+0+117,72+0=\unit{0}{\joule}\] + Le travail du poids pour un parcours fermé est donc nul. On dit d'une telle force qu'elle est ``conservative''. Ce n'est que pour de telles forces qu'il existe une énergie potentielle. + \end{sol} +\end{ex} + +\begin{ex} + Calculez le travail produit par une force de \unit{15}{\newton} s'exerçant constamment à \unit{10}{\degree} par rapport au déplacement. Celui-ci est rectiligne et d'une distance de \unit{20}{\metre}. Réponse : \unit{295,4}{\joule}. + \begin{sol} + Par définition du travail, on a : + \begin{align*} + A&=\overrightarrow F\cdot \overrightarrow d=F\cdot d\cdot \cos(\alpha)\\ + &=15\cdot 20\cdot \cos(10^{\circ})=\unit{295,4}{\joule} + \end{align*} + \end{sol} +\end{ex} + +\subsection{Relatifs à la conservation de l'énergie} + +\begin{ex} + Sans l'aide de l'accélération, calculez la vitesse d'une personne qui a sauté (sans vitesse initiale) d'un plongeoir de \unit{10}{\metre} et arrive dans l'eau. Réponse : \unit{50,4}{\kilo\metre\per\hour}. + \begin{sol} + Pour rappel, résolvons tout d'abord le problème à l'aide de l'accélération. Nous sommes dans le cas d'un MRUA. Donc, on peut écrire : + \begin{align*} + v^2&=v_o^2+2\cdot a\cdot h\;\Rightarrow\\ + v&=\sqrt{2\cdot 9,81\cdot 10}=14\,m/s=\unit{50,4}{\kilo\metre\per\hour} + \end{align*} + \medskip + + Pour résoudre le problème à l'aide de l'énergie, il faut considérer que toute l'énergie potentielle de la personne en haut du plongeoir se transforme graduellement en énergie cinétique pendant sa chute. Si le zéro de l'énergie potentielle est choisi au niveau de l'eau, le plongeur n'a plus d'énergie potentielle lorsqu'il l'atteint. Elle s'est entièrement transformée en énergie cinétique. On peut donc écrire : + \begin{align*} + m\cdot g\cdot h&=\frac{1}{2}\cdot m\cdot v^2\;\Rightarrow\\ + v&=\sqrt{2\cdot 9,81\cdot 10}=14\,m/s=\unit{50,4}{\kilo\metre\per\hour} + \end{align*} + et le problème est résolu. + \medskip + + Cependant, il faut remarquer que l'égalité de l'énergie potentielle et de l'énergie cinétique dérive du théorème de conservation de l'énergie mécanique. En effet, comme c'est le cas ici, en l'absence de forces non conservatives, on peut écrire : + \begin{align*} + \Delta E_{mec}&=0\\ + E_{mec f}-E_{mec i}&=0\\ + E_{pot f}+E_{cin f}-E_{pot i}-E_{cin i}&=0\\ + 0+\frac{1}{2}\cdot m\cdot v^2-m\cdot g\cdot h-0&=0\\ + \Rightarrow\;\frac{1}{2}\cdot m\cdot v^2=m\cdot g\cdot h& + \end{align*} + + Remarquons enfin, que le problème est ici tout aussi facile à résoudre des deux manières. Cependant, dans le premier cas, on a pu le faire car on sait qu'une chute libre est un MRUA. Pour d'autres types de mouvements, qui ne sont pas des MRUA, pour trouver la vitesse à partir de l'accélération, il faut résoudre une intégrale. Et là, cela peut être beaucoup plus difficile qu'en utilisant l'énergie. + \end{sol} +\end{ex} + +\begin{ex} + On lance verticalement vers le haut un objet de poids \unit{100}{\newton} avec une vitesse initiale \(v_{o}=\unit{10}{\metre\per\second}\). A quelle hauteur \(h\) est-il momentanément arrêté ? Réponse : \unit{5,1}{\metre}. + \begin{sol} + On n'utilisera pas ici la méthode newtonienne, même si le problème peut être résolu de cette manière. + + Fixons le zéro de l'énergie potentielle là où l'objet décolle. Alors, son énergie potentielle est nulle et son énergie cinétique maximale. En montant, il perd graduellement de l'énergie cinétique au profit de l'énergie potentielle. Arrivé au sommet de sa trajectoire, il s'arrête brièvement et son énergie potentielle est maximale alors que son énergie cinétique est nulle. On peut donc dire que toute son énergie cinétique du départ s'est transformée en énergie potentielle au sommet et écrire : + \begin{align*} + \frac{1}{2}\cdot m\cdot v^2&=m\cdot g\cdot h\;\Rightarrow\\ + h&=\frac{v^2}{2\cdot g}=\frac{10^2}{2\cdot 9,81}=\unit{5,1}{\metre} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Une tuile d'une masse de \unit{2}{\kilo\gram} se met à glisser du haut d'un toit dont le sommet se trouve à \unit{30}{\metre}. Au moment où elle quitte le toit, sa hauteur vaut \unit{25}{\metre} (on ne tient pas compte du frottement de la tuile sur le toit). Calculez la distance au pied du toit à laquelle la tuile arrivera au sol. L'angle du toit par rapport à l'horizontale vaut \unit{15}{\degree}. Réponse : \unit{19,12}{\metre}. + \begin{sol} + Commençons par calculer la vitesse à laquelle la tuile quitte le toit. La conservation de l'énergie mécanique implique que : + \begin{align*} + m\cdot g\cdot \Delta h&=\frac{1}{2}\cdot m\cdot v^2\;\Rightarrow\\ + v&=\sqrt{2\cdot g\cdot \Delta h}\\ + &=\sqrt{2\cdot 9,81\cdot (30-25)}=\unit{9.9}{\metre\per\second} + \end{align*} + Cette vitesse est un vecteur qui fait avec l'horizontale un angle de \unit{15}{\degree}. La