From d1c6ff02672ec13ed0a6c58434445702bc125d1e Mon Sep 17 00:00:00 2001 From: guyotv Date: Wed, 31 Aug 2022 22:38:59 +0200 Subject: [PATCH] =?UTF-8?q?R=C3=A9daction=20exos=20=C3=A9toile=20double?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Annexe-Exercices/Annexe-Exercices.tex | 16 ++++++++++++++-- Annexe-Exercices/Annexe-Exercices.tex.bak | 15 +++++++++++++-- CoursMecaniqueOS.pdf | Bin 12816662 -> 12826076 bytes CoursMecaniqueOSDF.pdf | Bin 12822610 -> 12826076 bytes Dynamique/Dynamique.tex | 6 +++++- Dynamique/Dynamique.tex.bak | 12 ++++++++---- SolutionsOS.tex | 14 +++++++++++++- 7 files changed, 53 insertions(+), 10 deletions(-) diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index 8793650..22bb77b 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -1871,9 +1871,21 @@ On lâche la première à vitesse initiale nulle. Calculez la vitesse de la seco \end{exos} \begin{exos} - Un système binaire est constitué de deux étoiles + Le système binaire à éclipse d'Algol (\(\beta\) Per) est constitué de deux étoiles dont la période relative est de \SI{2,8674}{jours}. Si le demi-grand axe du système vaut \SI{0,065}{\astronomicalunit}, calculez la masse totale des deux étoiles. On considère que les deux étoiles ont des masse non négligeables l'une par rapport à l'autre. \begin{solos} - Un autre corrigé de test. + La troisième loi de Kepler s'applique sous la forme donnée par l'équation \ref{keplerloitroisgenerale}, page \pageref{keplerloitroisgenerale}~: + \begin{equation*} + \frac{a^3}{T^2}=\frac{G}{4\cdot \pi^2}\cdot (M+m) + \end{equation*} + La période en seconde est~: \[T=2,8674\cdot 24\cdot 3600=\SI{247743.36}{\second}\] Le demi-grand axe en mètres vaut quant à lui~: \[a=0,065\cdot 149 597 870 700=\SI{9723861595.5}{\metre}\] + Ainsi, la masse totale du système est~: + \begin{align*} + M+m&=\frac{4\cdot \pi^2\cdot a^3}{G\cdot T^2}\\ + &=\frac{4\cdot \pi^2\cdot 9 723 861 595.5^3}{6,67\cdot 10^{-11}\cdot 247743.36^2}\\ + &=\SI{8,87e30}{\kilo\gram}=\frac{8,87\cdot 10^{30}}{1,9889\cdot 10^{30}}\\ + &=\SI{4,46}{M_{soleil}} + \end{align*} + Imaginez donc deux étoiles à une distance de 6,5\% de la distance Terre-Soleil (\SI{0,065}{\astronomicalunit}), dont la masse totale vaut plus de quatre masses solaires et qui tournent l'une autour de l'autre en un peu moins de trois jours ! \end{solos} \end{exos} diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index 8793650..ac5681e 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -1871,9 +1871,20 @@ On lâche la première à vitesse initiale nulle. Calculez la vitesse de la seco \end{exos} \begin{exos} - Un système binaire est constitué de deux étoiles + Le système binaire à éclipse d'Algol (\(\beta\) Per) est constitué de deux étoiles dont la période relative est de \SI{2,8674}{jours}. Si le demi-grand axe du système vaut \SI{0,065}{\astronomicalunit}, calculez la masse totale des deux étoiles. On considère que les deux étoiles ont des masse non négligeables l'une par rapport à l'autre. \begin{solos} - Un autre corrigé de test. + La troisième loi de Kepler s'applique sous la forme donnée par l'équation \ref{keplerloitroisgenerale}, page \pageref{keplerloitroisgenerale}~: + \begin{equation*} + \frac{a^3}{T^2}=\frac{G}{4\cdot \pi^2}\cdot (M+m) + \end{equation*} + La période en seconde est~: \[T=2,8674\cdot 24\cdot 3600=\SI{247743.36}{\second}\] Le demi-grand axe en mètres vaut quant à lui~: \[a=0,065\cdot 149 597 870 700=\SI{9723861595.5}{\metre}\] + Ainsi, la masse totale du système est~: + \begin{align*} + M+m&=\frac{4\cdot \pi^2\cdot a^3}{G\cdot T^2}\\ + &=\frac{4\cdot \pi^2\cdot 9 723 861 595.5^3}{6,67\cdot 10^{-11}\cdot 247743.36^2}\\ + &=\SI{8,87e30}{\kilo\gram}=\frac{8,87\cdot 10^{30}}{1,9889\cdot 10^{30}}=\SI{4,46}{M_{soleil}} + \end{align*} + Imaginez donc deux étoiles à une distance de 6,5\% de la distance Terre-Soleil (\SI{0,065}{\astronomicalunit}), dont la masse vaut \end{solos} \end{exos} diff --git a/CoursMecaniqueOS.pdf b/CoursMecaniqueOS.pdf index 5287365dd09ac06301f99a456ebc7af6bb37a84c..54aa1db730d1de45e0548d43821498ab5efe8b5b 100644 GIT binary patch delta 316626 zcmZU)Q*b6+w5^>twr!_l+qP|WY&&mkb!^+VZKFH3ZRh{?`S-ax7o+B?c{8ilsv0$B z)w5Cy;~+OL;vnng0T8TXE&>}#DgZhFQW6lL0=$#oIGO2kY;iJwXJqxsY^ce|K&T#3 z`KJr_*fO^^Dt4wQv1`(!wI272gFdtOFfy=8M^RB$8?sMVJ7gKDKVQ$=Jm^|BFmuL!lLI-xXO*@Yil8Le64Bz-BL)r3O}WUaQ3y#KX>{EQVL9$wf)$GE5H&&^R2yijN?O7@k}wL!FVOSwxyf^2n)5UUT2LCTU=<}W}hq+8n6=Y z1_mmfSn+%gi4v07>#b@-!p;RRLPoB3GbBn+7ma9rEq2JK>#AX?7AI*{ObmtV)9omVhT>C%_zd&BP|6ZpK7yo2uZakQ6%2Je0VF6%x!3?opE2YT4cEWp=IQC-WO z9A^R6iK$(!AflA>f>iR&XjHe)T|pRIF4R~NSJAx(|8?7{ z*|=&EyzMyMEMe+x7n(tb_r=BDwD7aLP(np(TQbNS2q>Y+$^6UFB8)*9Iz%D4w`c}7 z{o{m}o2xM(sttPN2Eo_02qR?ph5iZFhz?p1s@m^~|Dw7LmtsE@K27VKzxY5bMm3%Z zJa4|5WDMx2C?fB{ZkBkzBuc_AOhmA5*{5!}bS*NwFtSWMX-oYUUX-llS2XoM1+kX3 z0)dV_sz%k&o|-;k+Q{b|EQA4J1tAoI3izNLE&an2vqS7e>e~MobZ9Ww_BSWUku8;4 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L'équation \ref{troiskeplernewton} constitue la troisième loi de Kepler. -Mais la nouvelle forme de cette équation met en évidence le rôle de la masse \(M\) du corps central\footnote{Remarquons que cette forme de la troisième loi de Kepler où l'un des corps tourne de manière circulaire autour de l'autre dont la masse est très grande par rapport au premier trahit quelque peu la seconde loi qui porte sur le caractère elliptique de la trajectoire. Dans le cas où le rapport de masse est quelconque, la troisième loi s'exprime par~:\[\frac{a^3}{T^2}=M+m\]où \(a\) est le demi-grand axe de l'orbite et \(T\) la période. La démonstration de cette relation à l'aide de la loi de la gravitation universelle est plus complexe et ne sera pas abordée ici.} qui peut être calculée à partir du rayon de l'orbite et de la période\index{periode@période} du corps en rotation autour de lui. Cela constitue une méthode de calcul de la masse des astres\index{masse@masse!des astres}, comme celle de Jupiter à partir du rayon de l'orbite et de la période de Io, par exemple, ou celle de la Lune. +Mais la nouvelle forme de cette équation met en évidence le rôle de la masse \(M\) du corps central\footnote{Remarquons que cette forme de la troisième loi de Kepler où l'un des corps tourne de manière circulaire autour de l'autre dont la masse est très grande par rapport au premier trahit quelque peu la seconde loi qui porte sur le caractère elliptique de la trajectoire. Dans le cas où le rapport de masse est quelconque, la troisième loi s'exprime par~: +\begin{equation}\label{keplerloitroisgenerale} +\frac{a^3}{T^2}=\frac{G}{4\cdot \pi^2}\cdot (M+m) +\end{equation} +où \(a\) est le demi-grand axe de l'orbite et \(T\) la période. La démonstration de cette relation à l'aide de la loi de la gravitation universelle est plus complexe et ne sera pas abordée ici.