From 9484c0363f4b9dddf887f7c2a8cdad880742a3f0 Mon Sep 17 00:00:00 2001 From: guyotv Date: Sat, 13 Aug 2022 13:34:36 +0200 Subject: [PATCH] Ajout exos dilatation. --- Annexe-Exercices/Annexe-Exercices.tex | 35 ++++++++++++++++++++-- Annexe-Exercices/Annexe-Exercices.tex.bak | 35 ++++++++++++++++++++-- CoursMecaniqueOSDF.pdf | Bin 12800285 -> 12804664 bytes Solutions.tex | 5 ++++ SolutionsOS.tex | 14 +++++++++ Thermodynamique/Thermodynamique.tex | 1 + Thermodynamique/Thermodynamique.tex.bak | 1 + 7 files changed, 85 insertions(+), 6 deletions(-) diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index a2ab483..dd44328 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -2320,8 +2320,9 @@ Les exercices suivants doivent être réalisés si possible sans la théorie de \end{ex} \subsection{Relatifs à la thermodynamique} +Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \ref{coefdilat}, page \pageref{coefdilat}. -\begin{ex}\label{centaure} +\begin{ex}\label{regleallu} Une règle en aluminium est construite pour faire \SI{30}{\centi\metre} à \SI{20}{\celsius}. Si on la chauffe à \SI{40}{\celsius}, quelle sera alors sa longueur réelle ? Réponse~: \SI{30,014}{\centi\metre}. \begin{sol} La variation de longueur est aisément calculable par~: @@ -2332,8 +2333,8 @@ Les exercices suivants doivent être réalisés si possible sans la théorie de \end{sol} \end{ex} -\begin{ex}\label{centaure} - On veut faire entrer un objet en verre de \SI{20}{\metre} à \SI{15}{\celsius} dans une boite en fer de \SI{20,005}{\metre} à \SI{10}{\celsius} pour conserver le tout à \SI{20}{\celsius}. Est-ce possible ? +\begin{ex}\label{verrefer} + On veut faire entrer un objet en verre de \SI{20}{\metre} à \SI{15}{\celsius} dans une boite en fer de \SI{20,005}{\metre} à \SI{10}{\celsius} pour conserver le tout à \SI{20}{\celsius}. Est-ce possible ? Réponse~: oui. \begin{sol} En chauffant l'objet en verre de \SI{5}{\celsius}, il va se dilater de~: \[\Delta L=L_0\cdot\alpha\cdot\Delta\theta=20\cdot 68\cdot 10^{-6}\cdot (20-15)=\SI{6,8}{\milli\metre}\] @@ -2343,6 +2344,34 @@ Les exercices suivants doivent être réalisés si possible sans la théorie de \end{sol} \end{ex} +\begin{ex}\label{Eiffel} + La hauteur de la tour Eiffel est de \SI{330}{\metre}. Elle est construite en acier. Quelle est la variation de sa hauteur entre \SI{0}{\celsius} et \SI{30}{\celsius} ? Réponse~: \SI{11}{\centi\metre}. + \begin{sol} + La plage de température étant de \SI{30}{\celsius}, on a~: + \[\Delta L=330\cdot 11\cdot 10^{-6}\cdot (30-0)=\SI{0,11}{\metre}\] + \end{sol} +\end{ex} + +\optv{OS}{ + +\begin{exos} + Une passerelle constituée d'une poutre en acier est fixée entre deux rochers fixes. Si la température passe de \SI{-10}{\celsius} à \SI{30}{\celsius}, l'augmentation de la contrainte de compression de la poutre est alors de \SI{8,8e7}{\newton\per\metre\squared}. Quel est le module de Young de la poutre ? Réponse~: \SI{2e11}{\newton\per\metre\squared}. + \begin{solos} + Pour résoudre ce problème, il faut savoir ce qu'est le module de Young. En voici la définition~: + \[E=\frac{F_n/A}{\Delta L/L_0}\] + où le numérateur de la fraction est la contrainte de déformation, soit la force \(F_n\) par unité de surface \(A\) exercée longitudinalement (normalement à la section) sur la poutre et le dénominateur est le facteur de déformation constitué du rapport de la déformation à la longueur de la poutre. Les unités du numérateur sont des \si{\newton\per\metre\squared} et le dénominateur en est dépourvu. Ainsi, les unités du module de Young sont des \si{\newton\per\metre\squared}. + + \medskip + La passerelle se trouvant entre deux rochers fixes, la poutre ne peut se dilater. L'augmentation de température a donc pour conséquence d'augmenter la force exercée sur la poutre. Cette contrainte de compression est donnée~: + \[\frac{F_n}{A}=\SI{8,8e7}{\newton\per\metre\squared}\] + de même que la distance