balistique (voir annexe \ref{balistique}) nous apprend qu'il faut décomposer le mouvement sur chaque axe. En particulier la vitesse initiale a les composantes suivantes : + \begin{align*} + v_x&=v\cdot \cos(\alpha)=9,9\cdot \cos(-15^{\circ})=\unit{9,56}{\metre\per\second}\\ + v_y&=v\cdot \sin(\alpha)=9,9\cdot \sin(-15^{\circ})=\unit{-2,56}{\metre\per\second} + \end{align*} + puisque le vecteur vitesse pointe sous l'horizontale (\(\alpha<0\)). + + En choisissant un système d'axes dont l'origine est au sol et au pied du bord du toit, on peut alors écrire les équations de la position de la tuile au cours du temps : + \begin{align*} + x(t)&=v_x\cdot t=9,56\cdot t\\ + y(t)&=-\frac{1}{2}\cdot 9,81\cdot t^2+v_y\cdot t+y_o\\ + &=-\frac{1}{2}\cdot 9,81\cdot t^2-2,56\cdot t+25 + \end{align*} + On cherche alors la position horizontale de la tuile au moment où elle arrive au sol, soit \(x(t_{sol})\). Il faut donc calculer \(t_{sol}\). Or, à ce moment là, \(y(t_{sol})=0\). La seconde équation nous donne donc : + \begin{align*} + 0&=-\frac{1}{2}\cdot 9,81\cdot t_{sol}^2-2,56\cdot t_{sol}+25\\ + &\Rightarrow\;-4,905\cdot t_{sol}^2-2,56\cdot t_{sol}+25=0 + \end{align*} + C'est une équation du second degré à une inconnue \(t_{sol}\). Sa solution est : + \begin{align*} + t_{sol}&=\frac{2,56\pm \sqrt{2,56^2+4\cdot 4,905\cdot 25}}{2\cdot (-4,905)}\\ + &=\frac{2,56\pm 22,3}{-9,81}=\begin{cases}\unit{-2,53}{\second}\\\unit{2}{\second}\end{cases} + \end{align*} + Évidemment, le temps correct est celui qui est positif. + + Ainsi, \(t_{sol}=\unit{2}{\second}\). La distance au pied du toit à laquelle la tuile arrivera est donc : + \[x(t_{sol})=v_x\cdot t_{sol}=9,56\cdot 2=\unit{19,12}{\metre}\] + \end{sol} +\end{ex} + +\begin{ex} + Une voiture de deux tonnes roulant horizontalement augmente sa vitesse de \unit{5}{} à \unit{10}{\kilo\metre\per\hour}. Quelle est l'augmentation de son énergie cinétique ? Réponse : \unit{5787}{\joule}. + \begin{sol} + Simplement, on a : + \begin{align*} + \Delta E_{cin}&=\frac{1}{2}\cdot m\cdot v_f^2-\frac{1}{2}\cdot m\cdot v_i^2\\ + &=\frac{1}{2}\cdot 2000\cdot ((\frac{10}{3,6})^2-(\frac{5}{3,6}^2))=\unit{5787}{\joule} + \end{align*} + \end{sol} +\end{ex} + +\subsection{Relatifs à l'énergie hydraulique} + +\begin{ex} + Robinson Crusoé décide de s'éclairer en construisant une petite centrale électrique qui amène l'eau à travers une conduite forcée en bambou d'une hauteur de \unit{22}{\metre}. Le débit étant de \unit{30}{\litre\per\second} et le rendement faible de l'ordre de 40\%, calculez combien d'ampoules de \unit{40}{\watt} il pourra utiliser s'il branche simultanément un frigo de \unit{1}{\kilo\watt} et sa machine à laver de \unit{1,2}{\kilo\watt}. Réponse : 9. + \begin{sol} + La puissance de chute est : + \begin{align*} + P&=\eta\cdot Q\cdot g\cdot h=\\ + &=0,4\cdot 30\cdot 10^{-3}\cdot 9,81\cdot 22=\unit{2,59}{\kilo\watt} + \end{align*} + La chute à donc une puissance de \unit{2590}{\watt}. Son frigo et sa machine à laver ont une puissance totale de \unit{2,2}{\kilo\watt}. Reste donc \(2590-2200=\unit{390}{\watt}\) pour les ampoules. Or, \(390/40=9,75\). Robinson pourra donc utiliser \(9\) ampoules. + \end{sol} +\end{ex} + +\begin{ex} + Le même Robinson Crusoé veut prélever 5\% de l'eau de sa rivière (hauteur de chute \unit{22}{\metre}, débit \unit{30}{\litre\per\second} et \(\eta=40\%\)) pour faire monter mécaniquement un petit ascenseur lui permettant de s'élever en \unit{2}{\minute} de deux étages dans sa grotte. Si Robinson a une masse de \unit{60}{\kilo\gram}, que la hauteur d'un étage est de \unit{2}{\metre} et que les frottements représentent 50\% de l'énergie nécessaire pour monter, y parviendra-t-il ? Réponse : oui. + \begin{sol} + Le débit de la prise d'eau est de : + \[Q=0,05\cdot 30=1,5\,l/s=\unit{1,5\cdot 10^{-3}}{\metre\cubed\per\second}\] + La puissance de chute est donc de : + \begin{align*} + P&=\eta\cdot Q\cdot g\cdot h=\\ + &=0,4\cdot 1,5\cdot 9,81\cdot 22=\unit{129,5}{\watt} + \end{align*} + Or, faire monter une masse de \unit{60}{\kilo\gram} d'une hauteur de deux étages, soit \unit{4}{\metre}, demande une énergie de : + \[E=m\cdot g\cdot h=60\cdot 9,81\cdot 4=\unit{2'354,4}{\joule}\] + Compte tenu des frottements, l'énergie totale nécessaire est de \(2\cdot 2'354,4=\unit{4'708,8}{\joule}\). Soit, pour une montée en \unit{2}{\minute}, une puissance de : + \[P=\frac{E}{t}=\frac{4'708,8}{2\cdot 60}\simeq\unit{40}{\watt}\] + Ainsi, si \unit{40}{\watt} sont nécessaires, Robinson n'utilise certainement pas les quelques \unit{130}{\watt} fournis par la chute d'eau. L'ascenseur est donc réalisable. + \end{sol} +\end{ex} + +\begin{ex} + On désire fournir \unit{3'000}{\kilo\watt\hour} d'énergie pendant une année à chacun des ménages d'une ville de 30'000 