} qui peut être calculée à partir du rayon de l'orbite et de la période\index{periode@période} du corps en rotation autour de lui. Cela constitue une méthode de calcul de la masse des astres\index{masse@masse!des astres}, comme celle de Jupiter à partir du rayon de l'orbite et de la période de Io, par exemple, ou celle de la Lune. La masse de la Lune peut en effet être très simplement évaluée à partir de la troisième loi de Kepler. Pour cela, de l'équation \ref{troiskeplernewton}, on commence par tirer l'expression de la masse de la Lune~: \begin{equation}\label{masselunekepler} diff --git a/Dynamique/Dynamique.tex.bak b/Dynamique/Dynamique.tex.bak index 64a0c88..d93d13b 100644 --- a/Dynamique/Dynamique.tex.bak +++ b/Dynamique/Dynamique.tex.bak @@ -524,7 +524,11 @@ G\cdot \frac{M}{a^2}&=\frac{4\cdot \pi^2\cdot a}{T^2}\;\Rightarrow\nonumber \\ L'équation \ref{troiskeplernewton} constitue la troisième loi de Kepler. -Mais la nouvelle forme de cette équation met en évidence le rôle de la masse \(M\) du corps central\footnote{Remarquons que cette forme de la troisième loi de Kepler où l'un des corps tourne de manière circulaire autour de l'autre dont la masse est très grande par rapport au premier trahit quelque peu la seconde loi qui porte sur le caractère elliptique de la trajectoire. Dans le cas où le rapport de masse est quelconque, la troisième loi s'exprime par~:\[\frac{a^3}{T^2}=M+m\]où \(a\) est le demi-grand axe de l'orbite et \(T\) la période. La démonstration de cette relation à l'aide de la loi de la gravitation universelle est plus complexe et ne sera pas abordée ici.} qui peut être calculée à partir du rayon de l'orbite et de la période\index{periode@période} du corps en rotation autour de lui. Cela constitue une méthode de calcul de la masse des astres\index{masse@masse!des astres}, comme celle de Jupiter à partir du rayon de l'orbite et de la période de Io, par exemple, ou celle de la Lune. +Mais la nouvelle forme de cette équation met en évidence le rôle de la masse \(M\) du corps central\footnote{Remarquons que cette forme de la troisième loi de Kepler où l'un des corps tourne de manière circulaire autour de l'autre dont la masse est très grande par rapport au premier trahit quelque peu la seconde loi qui porte sur le caractère elliptique de la trajectoire. Dans le cas où le rapport de masse est quelconque, la troisième loi s'exprime par~: +\begin{equation}\label{keplerloitroisgenerale} +\frac{a^3}{T^2}=\frac{G}{4\cdot \pi^2}\cdot (M+m) +\end{equation} +où \(a\) est le demi-grand axe de l'orbite et \(T\) la période. La démonstration de cette relation à l'aide de la loi de la gravitation universelle est plus complexe et ne sera pas abordée ici.