From 33c40a0f08bafb4c69baa81a9dfee48be812dc68 Mon Sep 17 00:00:00 2001 From: guyotv Date: Fri, 26 Aug 2022 18:54:07 +0200 Subject: [PATCH] Corrections mineures dans l'annexe Incertitudes --- Annexe-Incertitudes/Annexe-Incertitudes.tex | 4 ++-- .../Annexe-Incertitudes.tex.bak | 22 +++++++++--------- CoursMecaniqueOS.pdf | Bin 12817205 -> 12816662 bytes CoursMecaniqueOSDF.pdf | Bin 12817205 -> 12816662 bytes 4 files changed, 13 insertions(+), 13 deletions(-) diff --git a/Annexe-Incertitudes/Annexe-Incertitudes.tex b/Annexe-Incertitudes/Annexe-Incertitudes.tex index b046e4e..620b69e 100644 --- a/Annexe-Incertitudes/Annexe-Incertitudes.tex +++ b/Annexe-Incertitudes/Annexe-Incertitudes.tex @@ -1,6 +1,6 @@ \myclearpage -\chapter{Ordre de grandeur, erreur et incertitudes} +\chapter{Erreur et incertitudes} \section{Ordre de grandeur} En physique la représentation des nombres est une chose complexe. Imaginez une mesure de la largeur d'une feuille A4 donnant le résultat de \SI{21,03544329}{\centi\metre}. Quelle en est sa précision ? On peut pour l'évaluer la représenter en millimètre, soit \SI{210,3544329}{\milli\metre}, en micromètre, soit \SI{210354,4329}{\micro\metre}, en nanomètre, soit \SI{210354432,9}{\nano\metre} ou même en Angstr\oe m, soit \SI{2103544329}{\angstrom}. @@ -195,7 +195,7 @@ Nous en resterons ici à un niveau aussi simple que possible en admettant qu'il Imaginons donc la mesure de la longueur d'une feuille A4 à l'aide d'une règle. La présentation de cette mesure est la suivante~: \[L=L_m\pm\, I(L_m)\] où, \(L\) est la grandeur, \(L_m\) sa valeur et \(I(L_m)\) son incertitude absolue. Par exemple, une mesure pourrait donner~: -\[L=\SI{29,0}\pm\, \SI{0,2}{\centi\metre}\] +\[L=\num{29,0}\pm\, \SI{0,2}{\centi\metre}\] L'origine de l'incertitude absolue\index{incertitude@incertitude!absolue} peut être de diverses nature~: \begin{enumerate} \item une estimation de la précision suite à la lecture visuelle sur la règle selon sa graduation, la distance à laquelle ou l'angle sous lequel on la regarde, \dots, diff --git a/Annexe-Incertitudes/Annexe-Incertitudes.tex.bak b/Annexe-Incertitudes/Annexe-Incertitudes.tex.bak index 2132963..620b69e 100644 --- a/Annexe-Incertitudes/Annexe-Incertitudes.tex.bak +++ b/Annexe-Incertitudes/Annexe-Incertitudes.tex.bak @@ -1,6 +1,6 @@ \myclearpage -\chapter{Ordre de grandeur, erreur et incertitudes} +\chapter{Erreur et incertitudes} \section{Ordre de grandeur} En physique la représentation des nombres est une chose complexe. Imaginez une mesure de la largeur d'une feuille A4 donnant le résultat de \SI{21,03544329}{\centi\metre}. Quelle en est sa précision ? On peut pour l'évaluer la représenter en millimètre, soit \SI{210,3544329}{\milli\metre}, en micromètre, soit \SI{210354,4329}{\micro\metre}, en nanomètre, soit \SI{210354432,9}{\nano\metre} ou même en Angstr\oe m, soit \SI{2103544329}{\angstrom}. @@ -13,7 +13,7 @@ Par ailleurs, faut-il écrire \SI{21,0}{\centi\metre}, \SI{21}{\centi\metre}, vo Ce qu'il faut garder à l'esprit, c'est que la représentation choisie peut être porteuse d'une information intéressante quant à la précision de la mesure qui est à son origine. Par exemple, entre \SI{21,0}{\centi\metre} et \SI{21}{\centi\metre}, si il n'y a aucune différence numérique, on peut considérer que la présence du zéro de la première expression signifie que sa précision est de l'ordre du millimètre, alors qu'elle est de l'ordre du centimètre dans la seconde. \subsection{Chiffres significatifs} -Si on réalise une mesure au centimètre près et qu'on