de déformation qui correspond à l'allongement thermique de la poutre~: + \[\Delta L=\alpha\cdot L_0\cdot\Delta\theta\] + Ainsi, on peut écrire~: + \[E=\frac{F_n/A}{\alpha\cdot\Delta\theta}=\frac{8,8\cdot 10^7}{11\cdot 10^{-6}\cdot 40}=\SI{2e11}{\newton\per\metre\squared}\] + \end{solos} +\end{exos} + +} \Closesolutionfile{ansos} % ferme le fichier SolutionsOS.tex qui contient les solutions OS. \Closesolutionfile{ans} % ferme le fichier Solutions.tex qui contient les solutions diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index a2ab483..dd44328 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -2320,8 +2320,9 @@ Les exercices suivants doivent être réalisés si possible sans la théorie de \end{ex} \subsection{Relatifs à la thermodynamique} +Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \ref{coefdilat}, page \pageref{coefdilat}. -\begin{ex}\label{centaure} +\begin{ex}\label{regleallu} Une règle en aluminium est construite pour faire \SI{30}{\centi\metre} à \SI{20}{\celsius}. Si on la chauffe à \SI{40}{\celsius}, quelle sera alors sa longueur réelle ? Réponse~: \SI{30,014}{\centi\metre}. \begin{sol} La variation de longueur est aisément calculable par~: @@ -2332,8 +2333,8 @@ Les exercices suivants doivent être réalisés si possible sans la théorie de \end{sol} \end{ex} -\begin{ex}\label{centaure} - On veut faire entrer un objet en verre de \SI{20}{\metre} à \SI{15}{\celsius} dans une boite en fer de \SI{20,005}{\metre} à \SI{10}{\celsius} pour conserver le tout à \SI{20}{\celsius}. Est-ce possible ? +\begin{ex}\label{verrefer} + On veut faire entrer un objet en verre de \SI{20}{\metre} à \SI{15}{\celsius} dans une boite en fer de \SI{20,005}{\metre} à \SI{10}{\celsius} pour conserver le tout à \SI{20}{\celsius}. Est-ce possible ? Réponse~: oui. \begin{sol} En chauffant l'objet en verre de \SI{5}{\celsius}, il va se dilater de~: \[\Delta L=L_0\cdot\alpha\cdot\Delta\theta=20\cdot 68\cdot 10^{-6}\cdot (20-15)=\SI{6,8}{\milli\metre}\] @@ -2343,6 +2344,34 @@ Les exercices suivants doivent être réalisés si possible sans la théorie de \end{sol} \end{ex} +\begin{ex}\label{Eiffel} + La hauteur de la tour Eiffel est de \SI{330}{\metre}. Elle est construite en acier. Quelle est la variation de sa hauteur entre \SI{0}{\celsius} et \SI{30}{\celsius} ? Réponse~: \SI{11}{\centi\metre}. + \begin{sol} + La plage de température étant de \SI{30}{\celsius}, on a~: + \[\Delta L=330\cdot 11\cdot 10^{-6}\cdot (30-0)=\SI{0,11}{\metre}\] + \end{sol} +\end{ex} + +\optv{OS}{ + +\begin{exos} + Une passerelle constituée d'une poutre en acier est fixée entre deux rochers fixes. Si la température passe de \SI{-10}{\celsius} à \SI{30}{\celsius}, l'augmentation de la contrainte de compression de la poutre est alors de \SI{8,8e7}{\newton\per\metre\squared}. Quel est le module de Young de la poutre ? Réponse~: \SI{2e11}{\newton\per\metre\squared}. + \begin{solos} + Pour résoudre ce problème, il faut savoir ce qu'est le module de Young. En voici la définition~: + \[E=\frac{F_n/A}{\Delta L/L_0}\] + où le numérateur de la fraction est la contrainte de déformation, soit la force \(F_n\) par unité de surface \(A\) exercée longitudinalement (normalement à la section) sur la poutre et le dénominateur est le facteur de déformation constitué du rapport de la déformation à la longueur de la poutre. Les unités du numérateur sont des \si{\newton\per\metre\squared} et le dénominateur en est dépourvu. Ainsi, les unités du module de Young sont des \si{\newton\per\metre\squared}. + + \medskip + La passerelle se trouvant entre deux rochers fixes, la poutre ne peut se dilater. L'augmentation de température a donc pour conséquence d'augmenter la force exercée sur la poutre. Cette contrainte de compression est donnée~: + \[\frac{F_n}{A}=\SI{8,8e7}{\newton\per\metre\squared}\] + de même que la distance de déformation qui correspond à l'allongement thermique de la poutre~: + \[\Delta L=\alpha\cdot L_0\cdot\Delta\theta\] + Ainsi, on peut écrire~: + \[E=\frac{F_n/A}{\alpha\cdot\Delta\theta}=\frac{8,8\cdot 10^7}{11\cdot 10^{-6}\cdot 40}=\SI{2e11}{\newton\per\metre\squared}\] + \end{solos} +\end{exos} + +} \Closesolutionfile{ansos} % ferme le fichier SolutionsOS.tex qui contient les solutions OS. \Closesolutionfile{ans} % ferme le fichier Solutions.tex qui contient les solutions diff --git a/CoursMecaniqueOSDF.pdf b/CoursMecaniqueOSDF.pdf index 2d251cd0dad75ec82c5479427287d0132b99f8df..43a901be9202a4ff79507f38acf6e569dd027e13 100644 GIT binary patch delta 88686 zcmV)PK()V}qKyW)#Ek~98v%d4ZWS>Qg?m4RD`Xct%#UYmw^7m{MUZeiM1d|LNpTZh zAO(-eqwowg8Smbo6$RSDqcg{IKHIyG2o!OAhjqIP`>0(Ju+Ks(q);hJ0YWKml!`=k 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zBc7lgi3d=cT)@L$z|giM2Ju=&vlfUxUGd|n)izWCh385mMwnJk3HtH_hVXEi$J%-D zbc+!Fx9&clI?Lk|A0))XG#!KbWJElDllR2bSKglC&pIRpJ1~BaSXB;XtPk%N@GecR z1q$UzW?d5aBBSC0_YeKXGBKLYx2d$tR9D-(iaGv}>ahHeAWCk#ujyhcaNQZ--9&$k3 z+|x(3h4}HMOkDT!fxt^LIA#0~5At*;AgoEa&7w2;sAA2~t2ils_z}8r46KyoBqneE z9}H+LUUb57_@9iPilD`;)!DJXv<CAGt2Z$Sk57jb-L zVc4FJHET_Wy+&}w|F=b`UvRMBpop-65x)-X V+RYDo2bP-2;#ah2jYc*7{tw;d>)`+Z diff --git a/Solutions.tex b/Solutions.tex index 3b47242..21efc8a 100644 --- a/Solutions.tex +++ b/Solutions.tex @@ -1039,3 +1039,8 @@ On a successivement~: Ainsi, la boite en fer fera \SI{20,0074}{\metre} et l'objet en verre pourra s'y placer. \end{Solution} +\begin{Solution}{73} + La plage de température étant de \SI{30}{\celsius}, on a~: + \[\Delta L=330\cdot 11\cdot 10^{-6}\cdot (30-0)=\SI{0,11}{\metre}\] + +\end{Solution} diff --git a/SolutionsOS.tex b/SolutionsOS.tex index 644faf7..4a2803d 100644 --- a/SolutionsOS.tex +++ b/SolutionsOS.tex @@ -753,3 +753,17 @@ Il faut conclure qu'obtenir les vitesses par la méthode de Newton est bien plus complexe que par l'énergie. \end{Solution OS} +\begin{Solution OS}{22} + Pour résoudre ce problème, il faut savoir ce qu'est le module de Young. En voici la définition~: + \[E=\frac{F_n/A}{\Delta L/L_0}\] + où le numérateur de la fraction est la contrainte de déformation, soit la force \(F_n\) par unité de surface \(A\) exercée longitudinalement (normalement à la section) sur la poutre et le dénominateur est le facteur de déformation constitué du rapport de la déformation à la longueur de la poutre. Les unités du numérateur sont des \si{\newton\per\metre\squared} et le dénominateur en est dépourvu. Ainsi, les unités du module de Young sont des \si{\newton\per\metre\squared}. + + \medskip + La passerelle se trouvant entre deux rochers fixes, la poutre ne peut se dilater. L'augmentation de température a donc pour conséquence d'augmenter la force exercée sur la poutre. Cette contrainte de compression est donnée~: + \[\frac{F_n}{A}=\SI{8,8e7}{\newton\per\metre\squared}\] + de même que la distance de déformation qui correspond à l'allongement thermique de la poutre~: + \[\Delta L=\alpha\cdot L_0\cdot\Delta\theta\] + Ainsi, on peut écrire~: + \[E=\frac{F_n/A}{\alpha\cdot\Delta\theta}=\frac{8,8\cdot 10^7}{11\cdot 10^{-6}\cdot 40}=\SI{2e11}{\newton\per\metre\squared}\] + +\end{Solution OS} diff --git a/Thermodynamique/Thermodynamique.tex b/Thermodynamique/Thermodynamique.tex index f86a319..97f9f68 100644 --- a/Thermodynamique/Thermodynamique.tex +++ b/Thermodynamique/Thermodynamique.tex @@ -74,6 +74,7 @@ Aluminium & \num{23,1e-6}\\ Béton & \num{10e-6}\\ Cuivre & \num{16,6e-6}\\ Fer & \num{12e-6}\\ +Acier & \num{11e-6}\\ Verre & \num{68e-6}\\ \hline \end{tabular} diff --git a/Thermodynamique/Thermodynamique.tex.bak b/Thermodynamique/Thermodynamique.tex.bak index f86a319..97f9f68 100644 --- a/Thermodynamique/Thermodynamique.tex.bak +++ b/Thermodynamique/Thermodynamique.tex.bak @@ -74,6 +74,7 @@ Aluminium & \num{23,1e-6}\\ Béton & \num{10e-6}\\ Cuivre & \num{16,6e-6}\\ Fer & \num{12e-6}\\ +Acier & \num{11e-6}\\ Verre & \num{68e-6}\\ \hline \end{tabular}