habitants. Si le nombre moyen de personnes par ménage est de deux et demi, quel doit être le débit de la conduite d'eau d'une hauteur de \unit{74}{\metre} nécessaire pour cela. Le rendement vaut \(\eta=90\%\). Quel type de turbine faut-il choisir ? Réponse : Francis. + \begin{sol} + Le nombre de ménages est de \(30'000/2,5=12'000\). Il faut donc fournir une énergie de : + \[E=12'000\cdot 3'000=\unit{36'000'000}{\kilowatthour}=\unit{36}{\giga\watt\hour}\] + Sur une année, cette énergie représente une puissance moyenne de : + \[P=\frac{E}{t}=\frac{36\cdot 10^9}{365\cdot 24}=\unit{4'110}{\watt}\] + On a alors : + \begin{align*} + P&=\eta\cdot Q\cdot g\cdot h\;\Rightarrow\\ + Q&=\frac{P}{\eta\cdot g\cdot h}\\ + &=\frac{4'110}{0,9\cdot 9,81\cdot 74}=\unit{6,3}{\metre\cubed\per\second} + \end{align*} + Il faut donc choisir une turbine Francis. + \end{sol} +\end{ex} + +\subsection{Relatifs à l'énergie éolienne} + +\begin{ex} + Un vent de force 4 sur l'échelle Beaufort, soufflant à \unit{21,6}{\kilo\metre\per\hour}, fait tourner une éolienne dont les pales ont une longueur de \unit{22}{\metre}. Quelle est la puissance du vent qui traverse cette éolienne ? La masse volumique de l'air est : \(\rho=\unit{1,293}{\kilo\gram\per\metre\cubed}\). Réponse : \unit{212,333}{\kilo\watt}. + \begin{sol} + La puissance du vent est donnée par la relation \ref{puissvent} : + \[P_{vent}=\frac{1}{2}\cdot \rho\cdot S\cdot v^3\] + Soit avec une surface $S=\pi\cdot R^2$, une vitesse en unités SI \(v=\unit{6}{\metre\per\second}\) et les valeurs données : + \begin{align*} + P_{vent}&=\frac{1}{2}\cdot 1,293\cdot \pi\cdot 22^2\cdot 6^3\\ + &=\unit{212'333}{\watt}=\unit{212,333}{\kilo\watt} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Une éolienne a des pales dont l'extrémité tourne à une vitesse de \unit{277,2}{\kilo\metre\per\hour} et dont la longueur vaut \unit{30}{\metre}. Sachant qu'elle tourne à 90\% de la limite de Betz et que la vitesse du vent est de \unit{32,4}{\kilo\metre\per\hour}, calculez sa vitesse de rotation en tr\per\second et la puissance qu'elle fournit. La masse volumique de l'air est : \(\rho=\unit{1,293}{\kilo\gram\per\metre\cubed}\). Réponses : \unit{0,41}{tr\per\second} et \unit{710}{\kilo\watt}. + \begin{sol} + La vitesse linéaire en bout de pale vaut donc : + \[v=\frac{277,2}{3,6}=\unit{77}{\metre\per\second}\] + Or, un tour représente une distance en bout de pale : \(d=2\cdot \pi\cdot 30=\unit{188,5}{\metre}\). La vitesse de rotation est donc de : + \[v=\frac{77}{188,5}=\unit{0,41}{tr\per\second}\] + + \medskip + + La puissance du vent vaut, selon l'équation \ref{puissvent} : + \begin{align*} + P_{vent}&=\frac{1}{2}\cdot 1,293\cdot \pi\cdot 30^2\cdot (\frac{32,4}{3,6})^3 \\ + &=\unit{1'332'565}{\watt}=\unit{1'332,6}{\kilo\watt} + \end{align*} + Comme l'éolienne tourne à \(\eta=90\%\) de la limite de Betz, on a, selon l'équation \ref{puissdebetz} : + \begin{align*} + \Delta P&=\eta\cdot \frac{16}{27}\cdot P_{vent} \\ + &=0,9\cdot 0,593\cdot 1'332'565= \\ + &=\unit{710'701}{\watt}=\unit{710}{\kilo\watt} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Un particulier désire alimenter la lampe de son petit cabanon de jardin avec une éolienne. En admettant que le vent moyen qui va la traverser souffle à \unit{14,4}{\kilo\metre\per\hour} (3 Beaufort), que la lampe nécessite une puissance de \unit{40}{\watt} et que l'éolienne tourne à 40\% de la limite de Betz, calculez la longueur des pales nécessaires. La masse volumique de l'air est : \(\rho=\unit{1,293}{\kilo\gram\per\metre\cubed}\). Réponse : \unit{1,14}{\metre}. + \begin{sol} + On a, selon l'équation \ref{puissdebetz}, que : + \begin{align*} + \Delta P&=\eta\cdot \frac{16}{27}\cdot P_{vent}\;\Rightarrow \\ + P_{vent}&=\frac{27\cdot \Delta P}{16\cdot \eta} \\ + &=\frac{27\cdot 40}{16\cdot 0,4}=\unit{168,75}{\watt} + \end{align*} + Il s'agit de la puissance du vent nécessaire pour alimenter l'ampoule. On a donc, selon l'équation \ref{puissvent} : + \begin{align*} + P_{vent}&=\frac{1}{2}\cdot \rho\cdot S\cdot v^3 \\ + &=\frac{1}{2}\cdot \rho\cdot \pi\cdot R^2\cdot v^3\;\Rightarrow \\ + R&=\sqrt{\frac{2\cdot P_{vent}}{\rho\cdot \pi\cdot v^3}} \\ + &=\sqrt{\frac{2\cdot 168,75}{1,293\cdot \pi\cdot 4^3}} \\ + &=\unit{1,14}{\metre} + \end{align*} + Il faut donc utiliser une petite éolienne de \unit{1,14}{\metre} de rayon. + \end{sol} +\end{ex} + +\subsection{Relatifs à l'énergie solaire} + +\begin{ex} + On désire chauffer une masse de \unit{100}{\litre} d'eau (\(c_{eau}=\unit{4'180}\joule\per\kilo\gram\celsius)\)) en quatre heures. Quelle est la surface de panneaux solaire nécessaire. On donne \(B=0,8\), \(K=\unit{3}{\watt\per\metre\squared\celsius}\) et la différence de température entre l'intérieur et l'extérieur des capteurs \(\Delta \theta=\unit{20}{\celsius}\). La puissance du rayonnement incident est de \unit{120}{\watt\per\metre\squared}. Réponse : \unit{16,2}{\metre\squared}. + \begin{sol} + L'équation \ref{thermique}, page \pageref{thermique}, permet de calculer la puissance \(\Delta P\) nécessaire pour chauffer une masse de \unit{100}{\kilo\gram} d'eau : + \begin{align*} + \Delta P&=\frac{E}{t}=\frac{m\cdot c_{eau}\cdot \Delta \theta}{t} \\ + &=\frac{100\cdot 4'180\cdot 20}{4\cdot 3600}=\unit{581}{\watt} + \end{align*} + L'équation \ref{puisssolthermique} nous donne alors la surface cherchée \(S\) : + \begin{align*} + \Delta P&=S\cdot (B\cdot P-K\cdot \Delta \theta)\;\Rightarrow \\ + S&=\frac{\Delta P}{B\cdot P-K\cdot \Delta \theta} \\ + &=\frac{581}{0,8\cdot 120-3\cdot 20}=\unit{16,2}{\metre\squared} + \end{align*} + \end{sol} +\end{ex} + +\begin{ex} + Quel est le débit d'eau (\(c_{eau}=\unit{4'180}{\joule\per\kilo\gram\celsius}\)) nécessaire pour évacuer l'énergie produite par \unit{5}{\metre\squared} de capteurs solaires thermiques caractérisés par \(B=0,7\), \(K=\unit{2}{\watt\per\metre\squared\celsius}\) et la différence de température entre l'intérieur et l'extérieur des capteurs \(\Delta \theta=\unit{15}{\celsius}\). La puissance du rayonnement incident est de \unit{150}{\watt\per\metre\squared}. Réponse : \unit{21,6}{\litre\per\hour}. + \begin{sol} + On calcule tout d'abord la puissance fournie par les capteurs : + \begin{align*} + \Delta P&=S\cdot (B\cdot P-K\cdot \Delta \theta)\;\Rightarrow \\ + &=5\cdot (0,7\cdot 150-2\cdot 15)=\unit{375}{\watt} + \end{align*} + Cela signifie que les capteurs produisent une énergie \(E=\unit{375}{\joule\per\second}\). Chaque seconde, il faut donc évacuer cette énergie. La masse d'eau nécessaire pour cela est donnée par l'équation \ref{thermique}, page \pageref{thermique} : + \begin{align*} + E&=m\cdot c_{eau}\cdot \Delta \theta\;\Rightarrow \\ + m&=\frac{E}{c_{eau}\cdot \Delta \theta} \\ + &=\frac{375}{4'180\cdot 15}=\unit{6}{\gram} + \end{align*} + C'est une masse qui correspond à un volume de : + \begin{align*} + \rho&=\frac{m}{V}\;\Rightarrow \\ + V&=\frac{m}{\rho}=\frac{6\cdot 10^{-3}}{1}=\unit{6}{\milli\litre} + \end{align*} + En effet, la masse volumique de l'eau est de \unit{1}{\kilo\gram\per\litre}. Le débit est donc très faible et vaut \(D=\unit{6}{\milli\litre\per\second}=\unit{21,6}{\litre\per\hour}\). + \end{sol} +\end{ex} + +\begin{ex} + Un particulier consomme \unit{2500}{\kilo\watt\hour\per an} d'énergie électrique. Il désire produire lui-même une partie de cette énergie à l'aide de cellules solaires électriques. Il installe donc \unit{8}{\metre\squared} de panneaux photoélectriques. Cela lui permet-il de subvenir à ses besoins ? Sinon, quelle est la puissance d'appoint dont il a besoin ? Il se trouve dans une région où la puissance solaire incidente est de \unit{160}{\watt\per\metre\squared}. Réponses : non et \unit{125,4}{\kilo\watt}. + \begin{sol} + L'énergie annuelle produite par les \unit{8}{\metre\squared} de cellules est, compte tenu d'un rendement d'environ 15\%, c'est-à-dire \unit{20}{\watt\per\metre\squared} : + \[E=8\cdot 20\cdot 24\cdot 365=\unit{1'401,6}{\kilowatthour}\] + La couverture solaire ne suffit donc pas. Il manque \(2'500-1'401,6=\unit{1'098,4}{\kilowatthour}\). Cela représente une puissance de : + \[P=\frac{E}{t}=\frac{1'098,4}{24\cdot 365}=\unit{125,4}{\kilo\watt}\] + \end{sol} +\end{ex} + +\Closesolutionfile{ans} % ferme le fichier Solutions.tex qui contient les solutions +\vfill +\pagebreak % met les solutions sur une autre page +\section{Solutions} +\Readsolutionfile{ans} % importe les données du fichier Solutions.tex + +\opt{OS}{ +\section{Solutions OS} +\Readsolutionfile{ansos} +} + +\newpage \ No newline at end of file diff --git a/Annexe-Exercices/Images/incline.eps b/Annexe-Exercices/Images/incline.eps new file mode 100644 index 0000000..0786673 --- /dev/null +++ b/Annexe-Exercices/Images/incline.eps @@ -0,0 +1,152 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: cairo 1.14.8 (http://cairographics.org) +%%CreationDate: Thu Jan 3 21:37:57 2019 +%%Pages: 1 +%%DocumentData: Clean7Bit +%%LanguageLevel: 2 +%%BoundingBox: 0 -1 107 139 +%%EndComments +%%BeginProlog +save +50 dict begin +/q { gsave } bind def +/Q { grestore } bind def +/cm { 6 array 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\@setckpt{Annexe-Incertitudes/Annexe-Incertitudes}{ -\setcounter{page}{223} +\setcounter{page}{225} \setcounter{equation}{1} \setcounter{enumi}{3} \setcounter{enumii}{0} @@ -46,7 +46,7 @@ \setcounter{lstnumber}{1} \setcounter{Solution}{26} \setcounter{exc}{71} -\setcounter{Solution OS}{2} -\setcounter{exosc}{2} +\setcounter{Solution OS}{3} +\setcounter{exosc}{3} \setcounter{lstlisting}{0} } diff --git a/Annexe-Maree/Annexe-Maree.tex