From e1e726a990eb8c74876711649bfc5a87791acd83 Mon Sep 17 00:00:00 2001 From: guyotv Date: Mon, 12 Sep 2022 22:00:55 +0200 Subject: [PATCH] =?UTF-8?q?Pr=C3=A9cisions=20sur=20quelques=20exos?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Annexe-Exercices/Annexe-Exercices.tex | 42 ++++++++++++++++++--- Annexe-Exercices/Annexe-Exercices.tex.bak | 44 ++++++++++++++++++---- CoursMecaniqueOS.pdf | Bin 12831037 -> 12836874 bytes CoursMecaniqueOSDF.pdf | Bin 12831037 -> 12836874 bytes SolutionsOS.tex | 42 ++++++++++++++++++--- 5 files changed, 108 insertions(+), 20 deletions(-) diff --git a/Annexe-Exercices/Annexe-Exercices.tex b/Annexe-Exercices/Annexe-Exercices.tex index 4eb0321..f4fc40f 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex +++ b/Annexe-Exercices/Annexe-Exercices.tex @@ -2127,7 +2127,12 @@ Les exercices suivants doivent être réalisés si possible sans la théorie de \begin{equation*} a_{//}=g\cdot\sin(\alpha)=9,81\cdot \sin(15)=\SI{2,54}{\metre\per\second\squared} \end{equation*} - Ainsi, considérant que le mouvement sans frottement d'une tuile sur un plan incliné est un MRUA, on peut déterminer la vitesse parallèle au toit, en bordure de celui-ci à l'aide soit du temps de descente soit de la distance parcourue d. Considérant une hauteur de \SI{5}{\metre} et un angle de \SI{15}{\degree}, on a~: + Ainsi, considérant que le mouvement sans frottement d'une tuile sur un plan incliné est un MRUA, on peut déterminer la vitesse parallèle au toit, en bordure de celui-ci à l'aide soit du temps de descente soit de la distance parcourue d\footnote{La considération suivante permet d'obtenir la vitesse au bord du toit d'une manière très simple. En effet, pour un objet en MRUA, on sait que la vitesse de chute verticale au bout d'une hauteur h vaut \(v=\sqrt{2\cdot g\cdot h}\). Or, pour un objet en MRUA sur un plan incliné d'un angle \(\alpha\), la vitesse au bout d'une distance d vaut \(v=\sqrt{2\cdot a\cdot d}\), avec une accélération le long du plan valant \(a=g\cdot \sin(\alpha)\) et une distance \(d=h/\sin(\alpha)\). On a donc~: + \begin{align*} + v&=\sqrt{2\cdot g\cdot \sin(\alpha)\cdot \frac{h}{\sin(\alpha)}}\\ + &=\sqrt{2\cdot g\cdot h} + \end{align*} + La vitesse au bord du toit est donc la même en grandeur que celle d'un objet en chute libre verticale.}. Considérant une hauteur de \SI{5}{\metre} et un angle de \SI{15}{\degree}, on a~: \begin{equation*} d=\frac{h}{\sin(\alpha)}=\frac{5}{\sin(15^{\circ})}=\SI{19,32}{\metre} \end{equation*} @@ -2541,25 +2546,50 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \item La vitesse \[v=\frac{x-x_0}{t}\] \end{itemize} \begin{solos} - Pour chaque cas, on a~: + Procédons simplement au calcul de l'incertitude absolue. + + \smallskip + Pour l'accélération centripète, on a~: \begin{align*} I(a)&=I(v^2/R)=v^2/R\cdot i(v^2/R)\\ &=\frac{v^2}{R}\cdot (2\cdot i(v) + i(R))\\ - &=\frac{v^2}{R}\cdot (2\cdot \frac{I(v)}{v} + \frac{I(R)}{R})\\ + &=\frac{v^2}{R}\cdot (2\cdot \frac{I(v)}{v} + \frac{I(R)}{R}) + \end{align*} + Pour la force de gravitation, on a~: + \begin{align*} I(F)&=I(G\cdot M\cdot m/d^2)\\ &=G\cdot M\cdot m/d^2\cdot i(G\cdot M\cdot m/d^2)\\ &=G\cdot \frac{M\cdot m}{d^2}\cdot (i(G)+i(M)+i(m)+2\cdot i(d))\\ - &=G\cdot \frac{M\cdot m}{d^2}\cdot (\frac{I(G)}{G}+\frac{I(M)}{M}+\frac{I(m)}{m}+2\cdot \frac{I(d)}{d})\\ + &=G\cdot \frac{M\cdot m}{d^2}\cdot (\frac{I(G)}{G}+\frac{I(M)}{M}+\frac{I(m)}{m}+2\cdot \frac{I(d)}{d}) + \end{align*} + Pour la position d'un MRUA, on a~: + \begin{align*} I(x)&=I(1/2\cdot a\cdot t^2 +v_0\cdot t+ x_0)\\ &=I(1/2\cdot a\cdot t^2)+I(v_0\cdot t)+I(x_0)\\ &=\frac{1}{2}\cdot a\cdot t^2\cdot (i(1/2)+i(a)+2\cdot i(t))\\ &+v_0\cdot t\cdot (i(v_0)+i(t))+I(x_0)\\ &=\frac{1}{2}\cdot a\cdot t^2\cdot (0+\frac{I(a)}{a}+2\cdot \frac{I(t)}{t})\\ - &+v_0\cdot t\cdot (\frac{I(v_0}{v_0}+\frac{I(t)}{t})+I(x_0))\\ + &+v_0\cdot t\cdot (\frac{I(v_0}{v_0}+\frac{I(t)}{t})+I(x_0)) + \end{align*} + Pour la vitesse, on a~: + \begin{align*} I(v)&=I((x-x_0)/t)=\frac{x-x_0}{t}\cdot (i(x-x_0)+i(t))\\ &=\frac{x-x_0}{t}\cdot (\frac{I(x-x_0)}{x-x_0}+\frac{I(t)}{t})\\ - &=\frac{x-x_0}{t}\cdot (\frac{I(x)+I(x_0)}{x-x_0}+\frac{I(t)}{t}) + &=\frac{x-x_0}{t}\cdot (\frac{I(x)+I(x_0)}{x-x_0}+\frac{I(t)}{t})\\ + &=\frac{I(x)+I(x_0)}{t}+(x-x_0)\cdot \frac{I(t)}{t^2} \end{align*} + Dans le dernier cas, on peut aussi procéder différemment~: + \begin{align*} + I(v)&=I((x-x_0)/t)=I(x/t-x_0/t)\\ + &=I(x/t)+I(x_0/t)\\ + &=\frac{x}{t}\cdot i(x/t)+\frac{x_0}{t}\cdot i(x_0/t)\\ + &=\frac{x}{t}\cdot (i(x)+i(t))+\frac{x_0}{t}\cdot (i(x_0)+i(t))\\ + &=\frac{x}{t}\cdot (\frac{I(x)}{x}+\frac{I(t)}{t})\\ + &+\frac{x_0}{t}\cdot (\frac{I(x_0)}{x_0}+\frac{I(t)}{t})\\ + &=\frac{I(x)}{t}+\frac{x\cdot I(t)}{t^2}+\frac{I(x_0)}{t}+\frac{x_0\cdot I(t)}{t^2}\\ + &=\frac{I(x)+I(x_0)}{t}+(x+x_0)\cdot \frac{I(t)}{t^2} + \end{align*} + Cette expression est légèrement différente de la précédente. Mais, le second terme est négligeable en raison de la présence de l'incertitude sur le temps au carré. On voit ainsi qu'il est nécessaire de faire attention aux ordres de grandeurs. \end{solos} \end{exos} diff --git a/Annexe-Exercices/Annexe-Exercices.tex.bak b/Annexe-Exercices/Annexe-Exercices.tex.bak index c0d465d..e26bfb8 100644 --- a/Annexe-Exercices/Annexe-Exercices.tex.bak +++ b/Annexe-Exercices/Annexe-Exercices.tex.bak @@ -2127,7 +2127,12 @@ Les exercices suivants doivent être réalisés si possible sans la théorie de \begin{equation*} a_{//}=g\cdot\sin(\alpha)=9,81\cdot \sin(15)=\SI{2,54}{\metre\per\second\squared} \end{equation*} - Ainsi, considérant que le mouvement sans frottement d'une tuile sur un plan incliné est un MRUA, on peut déterminer la vitesse parallèle au toit, en bordure de celui-ci à l'aide soit du temps de descente soit de la distance parcourue d. Considérant une hauteur de \SI{5}{\metre} et un angle de \SI{15}{\degree}, on a~: + Ainsi, considérant que le mouvement sans frottement d'une