From 2ba442d40641089bc51556901e1449724a988104 Mon Sep 17 00:00:00 2001 From: guyotv Date: Sun, 26 Jan 2025 09:42:31 +0100 Subject: [PATCH] Report d'un exo moteur --- Annexe-Exercices/Annexe-Exercices.tex | 67 ++++ Annexe-Exercices/Annexe-Exercices.tex.bak | 64 +++- .../Images/Climatiseur.ps_tex.bak | 30 +- Annexe-Exercices/Images/Lemoteur.ps_tex | 69 ++++ Annexe-Exercices/Images/Lemoteur.ps_tex.bak | 69 ++++ Annexe-Exercices/Images/Lemoteur.svg | 323 ++++++++++++++++++ CoursMecaniqueOSDF.out | 6 +- CoursMecaniqueOSDF.pdf | Bin 13479264 -> 13487232 bytes CoursMecaniqueOSDF.tks | 3 +- SolutionsOS.tex | 53 ++- 10 files changed, 662 insertions(+), 22 deletions(-) create mode 100644 Annexe-Exercices/Images/Lemoteur.ps_tex create mode 100644 Annexe-Exercices/Images/Lemoteur.ps_tex.bak create mode 100644 Annexe-Exercices/Images/Lemoteur.svg diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index 2011be1..c89bab9 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -2939,6 +2939,73 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + \begin{figure}[t] + \centering + \caption{Diagramme PV du moteur.\label{lemoteur}} + \centering + \def\svgwidth{0.9\columnwidth} + \input{Annexe-Exercices/Images/Lemoteur.ps_tex} + \end{figure} + On donne le diagramme PV d'un moteur thermique qui fonctionne avec un gaz diatomique dans le sens A → B → C. + \begin{enumerate} + \item Calculez les caractéristiques manquantes. + \item Déterminez \(\Delta U\), \(A\) et \(Q\) pour chaque étape. + \item Calculez le travail que ce moteur fournit à chaque cycle. + \item Calculez sa puissance s'il effectue 3000 cycles par minute. + Indication. Pour une transformation isotherme, le travail se calcule par~: + \[A=nR\cdot T\cdot \ln(V_2/V_1)\] + \end{enumerate} + \begin{solos} + Au préalable, calculons nR. + \begin{equation*} + nR=\frac{p_B\cdot V_B}{T_B}=\frac{10\cdot 10^5\cdot 2\cdot 10^{-3}}{1500}=1,33 + \end{equation*} + \begin{enumerate} + \item Les grandeurs manquantes. + \begin{align*} + T_1&=\frac{p_1\cdot V_1}{nR}=\frac{2\cdot 10^5\cdot 2\cdot 10^{-3}}{1,33}=\SI{300}{\kelvin}\\ + V_C&=\frac{nR\cdot T_C}{p_C}=\frac{1,33\cdot 1500}{2\cdot 10^5}=\SI{10}{\deci\metre\cubed} + \end{align*} + \item Les grandeurs des transformations. + \begin{description} + \item[AB] Isochore + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T\\ + &=\frac{5}{2}\cdot 1,33\cdot (1500-300)=\SI{4000}{\joule}\\ + A&=0 \\ + Q&= \Delta U+A=\Delta U=\SI{4000}{\joule} + \end{align*} + \item[BC] Isotherme + \begin{align*} + \Delta U&=0 \\ + A&=nR\cdot T\cdot \ln(V_C/V_B)\\ + &=1,33\cdot 1500\cdot \ln(10/2)=\SI{3218,9}{\joule}\\ + Q&= \Delta U+A=A=\SI{3218,9}{\joule} + \end{align*} + \item[CA] Isobare + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T\\ + &=\frac{5}{2}\cdot 1,33\cdot (300-1500)=\SI{-4000}{\joule}\\ + A&= p\cdot \Delta V\\ + &=2\cdot 10^5\cdot (2\cdot 10^{-3}-10\cdot 10^{-3})\\ + &=\SI{-1600}{\joule}\\ + Q&= \Delta U+A=-4000-1600\\ + &=\SI{-5600}{\joule} + \end{align*} + \end{description} + \item On peut calculer le travail total de deux manières~: + \begin{align*} + A_{tot}&=\sum A=0+3218,9-1600=\SI{1618,9}{\joule}\\ + A_{tot}&=\sum Q=4000+3218,9-5600\\ + &=\SI{1618,9}{\joule} + \end{align*} + \item La puissance est simplement~: + \[P=\frac{A}{t}=\frac{3000\cdot 1618,9}{60}=\SI{80985}{\watt}\] + \end{enumerate} + \end{solos} +\end{exos} + \begin{exos} Le climatiseur d'un local fonctionne avec un gaz monoatomique qui subit, dans un circuit fermé, trois transformations au cours de son cycle. On prendra \(nR=10\). \begin{description} diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index 1253bdd..f36c193 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -2939,6 +2939,66 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \end{solos} \end{exos} +\begin{exos} + \begin{figure}[t] + \centering + \caption{Diagramme PV du moteur.