b/Annexe-Maree/Annexe-Maree.tex index 7355eb0..de0cc46 100644 --- a/Annexe-Maree/Annexe-Maree.tex +++ b/Annexe-Maree/Annexe-Maree.tex @@ -81,7 +81,7 @@ P-R-\nonumber\\ G\frac{M_L\cdot m}{d_{T-L}^2}\cdot (1+\frac{2\cdot R_T}{d_{T-L}})+G\frac{M_L}{d_{T-L}^2}&=0\label{DLmaree}\\ P-R-G\frac{M_L\cdot m}{d_{T-L}^2}-\nonumber\\ 2\dot G\frac{M_L\cdot m}{d_{T-L}^3}\cdot R_T+G\frac{M_L\cdot m}{d_{T-L}^2}&=0\nonumber\\ -P-R-2\dot G\frac{M_L\cdot m}{d_{T-L}^3}\cdot R_T&=0\label{secondeloimaree} +P-R-2\cdot G\frac{M_L\cdot m}{d_{T-L}^3}\cdot R_T&=0\label{secondeloimaree} \end{align} Pour établir l'équation \ref{DLmaree}, on a utilisé le développement limité : \((1+x)^p\simeq1+p\cdot x\). \begin{align*} diff --git a/CoursMecaniqueOSDF.dvi b/CoursMecaniqueOSDF.dvi index f04cb1b94e42df91e6be816a4904918a156fa072..9a2a66d571edb8b25e778759d9f8b5448aff4365 100644 GIT binary patch delta 11159 zcmbta2~<;8x8|PQBqSt2lA91F#X8^w0#-#7CseeeIM=y`5H%JOOo)Qk28aH(wM1iT zPAdIHtgWIQ6vZP}t#-0b72Dc6^jWlO!P&OjTCKc&?hS$3zW=?oUe09)@khGB3i# za<;4X8G~i@o_dGDR_ql`hCotQn>QoGOW)t%~ z(XbJ?VuePB?^LmE3EZ5k!39<%jmb*v;c|EXvto3f!Quudn=QsXv!$fLEqXO$r)tJN z2G2Z;^!V1#BDKe5l;Z7TWrSeW3$>#0ObG!aVxJP`!2TkdFBqPT-~ClI27R>!mLUT# zUZ5C^|8-9^9^L%psRE2)qN$o#MJ`ZNS8YKIg+TI^Op2NOO z+Q+g{;iy#)0s@1ysGGjUv#bnufea0vMh73kwOb@|;v31fk4B^3YQ9UM)HjA}DoUU@wkAKe(Ti>Y6NW(4rIwXRw{r zWr{OH(Xd;1$7RWSa(JS6W+<8zgzvh=X~bv}m+Fq=8 zP|}`Ed{O*{204!q5I@f$Pc7B89oN2*z$9&O@zYTL>_# zB$1J5^;3et*)5W*I#l_PfQDP)=+FiNidRa}SF8CUO`y9ubc*#;dA7oH_Oqg0Y}*QB>ObiL7!o+*}8FA0`LiGrD}f>iCc 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{figure}{\numberline {L.5}{\ignorespaces Plan inclin\IeC {\'e}}}{221} \addvspace {10\p@ } diff --git a/CoursMecaniqueOSDF.log b/CoursMecaniqueOSDF.log index 879ccd4..f6a0bbe 100644 --- a/CoursMecaniqueOSDF.log +++ b/CoursMecaniqueOSDF.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex 2018.12.20) 21 DEC 2018 21:05 +This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016/Debian) (preloaded format=latex 2018.12.20) 4 JAN 2019 10:38 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -3229,58 +3229,58 @@ Output from handle ans going to Solutions.tex \openout12 = `Solutions.tex'. -Underfull \hbox (badness 1178) in paragraph at lines 974--985 +Underfull \hbox (badness 1178) in paragraph at lines 1042--1053 \T1/cmr/m/n/10 teur de la Lune avec les don-nées des ta-bles. [] -Underfull \hbox (badness 1609) in paragraph at lines 1025--1026 +Underfull \hbox (badness 1609) in paragraph at lines 1093--1094 []\T1/cmr/m/n/10 Une voiture a une masse $\OML/cmm/m/it/10 m \OT1/cmr/m/n/10 = [] -Underfull \hbox (badness 10000) in paragraph at lines 1033--1053 +Underfull \hbox (badness 10000) in paragraph at lines 1101--1121 []\T1/cmr/m/n/10 Réponses : $[]$, $[]$ et [] [212] -Underfull \hbox (badness 2343) in paragraph at lines 1234--1242 +Underfull \hbox (badness 2343) in paragraph at lines 1302--1310 \T1/cmr/m/n/10 hor-i-zon-tale-ment aug-mente sa vitesse de $[]$ à [] -Underfull \hbox (badness 10000) in paragraph at lines 1244--1244 +Underfull \hbox (badness 10000) in paragraph at lines 1312--1312 []\T1/cmr/bx/n/12 Relatifs à l'énergie hy- [] -Underfull \hbox (badness 2245) in paragraph at lines 1259--1274 +Underfull \hbox (badness 2245) in paragraph at lines 1327--1342 []\T1/cmr/m/n/10 Le même Robin-son Cru-soé veut [] -Underfull \hbox (badness 10000) in paragraph at lines 1277--1291 +Underfull \hbox (badness 10000) in paragraph at lines 1345--1359 []\T1/cmr/m/n/10 On désire fournir $[]$ [] - -Underfull \hbox (badness 4144) in paragraph at lines 1309--1330 +[213] +Underfull \hbox (badness 4144) in paragraph at lines 1377--1398 []\T1/cmr/m/n/10 Une éoli-enne a des pales dont [] -[213] -Underfull \hbox (badness 1005) in paragraph at lines 1373--1393 + +Underfull \hbox (badness 1005) in paragraph at lines 1441--1461 []\T1/cmr/m/n/10 Quel est le débit d'eau ($\OML/cmm/m/it/10 c[] \OT1/cmr/m/n/10 = [] -Underfull \hbox (badness 10000) in paragraph at lines 1396--1403 +Underfull \hbox (badness 10000) in paragraph at lines 1464--1471 []\T1/cmr/m/n/10 Un par-ti-c-ulier con-somme [] -Underfull \hbox (badness 3271) in paragraph at lines 1396--1403 +Underfull \hbox (badness 3271) in paragraph at lines 1464--1471 []$ \T1/cmr/m/n/10 d'énergie élec-trique. Il désire [] @@ -3329,11 +3329,22 @@ Underfull \hbox (badness 1769) in paragraph at lines 283--285 \T1/cmr/m/n/10 Compte tenu des frot-te-ments, l'énergie to-tale [] -[218] [219]) (./SolutionsOS.tex) [220]) +[218] [219]) (./SolutionsOS.tex (./Annexe-Exercices/Images/incline.eps_tex +File: ./Annexe-Exercices/Images/incline.eps Graphic file (type eps) + +<./Annexe-Exercices/Images/incline.eps>) + +LaTeX Warning: `h' float specifier changed to `ht'. + +[220]) [221]) \openout2 = `Annexe-Incertitudes/Annexe-Incertitudes.aux'. + (./Annexe-Incertitudes/Annexe-Incertitudes.tex [222 -(./Annexe-Incertitudes/Annexe-Incertitudes.tex + + + +] Appendix M. Underfull \vbox (badness 4205) has occurred while \output is active [] @@ -3341,13 +3352,11 @@ Underfull \vbox (badness 4205) has occurred while \output is active [] Underfull \vbox (badness 4429) has occurred while \output is active [] - [221 - - + [223 ]) -[222] (./CoursMecaniqueOSDF.ent +[224] (./CoursMecaniqueOSDF.ent Underfull \hbox (badness 7415) in paragraph at lines 2--22 \T1/cmtt/m/n/8 hubble . nasa . gov / multimedia / astronomy . php$ \T1/cmr/m/n/ 8 no-tam-ment @@ -3555,7 +3564,7 @@ Underfull \hbox (badness 2057) in paragraph at lines 616--624 ommons . [] -[223 +[225 ] Underfull \hbox (badness 10000) in paragraph at lines 626--650 @@ -3690,7 +3699,7 @@ Underfull \hbox (badness 5077) in paragraph at lines 868--883 cmr/m/n/8 ou [] -) [224 +) [226 ] (./CoursMecaniqueOSDF.bbl Underfull \hbox (badness 5954) in paragraph at lines 37--43 @@ -3713,15 +3722,15 @@ Underfull \hbox (badness 1325) in paragraph at lines 140--152 \T1/cmr/m/n/10 Ou-vrage très com-plet et très math-é-ma-tique [] -[225 +[227 -]) (./CoursMecaniqueOSDF.ind [226 +]) (./CoursMecaniqueOSDF.ind [228 -] [227 +] [229 -] [228] [229] [230] [231] [232] -[233] [234 +] [230] [231] [232] [233] [234] +[235] [236 ]) (./CoursMecaniqueOSDF.aux (./Prefaces/Prefaces.aux) (./Introduction/Introduction.aux) (./Cinematique/Cinematique.aux) @@ -3753,12 +3762,12 @@ LaTeX Warning: There were multiply-defined labels. ) Here is how much of TeX's memory you used: - 19445 strings out of 494830 - 307201 string characters out of 6176634 - 446163 words of memory out of 5000000 - 21936 multiletter 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OS} +\newenvironment{exos}{\begin{exosc}\normalfont}{\end{exosc}} + +\setlength{\headheight}{12.2pt} % spécifie la hauteur de l'entête pour qu'elle ne couvre pas le titre + +\setcounter{topnumber}{3} % nombre maximal de flottant en haut d'une page (par défaut 2) +\setcounter{bottomnumber}{2} % nombre maximal de flottant en bas d'une page (par défaut 1) +%\setcounter{totalnumber}{3} % nombre maximal de flottant sur une page (par défaut 3) +\renewcommand{\topfraction}{0.9} % proportion maximale de la page qui peut recevoir des flottants en haut de la page (par défaut 0,7) +\renewcommand{\textfraction}{0.1} % proportion minimale d'une page qui reçoit du texte (sans les pages de flottants) +\renewcommand{\floatpagefraction}{0.8} % proportion minimale de flottants sur une page pour qu'elle devienne une page uniquement de flottants +\renewcommand{\bottomfraction}{0.5} % proportion maximale de la page qui peut recevoir des flottants en bas de la page (par défaut 0,3) + +%\newcommand\relphantom[1]{\mathrel{\phantom{#1}}} % sert à afficher un = invisible +%---------------------------------------------------------------- + +\graphicspath{{./Images/}{./Introduction/Images/}{./Cinematique/Images/}{./Dynamique/Images/}{./MecaniqueDim/Images/}{./MecaniqueDifferentielle/Images/}{./QtiteMvt/Images/}{./Energie/Images/}{./EnergieOS/Images/}{./Thermodynamique/Images/}{./ThermodynamiqueOS/Images/}{./Annexe-UnitesInternationales/Images/}{./Annexe-SystemeCoordonnees/Images/}{./Annexe-MesuresDistances/Images/}{./Annexe-TravauxPratiques/Images/}{./Annexe-Rotations/Images/}{./Annexe-MRUA/Images/}{./Annexe-ChuteLune/Images/}{./Annexe-Satellites/Images/}{./Annexe-Relativite/Images/}{./Annexe-Maree/Images/}{./Annexe-Energies/Images/}{./Annexe-Exercices/Images/}{./Annexe-Incertitudes/Images/}} % spécifie les dossiers dans lesquels sont les images + +%-------------------------------------------------------------- +% définition de l'image de la page de titre +\usepackage{eso-pic} % pour mettre une image en fond de première page +\newcommand\BackgroundPic{ +\put(0,0){ +\parbox[b][\paperheight]{\paperwidth}{% +\vfill +\centering +%\includegraphics[width=\paperwidth,height=\paperheight,keepaspectratio]{TrouNoir.eps}% +\includegraphics[width=\paperwidth,height=\paperheight]{TrouNoir.eps}% +\vfill +}}} +%-------------------------------------------------------------- +% inclusion des macros +\include{Macros} + +\begin{document} + +%-------------------------------------------------------------- +% inclusion des Préfaces +\include{Prefaces/Prefaces} +%-------------------------------------------------------------- + +\twocolumn + +%-------------------------------------------------------------- +% table des matières, liste des figures et liste des tables +\tableofcontents{} + +\renewcommand{\listfigurename}{Liste des figures} +\listoffigures + +\myclearpage + +\listoftables +%-------------------------------------------------------------- + +\myclearpage + +%-------------------------------------------------------------- +% inclusion du chapitre