tuile sur un plan incliné est un MRUA, on peut déterminer la vitesse parallèle au toit, en bordure de celui-ci à l'aide soit du temps de descente soit de la distance parcourue d\footnote{La considération suivante permet d'obtenir la vitesse au bord du toit d'une manière très simple. En effet, pour un objet en MRUA, on sait que la vitesse de chute verticale au bout d'une hauteur h vaut \(v=\sqrt{2\cdot g\cdot h}\). Or, pour un objet en MRUA sur un plan incliné d'un angle \(\alpha\), la vitesse au bout d'une distance d vaut \(v=\sqrt{2\cdot a\cdot d}\), avec une accélération le long du plan valant \(a=g\cdot \sin(\alpha)\) et une distance \(d=h/\sin(\alpha)\). On a donc~: + \begin{align*} + v&=\sqrt{2\cdot g\cdot \sin(\alpha)\cdot \frac{h}{\sin(\alpha)}}\\ + &=\sqrt{2\cdot g\cdot h} + \end{align*} + La vitesse au bord du toit est donc la même en grandeur que celle d'un objet en chute libre verticale.}. Considérant une hauteur de \SI{5}{\metre} et un angle de \SI{15}{\degree}, on a~: \begin{equation*} d=\frac{h}{\sin(\alpha)}=\frac{5}{\sin(15^{\circ})}=\SI{19,32}{\metre} \end{equation*} @@ -2541,32 +2546,55 @@ Les valeurs des c\oe fficients de dilatation thermique sont celles du tableau \r \item La vitesse \[v=\frac{x-x_0}{t}\] \end{itemize} \begin{solos} - Pour chaque cas, on a~: + Procédons simplement au calcul de l'incertitude absolue. + Pour l'accélération centripète, on a~: \begin{align*} I(a)&=I(v^2/R)=v^2/R\cdot i(v^2/R)\\ &=\frac{v^2}{R}\cdot (2\cdot i(v) + i(R))\\ - &=\frac{v^2}{R}\cdot (2\cdot \frac{I(v)}{v} + \frac{I(R)}{R})\\ + &=\frac{v^2}{R}\cdot (2\cdot \frac{I(v)}{v} + \frac{I(R)}{R}) + \end{align*} + Pour la force de gravitation, on a~: + \begin{align*} I(F)&=I(G\cdot M\cdot m/d^2)\\ &=G\cdot M\cdot m/d^2\cdot i(G\cdot M\cdot m/d^2)\\ &=G\cdot \frac{M\cdot m}{d^2}\cdot (i(G)+i(M)+i(m)+2\cdot i(d))\\ - &=G\cdot \frac{M\cdot m}{d^2}\cdot (\frac{I(G)}{G}+\frac{I(M)}{M}+\frac{I(m)}{m}+2\cdot \frac{I(d)}{d})\\ + &=G\cdot \frac{M\cdot m}{d^2}\cdot (\frac{I(G)}{G}+\frac{I(M)}{M}+\frac{I(m)}{m}+2\cdot \frac{I(d)}{d}) + \end{align*} + Pour la position d'un MRUA, on a~: + \begin{align*} I(x)&=I(1/2\cdot a\cdot t^2 +v_0\cdot t+ x_0)\\ &=I(1/2\cdot a\cdot t^2)+I(v_0\cdot t)+I(x_0)\\ &=\frac{1}{2}\cdot a\cdot t^2\cdot (i(1/2)+i(a)+2\cdot i(t))\\ &+v_0\cdot t\cdot (i(v_0)+i(t))+I(x_0)\\ &=\frac{1}{2}\cdot a\cdot t^2\cdot (0+\frac{I(a)}{a}+2\cdot \frac{I(t)}{t})\\ - &+v_0\cdot t\cdot (\frac{I(v_0}{v_0}+\frac{I(t)}{t})+I(x_0))\\ + &+v_0\cdot t\cdot (\frac{I(v_0}{v_0}+\frac{I(t)}{t})+I(x_0)) + \end{align*} + Pour la vitesse, on a~: + \begin{align*} I(v)&=I((x-x_0)/t)=\frac{x-x_0}{t}\cdot (i(x-x_0)+i(t))\\ &=\frac{x-x_0}{t}\cdot (\frac{I(x-x_0)}{x-x_0}+\frac{I(t)}{t})\\ - &=\frac{x-x_0}{t}\cdot (\frac{I(x)+I(x_0)}{x-x_0}+\frac{I(t)}{t}) + &=\frac{x-x_0}{t}\cdot (\frac{I(x)+I(x_0)}{x-x_0}+\frac{I(t)}{t})\\ + &=\frac{I(x)+I(x_0)}{t}+(x-x_0)\cdot \frac{I(t)}{t^2} \end{align*} + Dans le dernier cas, on peut aussi