\label{lemoteur}} + \centering + \def\svgwidth{0.9\columnwidth} + \input{Annexe-Exercices/Images/Lemoteur.ps_tex} + \end{figure} + On donne le diagramme PV d'un moteur thermique qui fonctionne avec un gaz diatomique dans le sens A → B → C. + \begin{enumerate} + \item Calculez les caractéristiques manquantes. + \item Déterminez \(\Delta U\), \(A\) et \(Q\) pour chaque étape. + \item Calculez le travail que ce moteur fournit à chaque cycle. + \item Calculez sa puissance s'il effectue 3000 cycles par minute. + Indication. Pour une transformation isotherme, le travail se calcule par~: + \[A=nR\cdot T\cdot \ln(V_2/V_1)\] + \end{enumerate} + \begin{solos} + Au préalable, calculons nR. + \begin{equation*} + nR=\frac{p_B\cdot V_B}{T_B}=\frac{10\cdot 10^5\cdot 2\cdot 10^{-3}}{1500}=1,33 + \end{equation*} + \begin{enumerate} + \item Les grandeurs manquantes. + \begin{align*} + T_1&=\frac{p_1\cdot V_1}{nR}=\frac{2\cdot 10^5\cdot 2\cdot 10^{-3}}{1,33}=\SI{300}{\kelvin}\\ + V_C&=\frac{nR\cdot T_C}{p_C}=\frac{1,33\cdot 1500}{2\cdot 10^5}=\SI{10}{\deci\metre\cubed} + \end{align*} + \item Les grandeurs des transformations. + \begin{description} + \item[AB] Isochore + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T\\ + &=\frac{5}{2}\cdot 1,33\cdot (1500-300)=\SI{4000}{\joule}\\ + A&=0 \\ + Q&= \Delta U+A=\Delta U=\SI{4000}{\joule} + \end{align*} + \item[BC] Isotherme + \begin{align*} + \Delta U&=0 \\ + A&=nR\cdot T\cdot \ln(V_C/V_B)\\ + &=1,33\cdot 1500\cdot \ln(10/2)=\SI{3218,9}{\joule}\\ + Q&= \Delta U+A=A=\SI{3218,9}{\joule} + \end{align*} + \item[CA] Isobare + \begin{align*} + \Delta U&=\frac{i}{2}\cdot nR\cdot \Delta T\\ + &=\frac{5}{2}\cdot 1,33\cdot (300-1500)=\SI{-4000}{\joule}\\ + A&= p\cdot \Delta V\\ + &=2\cdot 10^5\cdot (2\cdot 10^{-3}-10\cdot 10^{-3})\\ + &=\SI{-1600}{\joule}\\ + Q&= \Delta U+A=-4000-1600\\ + &=\SI{-5600}{\joule} + \end{align*} + \end{description} + \item + \end{enumerate} + \end{solos} +\end{exos} + \begin{exos} Le climatiseur d'un local fonctionne avec un gaz monoatomique qui subit, dans un circuit fermé, trois transformations au cours de son cycle. On prendra \(nR=10\). \begin{description} @@ -3070,7 +3130,9 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r &=p_A\cdot V_A^{3/2}\cdot\int_A^B V^{-3/2}\cdot dV\\ &=p_A\cdot V_A^{3/2}\cdot[\frac{1}{-3/2+1}\cdot V^{-3/2+1}]_A^B\\ &=p_A\cdot V_A^{3/2}\cdot[-2\cdot V^{-1/2}]_A^B\\ - &=-2\cdot p_A\cdot V_A^{3/2}\cdot (V_B + &=-2\cdot p_A\cdot V_A^{3/2}\cdot (V_B^{-1/2}-V_A^{-1/2})\\ + &=-2\cdot 10^5\cdot 0,029^{1,5}\cdot (\frac{1}{\sqrt{0,00725}}-\\ + &\;\frac{1}{\sqrt{0,029}}=\SI{-5800}{\joule} \end{align*} \end{enumerate} \end{solos} diff --git a/Annexe-Exercices/Images/Climatiseur.ps_tex.bak b/Annexe-Exercices/Images/Climatiseur.ps_tex.bak index ee1d005..178c35a 100644 --- a/Annexe-Exercices/Images/Climatiseur.ps_tex.bak +++ b/Annexe-Exercices/Images/Climatiseur.ps_tex.bak @@ -50,22 +50,22 @@ \global\let\svgwidth\undefined% \global\let\svgscale\undefined% \makeatother% - \begin{picture}(1,0.962919)% + \begin{picture}(1,0.967725)% \lineheight{1}% 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