} qui peut être calculée à partir du rayon de l'orbite et de la période\index{periode@période} du corps en rotation autour de lui. Cela constitue une méthode de calcul de la masse des astres\index{masse@masse!des astres}, comme celle de Jupiter à partir du rayon de l'orbite et de la période de Io, par exemple, ou celle de la Lune. La masse de la Lune peut en effet être très simplement évaluée à partir de la troisième loi de Kepler. Pour cela, de l'équation \ref{troiskeplernewton}, on commence par tirer l'expression de la masse de la Lune~: \begin{equation}\label{masselunekepler} @@ -664,10 +668,10 @@ Le résultat correspond au graphe du champ vectoriel\index{maree@marée!champ ve On peut montrer sur la base de la deuxième loi de Newton et de la loi de la gravitation universelle (voir \cite[p. 416]{GC88}) que le vecteur \(\overrightarrow F_{mar\acute ee}\) correspondant à la force de marée s'exerçant sur une masse \(m\) d'eau située au point de coordonnées \((x,y)\) d'un référentiel cartésien lié au centre de la Terre (dans le plan d'un méridien) vaut~: \begin{equation}\label{eqmareevect} -\overrightarrow F_{mar\acute ee}=G\cdot \frac{M_L\cdot m}{d_{T-L}^3}\cdot {2\cdot x \choose -y} +\overrightarrow F_{mar\acute ee}=G\cdot \frac{M_L\cdot m}{d_{T-L}^3}\cdot \binom{2\cdot x}{-y} \end{equation} -% Attention, dans l'équation ci-dessus, il ne faut pas remplacer \binom{x}{-y} par x \choose -y -% car cela produit un warning +% Attention, dans l'équation ci-dessus, il ne faut pas remplacer \binom{2\cdot x}{-y} par 2\cdot x \choose -y +% car cela produit un warning de foreign command \atopwithdemims. Où \(M_L\) est la masse de la Lune et \(d_{T-L}\) la distance Terre-Lune. L'expression de la force \(\overrightarrow F_{mar\acute ee}\) donnée par \ref{eqmareevect} trouve sa représentation graphique dans le champ de vecteurs représenté à la figure \ref{mareechampforce}. On y voit l'attraction de la Lune sur les masses d'eau qui se trouvent de son côté et, de l'autre côté, l'effet de la force d'inertie (pseudo-force centrifuge). Deux marées hautes se trouvent donc aux antipodes l'une de l'autre. La Terre tournant sur elle-même bien plus rapidement que la Lune tourne autour d'elle, l'existence de deux marées par jour trouve donc ici une explication à travers l'application de la deuxième loi de Newton. diff --git a/SolutionsOS.tex b/SolutionsOS.tex index 29170dc..afff4de 100644 --- a/SolutionsOS.tex +++ b/SolutionsOS.tex @@ -655,7 +655,19 @@ \end{Solution OS} \begin{Solution OS}{18} - Un autre corrigé de test. + La troisième loi de Kepler s'applique sous la forme donnée par l'équation \ref{keplerloitroisgenerale}, page \pageref{keplerloitroisgenerale}~: + \begin{equation*} + \frac{a^3}{T^2}=\frac{G}{4\cdot \pi^2}\cdot (M+m) + \end{equation*} + La période en seconde est~: \[T=2,8674\cdot 24\cdot 3600=\SI{247743.36}{\second}\] Le demi-grand axe en mètres vaut quant à lui~: \[a=0,065\cdot 149 597 870 700=\SI{9723861595.5}{\metre}\] + Ainsi, la masse totale du système est~: + \begin{align*} + M+m&=\frac{4\cdot \pi^2\cdot a^3}{G\cdot T^2}\\ + &=\frac{4\cdot \pi^2\cdot 9 723 861 595.5^3}{6,67\cdot 10^{-11}\cdot 247743.36^2}\\ + &=\SI{8,87e30}{\kilo\gram}=\frac{8,87\cdot 10^{30}}{1,9889\cdot 10^{30}}\\ + &=\SI{4,46}{M_{soleil}} + \end{align*} + Imaginez donc deux étoiles à une distance de 6,5\% de la distance Terre-Soleil (\SI{0,065}{\astronomicalunit}), dont la masse totale vaut plus de quatre masses solaires et qui tournent l'une autour de l'autre en un peu moins de trois jours ! \end{Solution OS} \begin{Solution OS}{19}