procéder différemment~: + \begin{align*} + I(v)&=I((x-x_0)/t)=I(x/t-x_0/t)\\ + &=I(x/t)+I(x_0/t)\\ + &=\frac{x}{t}\cdot i(x/t)+\frac{x_0}{t}\cdot i(x_0/t)\\ + &=\frac{x}{t}\cdot (i(x)+i(t))+\frac{x_0}{t}\cdot (i(x_0)+i(t))\\ + &=\frac{x}{t}\cdot (\frac{I(x)}{x}+\frac{I(t)}{t})\\ + &+\frac{x_0}{t}\cdot (\frac{I(x_0)}{x_0}+\frac{I(t)}{t})\\ + &=\frac{I(x)}{t}+\frac{x\cdot I(t)}{t^2}+\frac{I(x_0)}{t}+\frac{x_0\cdot I(t)}{t^2}\\ + &=\frac{I(x)+I(x_0)}{t}+(x+x_0)\cdot \frac{I(t)}{t^2} + \end{align*} + Cette expression est légèrement différente de la précédente. Mais, le second terme est négligeable en raison de la présence de l'incertitude sur le temps au carré. On voit ainsi qu'il est nécessaire de faire attention aux ordres de grandeurs. \end{solos} \end{exos} \begin{exos} - À l'aide des mesures faites sur le pendule simple pour déterminer la dépendance de sa période en fonction des paramètres masse, angle et longueur, faites un graphe de la période par paramètre avec Gnuplot à l'intérieur du modèle \latex de TP. Inspirez-vous des graphes présentez dans ce modèle. En déterminant les incertitudes absolues de chaque paramètres, reportez-les sur + À l'aide des mesures faites sur le pendule simple pour déterminer la dépendance de sa période en fonction des paramètres masse, angle et longueur, faites un graphe de la période par paramètre avec Gnuplot à l'intérieur du modèle \LaTeX{} de TP. Inspirez-vous des graphes présentez dans ce modèle. 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zH67$e>`s5D;Lwx?#Ilbgqzq3$4{C=A*xG@k#+?Cgg=d~_Wr{5pO}bK$m1K%9wDC5A%Xp!mCfaKd<`_?=H@F&#(KB+sk_t!@JE7-tzqSxn7-}-T&IG{sAIxwLbs= diff --git a/SolutionsOS.tex b/SolutionsOS.tex index 8d7b40f..7c60253 100644 --- a/SolutionsOS.tex +++ b/SolutionsOS.tex @@ -721,7 +721,12 @@ \begin{equation*} a_{//}=g\cdot\sin(\alpha)=9,81\cdot \sin(15)=\SI{2,54}{\metre\per\second\squared} \end{equation*} - Ainsi, considérant que le mouvement sans frottement d'une tuile sur un plan incliné est un MRUA, on peut déterminer la vitesse parallèle au toit, en bordure de celui-ci à l'aide soit du temps de descente soit de la distance parcourue d. Considérant une hauteur de \SI{5}{\metre} et un angle de \SI{15}{\degree}, on a~: + Ainsi, considérant que le mouvement sans frottement d'une tuile sur un plan incliné est un MRUA, on peut déterminer la vitesse parallèle au toit, en bordure de celui-ci à l'aide soit du temps de descente soit de la distance parcourue d\footnote{La considération suivante permet d'obtenir la vitesse au bord du toit d'une manière très simple. En effet, pour un objet en MRUA, on sait que la vitesse de chute verticale au bout d'une hauteur h vaut \(v=\sqrt{2\cdot g\cdot h}\). Or, pour un objet en MRUA sur un plan incliné d'un angle \(\alpha\), la vitesse au bout d'une distance d vaut \(v=\sqrt{2\cdot a\cdot d}\), avec une accélération le long du plan valant \(a=g\cdot \sin(\alpha)\) et une distance \(d=h/\sin(\alpha)\). On a donc~: + \begin{align*} + v&=\sqrt{2\cdot g\cdot \sin(\alpha)\cdot \frac{h}{\sin(\alpha)}}\\ + &=\sqrt{2\cdot g\cdot h} + \end{align*} + La vitesse au bord du toit est donc la même en grandeur que celle d'un objet en chute libre verticale.