d'introduction +\include{Introduction/Introduction} + +% inclusion du chapitre de cinématique +\include{Cinematique/Cinematique} + +% inclusion du chapitre de dynamique +\include{Dynamique/Dynamique} + +% inclusion de la Mécanique en plusieurs dimensions (OS) +\opt{OS}{\include{MecaniqueDim/MecaniqueDim}} + +% inclusion de la Mécanique différentielle (OS) +\opt{OS}{\include{MecaniqueDifferentielle/MecaniqueDifferentielle}} + +% inclusion de la Quantité de mouvement (OS) +\opt{OS}{\include{QtiteMvt/QtiteMvt}} + +% inclusion du chapitre d'énergie +\include{Energie/Energie} + +% inclusion de lénergie (OS) +\opt{OS}{\include{EnergieOS/EnergieOS}} + +% inclusion du chapitre de thermodynamique +\include{Thermodynamique/Thermodynamique} + +% inclusion de la Thermodynamique (OS) +\opt{OS}{\include{ThermodynamiqueOS/ThermodynamiqueOS}} +%-------------------------------------------------------------- + +\appendix + +\onecolumn + +%-------------------------------------------------------------- +% inclusion de l'annexe Unités du système international +\include{Annexe-UnitesInternationales/Annexe-UnitesInternationales} + +% inclusion de l'annexe Deux systèmes de coordonnées +\include{Annexe-SystemeCoordonnees/Annexe-SystemeCoordonnees} + +% inclusion de l'annexe Mesures de distances +\include{Annexe-MesuresDistances/Annexe-MesuresDistances} + +% inclusion de l'annexe Travaux pratiques +\include{Annexe-TravauxPratiques/Annexe-TravauxPratiques} + +% inclusion de l'annexe Rotations +\include{Annexe-Rotations/Annexe-Rotations} + +% inclusion de l'annexe MRUA développements +\include{Annexe-MRUA/Annexe-MRUA} + +% inclusion de l'annexe Chute de la lune +\include{Annexe-ChuteLune/Annexe-ChuteLune} + +% inclusion de l'annexe Satellite en orbite géostationnaire +\include{Annexe-Satellites/Annexe-Satellites} + +% inclusion de l'annexe Relativité +\include{Annexe-Relativite/Annexe-Relativite} + +% inclusion de l'annexe Marées +\include{Annexe-Maree/Annexe-Maree} + +% inclusion de l'annexe Énergies +\include{Annexe-Energies/Annexe-Energies} + +% inclusion de l'annexe Exercices +\include{Annexe-Exercices/Annexe-Exercices} + +% inclusion de l'annexe Incertitudes (OS) +\opt{OS}{\include{Annexe-Incertitudes/Annexe-Incertitudes}} +%-------------------------------------------------------------- + +\theendnotes + +\myclearpage + +%\nocite{*} % pour mettre toutes les entrées dans la biblio, y compris celles non citées ds le texte +\bibliographystyle{apalike-fr} % le style de bibliographie apalike francisé. Attention, il ne faut pas le module apalike, mais il faut le module natbib. +%\bibliographystyle{apalike} % le style de bibliographie +\bibliography{../Bibliographies/BiblioCours} % définit le fichier des réf. biblio. nom.bib + % le logiciel utilisé pour faire la biblio est pybliographic (pybliographer en ligne de commande) !!! suite à la suppression de + % pybliographic (pybliographer a maintenant une ui) de Lenny, j'utilise JABREF <-- très très bien. +\printindex{} % attention, pour que l'index soit mis à jour il faut lancer la commande de konsole : makeindex -s ../Perso.ist nom_du_cours.idx + % le Perso.ist est un fichier disant qu'il faut mettre des lettres entre les parties de l'index + +%\renewcommand{\listfigurename}{Liste des figures} +%\listoffigures % ai tout basculé après la table des matières car je n'arrivais pas à éliminer le petit bug ci-dessous. Mais le compagnon latex fait idem. +%\listoftables % un petit problème (bug ?) d'alignement m'a fait retirer provisoirement cette liste + +\end{document} +\endinput % termine l'importation des solutions \ No newline at end of file diff --git a/CoursMecaniqueOSDF.toc b/CoursMecaniqueOSDF.toc index b3416be..d5aefc6 100644 --- a/CoursMecaniqueOSDF.toc +++ b/CoursMecaniqueOSDF.toc @@ -389,14 +389,14 @@ \contentsline {subsection}{\numberline {L.1.7}Relatifs \IeC {\`a} la physique aristot\IeC {\'e}licienne}{210} \contentsline {subsection}{\numberline {L.1.8}Relatifs \IeC {\`a} la physique newtonienne}{210} \contentsline {subsection}{\numberline {L.1.9}Relatifs aux forces}{212} -\contentsline {subsection}{\numberline {L.1.10}Relatifs \IeC {\`a} l'\IeC {\'e}nergie}{212} +\contentsline {subsection}{\numberline {L.1.10}Relatifs \IeC {\`a} l'\IeC {\'e}nergie}{213} \contentsline {subsection}{\numberline {L.1.11}Relatifs \IeC {\`a} la conservation de l'\IeC {\'e}nergie}{213} \contentsline {subsection}{\numberline {L.1.12}Relatifs \IeC {\`a} l'\IeC {\'e}nergie hydraulique}{213} \contentsline {subsection}{\numberline {L.1.13}Relatifs \IeC {\`a} l'\IeC {\'e}nergie \IeC {\'e}olienne}{213} \contentsline {subsection}{\numberline {L.1.14}Relatifs \IeC {\`a} l'\IeC {\'e}nergie solaire}{214} \contentsline {section}{\numberline {L.2}Solutions}{214} \contentsline {section}{\numberline {L.3}Solutions OS}{220} -\contentsline {chapter}{\numberline {M}Ordre de grandeur, erreur et incertitudes}{221} -\contentsline {section}{\numberline {M.1}Ordre de grandeur}{221} -\contentsline {section}{\numberline {M.2}\IeC {\'E}cart et erreur}{221} -\contentsline {section}{\numberline {M.3}Incertitude}{222} +\contentsline {chapter}{\numberline {M}Ordre de grandeur, erreur et incertitudes}{223} +\contentsline {section}{\numberline {M.1}Ordre de grandeur}{223} +\contentsline {section}{\numberline {M.2}\IeC {\'E}cart et erreur}{223} +\contentsline {section}{\numberline {M.3}Incertitude}{224} diff --git a/SolutionsOS.tex b/SolutionsOS.tex index 4c980eb..174ff9c 100644 --- a/SolutionsOS.tex +++ b/SolutionsOS.tex @@ -20,6 +20,71 @@ \end{Solution OS} \begin{Solution OS}{2} + Deux forces agissent ici : la réaction du plan, qui lui est normale (c'est-à-dire perpendiculaire), et le poids de la masse. Comme la réaction du plan n'a aucune composante parallèlement au plan, elle ne peut être responsable de l'accélération de la masse le long de celui-ci. + + Il faut donc trouver la composante du poids qui est parallèle au plan incliné. L'angle entre le poids et un plan horizontal est de \unit{90}{\degree}. Quand le plan est incliné, cet angle diminue de la valeur de l'inclinaison. L'angle \(\beta\) entre le plan incliné et le poids et donc \(\beta=90-\alpha\). + + Comme la projection du poids selon l'angle \(\beta\) correspond à sa composante parallèle au plan, dans le triangle rectangle composé du poids comme hypoténuse et de ses composantes parallèle et perpendiculaire au plan, la composante parallèle au plan correspond au côté adjacent. Ainsi, on peut écrire~: + \begin{align*} + P_{//}&=P\cdot\cos(\beta)=P\cdot\cos(90-\alpha)\\ + &=P\cdot\sin(\alpha) + \end{align*} + La seconde loi de Newton s'écrit donc le long du plan incliné~: + \begin{align*} + \sum F^{ext}=P\cdot\sin(\alpha)&=m\cdot a\;\Rightarrow\\ + m\cdot g\cdot\sin(\alpha)&=m\cdot a\;\Rightarrow\\ + a&=g\cdot\sin(\alpha)\;\Rightarrow\\ + a&=9,81\cdot\sin(20)=\unit{3,36}{\metre\per\second\squared} + \end{align*} + Avec une vitesse initiale nulle, pour un MRUA d'accélération calculée ci-dessus, la vitesse au bout d'un temps t=\unit{2}{\second} s'obtient par~: + \[v=a\cdot t+v_0=3,36\cdot 2=\unit{9,72}{\metre\per\second}\] + + \medskip + Une autre manière de résoudre le problème est de procéder avec méthode. Le système qu'on doit choisir est bien évidemment la masse m, puisque c'est de celle-ci qu'on cherche l'accélération pour en trouver la vitesse au bout de \unit{2}{\second}. + \begin{figure}[h] + \centering + \caption[Plan incliné]{Le plan incliné}\label{incline} + \medskip + \def\svgwidth{6cm} + \input{Annexe-Exercices/Images/incline.eps_tex} + \end{figure} + + La figure \ref{incline} présente ensuite le dessin des forces extérieures et le choix du système d'axes. Remarquez que ce dernier l'a été selon l'inclinaison du plan. Il aurait pu ne pas en être ainsi, mais ce choix simplifie les calculs, car la masse étant contrainte à se déplacer le long du plan, son accélération perpendiculairement est nulle. Les équations de la seconde loi de Newton, obtenues par projection des forces extérieures et de l'accélération selon les axes, peuvent alors s'écrire : + \begin{align*} + \sum F^{ext}_x&=P_x=m\cdot a_x &\text{sur l'axe x}\\ + \sum F^{ext}_y&=R-P_y=m\cdot a_y=0 &\text{sur l'axe y} + \end{align*} + car l'accélération perpendiculairement au plan est nulle, comme déjà mentionné. + + \smallskip + Si on considére l'angle \(\alpha\), en s'imaginant le plan incliné horizontal, on comprends qu'il se reporte entre le vecteur poids \(\overleftarrow{P}\) et sa compostante selon y \(P_y\). + + Avec le triangle rectangle formé par le poids et ses composantes et un peu de trigonométrie, on peut en déduire~: + \begin{align*} + P_x &= P\cdot \sin(\alpha)\\ + P_y &= P\cdot \cos(\alpha) + \end{align*} + Comme par ailleurs on sait que \(P=m\cdot g\), on peut réécrire les équations de Newton sur les axes comme~: + \begin{align*} + \sum F^{ext}_x&=m\cdot g\cdot \sin(\alpha)=m\cdot a_x\\ + \sum F^{ext}_y&=R-m\cdot g\cdot \cos(\alpha)=m\cdot a_y=0 + \end{align*} + La première de ces équations permet de trouver l'accélération du bloc selon le plan incliné~: + \begin{align*} + a=a_x&=g\cdot \sin(\alpha)\\ + &=9,81\cdot\sin(20)=\unit{3,36}{\metre\per\second\squared} + \end{align*} + + \smallskip + Mais une information supplémentaire nous est donnée par la seconde équation, c'est la valeur de la réaction R au plan~: + \begin{align*} + R&=m\cdot g\cdot \cos(\alpha)\\ + &=3\cdot 9,81\cdot \cos(20)=\unit{27,67}{\newton} + \end{align*} + + +\end{Solution OS} +\begin{Solution OS}{3} Un corrigé de test. \end{Solution OS}

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