}. Considérant une hauteur de \SI{5}{\metre} et un angle de \SI{15}{\degree}, on a~: \begin{equation*} d=\frac{h}{\sin(\alpha)}=\frac{5}{\sin(15^{\circ})}=\SI{19,32}{\metre} \end{equation*} @@ -867,25 +872,50 @@ \end{Solution OS} \begin{Solution OS}{24} - Pour chaque cas, on a~: + Procédons simplement au calcul de l'incertitude absolue. + + \smallskip + Pour l'accélération centripète, on a~: \begin{align*} I(a)&=I(v^2/R)=v^2/R\cdot i(v^2/R)\\ &=\frac{v^2}{R}\cdot (2\cdot i(v) + i(R))\\ - &=\frac{v^2}{R}\cdot (2\cdot \frac{I(v)}{v} + \frac{I(R)}{R})\\ + &=\frac{v^2}{R}\cdot (2\cdot \frac{I(v)}{v} + \frac{I(R)}{R}) + \end{align*} + Pour la force de gravitation, on a~: + \begin{align*} I(F)&=I(G\cdot M\cdot m/d^2)\\ &=G\cdot M\cdot m/d^2\cdot i(G\cdot M\cdot m/d^2)\\ &=G\cdot \frac{M\cdot m}{d^2}\cdot (i(G)+i(M)+i(m)+2\cdot i(d))\\ - &=G\cdot \frac{M\cdot m}{d^2}\cdot (\frac{I(G)}{G}+\frac{I(M)}{M}+\frac{I(m)}{m}+2\cdot \frac{I(d)}{d})\\ + &=G\cdot \frac{M\cdot m}{d^2}\cdot (\frac{I(G)}{G}+\frac{I(M)}{M}+\frac{I(m)}{m}+2\cdot \frac{I(d)}{d}) + \end{align*} + Pour la position d'un MRUA, on a~: + \begin{align*} I(x)&=I(1/2\cdot a\cdot t^2 +v_0\cdot t+ x_0)\\ &=I(1/2\cdot a\cdot t^2)+I(v_0\cdot t)+I(x_0)\\ &=\frac{1}{2}\cdot a\cdot t^2\cdot (i(1/2)+i(a)+2\cdot i(t))\\ &+v_0\cdot t\cdot (i(v_0)+i(t))+I(x_0)\\ &=\frac{1}{2}\cdot a\cdot t^2\cdot (0+\frac{I(a)}{a}+2\cdot \frac{I(t)}{t})\\ - &+v_0\cdot t\cdot (\frac{I(v_0}{v_0}+\frac{I(t)}{t})+I(x_0))\\ + &+v_0\cdot t\cdot (\frac{I(v_0}{v_0}+\frac{I(t)}{t})+I(x_0)) + \end{align*} + Pour la vitesse, on a~: + \begin{align*} I(v)&=I((x-x_0)/t)=\frac{x-x_0}{t}\cdot (i(x-x_0)+i(t))\\ &=\frac{x-x_0}{t}\cdot (\frac{I(x-x_0)}{x-x_0}+\frac{I(t)}{t})\\ - &=\frac{x-x_0}{t}\cdot (\frac{I(x)+I(x_0)}{x-x_0}+\frac{I(t)}{t}) + &=\frac{x-x_0}{t}\cdot (\frac{I(x)+I(x_0)}{x-x_0}+\frac{I(t)}{t})\\ + &=\frac{I(x)+I(x_0)}{t}+(x-x_0)\cdot \frac{I(t)}{t^2} \end{align*} + Dans le dernier cas, on peut aussi procéder différemment~: + \begin{align*} + I(v)&=I((x-x_0)/t)=I(x/t-x_0/t)\\ + &=I(x/t)+I(x_0/t)\\ + &=\frac{x}{t}\cdot i(x/t)+\frac{x_0}{t}\cdot i(x_0/t)\\ + &=\frac{x}{t}\cdot (i(x)+i(t))+\frac{x_0}{t}\cdot (i(x_0)+i(t))\\ + &=\frac{x}{t}\cdot (\frac{I(x)}{x}+\frac{I(t)}{t})\\ + &+\frac{x_0}{t}\cdot (\frac{I(x_0)}{x_0}+\frac{I(t)}{t})\\ + &=\frac{I(x)}{t}+\frac{x\cdot I(t)}{t^2}+\frac{I(x_0)}{t}+\frac{x_0\cdot I(t)}{t^2}\\ + &=\frac{I(x)+I(x_0)}{t}+(x+x_0)\cdot \frac{I(t)}{t^2} + \end{align*} + Cette expression est légèrement différente de la précédente. Mais, le second terme est négligeable en raison de la présence de l'incertitude sur le temps au carré. On voit ainsi qu'il est nécessaire de faire attention aux ordres de grandeurs. \end{Solution OS} \begin{Solution OS}{25}