l'écrit comme SI{21,0}{\centi\metre}, on ne peut dire que le chiffre zéro signifie quelque chose. On dira donc que la mesure a été faite avec deux chiffres significatifs et on s'abstiendra d'écrire le zéro. Le nombre de chiffres significatifs dépend donc de l'estimation de la qualité de la mesure, de ce qu'on nomme incertitude et qu'on présentera plus loin. Ainsi, quand cette estimation s'exprime correctement dans l'expression d'un nombre, le nombre de chiffres significatifs \index{chiffres@chiffres!significatifs} est simplement le nombre de chiffres présents, à l'exception des zéros qui pourraient se trouver à gauche du nombre. +Si on réalise une mesure au centimètre près et qu'on l'écrit comme \SI{21,0}{\centi\metre}, on ne peut dire que le chiffre zéro signifie quelque chose. On dira donc que la mesure a été faite avec deux chiffres significatifs et on s'abstiendra d'écrire le zéro. Le nombre de chiffres significatifs dépend donc de l'estimation de la qualité de la mesure, de ce qu'on nomme incertitude et qu'on présentera plus loin. Ainsi, quand cette estimation s'exprime correctement dans l'expression d'un nombre, le nombre de chiffres significatifs \index{chiffres@chiffres!significatifs} est simplement le nombre de chiffres présents, à l'exception des zéros qui pourraient se trouver à gauche du nombre. \smallskip Ainsi, écrire \SI{21}{\centi\metre} ou \SI{2,1}{\deci\metre} ne change pas le nombre de chiffres significatifs qui est ici de deux. @@ -31,10 +31,10 @@ Généralement, la notation scientifique se définit par la présence d'une mant Si on utilise la notation scientifique particulière présentée ci-dessus, l'ordre de grandeur \index{ordre@ordre!de grandeur} d'un chiffre exprimé dans une unité donnée est son exposant. \medskip -Relevez enfin qu'il ne faut pas confondre notation scientifique et notation d'ingénieur \index{notation@notation!d'ingénieur}. Cette dernière est une notation scientifique dont les exposants sont des multiples de trois. La raison en est que les préfixes des unités changent généralement leurs ordres de grandeur par trois : \SI{1e-3}{\milli\metre}, \SI{1e-6}{\micro\metre} ou \SI{1e-9}{\nano\metre}, etc. +Relevez enfin qu'il ne faut pas confondre notation scientifique et notation d'ingénieur \index{notation@notation!d'ingénieur}. Cette dernière est une notation scientifique dont les exposants sont des multiples de trois. La raison en est que les préfixes des unités changent généralement leurs ordres de grandeur par trois~: \SI{1e-3}{\milli\metre}, \SI{1e-6}{\micro\metre} ou \SI{1e-9}{\nano\metre}, etc. \section{Écart et erreur} -On peut facilement déterminer l'écart\index{ecart@écart} entre deux valeurs $a$ et $b$ par leur différence $a-b$. On peut, par exemple, mesurer la longueur $L$ des baguettes de pain vendues par un boulanger et déterminer les différents écarts entre elles. Par exemple, on pourrait avoir une série de mesures telles que celle données dans le tableau \ref{baguettepain}. +On peut facilement déterminer l'écart\index{ecart@écart} entre deux valeurs a et b par leur différence a-b. On peut, par exemple, mesurer la longueur L des baguettes de pain vendues par un boulanger et déterminer les différents écarts entre elles. Par exemple, on pourrait avoir une série de mesures telles que celle données dans le tableau \ref{baguettepain}. \begin{table}[ht] \centering @@ -66,15 +66,15 @@ On peut facilement déterminer l'écart\index{ecart@écart} entre deux valeurs $ \end{tabular} \end{table} -On voit immédiatement que le calcul des écarts pose un problème : il faut déterminer les écarts entre chaque baguettes deux par deux. On peut le faire. Mais quel sens cela a-t-il ? +On voit immédiatement que le calcul des écarts pose un problème~: il faut déterminer les écarts entre chaque baguettes deux par deux. On peut le faire. Mais quel sens cela a-t-il ? Par contre, déterminer quel est l'écart à la moyenne des baguettes est plus instructif. La moyenne vaut \SI{62}{\centi\metre} et la seconde colonne du tableau \ref{baguettepain} présente les écarts. On voit alors facilement que l'écart ne dépasse pas \SI{5}{\centi\metre}. Ce qui peut avoir de l'importance si on a faim. -Par ailleurs, si on sait que le boulanger avait décidé de faire des baguettes de \SI{60}{\centi\metre}, on peut se poser une autre question : quel est l'écart à cette valeur ? La quatrième colonne y apporte une réponse. Comme on utilise ce qu'on peut appeler une référence, on parlera d'\emph{erreur}\index{erreur@erreur}, plutôt que d'écart. Ce qui est alors intéressant, c'est qu'on voit que le boulanger à tendance à faire des baguettes trop grandes. Cela peut avoir une importance pour lui s'il a prévu un budget précis de matières premières pour des baguettes de \SI{60}{\centi\metre}. Cela permet aussi de s'interroger sur la règle utilisée par le boulanger pour estimer la longueur des baguettes. Comme ses baguettes sont trop longues, on peut penser que sa règle est aussi trop longue, ce qui peut avoir pour conséquence une mauvaise estimation de la longueur de la baguette par le boulanger. On parlera alors d'une \emph{erreur systèmatique}\index{erreur@erreur!systématique} induite par un matériel mal calibré. Ce type d'erreur se détecte par la présence d'une importante quantité de signes\index{signe@signe!des écarts} systématiquement positifs ou systématiquement négatifs dans les écarts. En effet, si la règle avait une longueur correcte, l'erreur faite par le boulanger devrait être aléatoirement répartie autour de la valeur de \SI{60}{\centi\metre} et les signes positifs et négatifs des écarts devraient être en nombres à peu près identiques. +Par ailleurs, si on sait que le boulanger avait décidé de faire des baguettes de \SI{60}{\centi\metre}, on peut se poser une autre question~: quel est l'écart à cette valeur ? La quatrième colonne y apporte une réponse. Comme on utilise ce qu'on peut appeler une référence, on parlera d'\emph{erreur}\index{erreur@erreur}, plutôt que d'écart. Ce qui est alors intéressant, c'est qu'on voit que le boulanger à tendance à faire des baguettes trop grandes. Cela peut avoir une importance pour lui s'il a prévu un budget précis de matières premières pour des baguettes de \SI{60}{\centi\metre}. Cela permet aussi de s'interroger sur la règle utilisée par le boulanger pour estimer la longueur des baguettes. Comme ses baguettes sont trop longues, on peut penser que sa règle est aussi trop longue, ce qui peut avoir pour conséquence une mauvaise estimation de la longueur de la baguette par le boulanger. On parlera alors d'une \emph{erreur systématique}\index{erreur@erreur!systématique} induite par un matériel mal calibré. Ce type d'erreur se détecte par la présence d'une importante quantité de signes\index{signe@signe!des écarts} systématiquement positifs ou systématiquement négatifs dans les écarts. En effet, si la règle avait une longueur correcte, l'erreur faite par le boulanger devrait être aléatoirement répartie autour de la valeur de \SI{60}{\centi\metre} et les signes positifs et négatifs des écarts devraient être en nombres à peu près identiques. Évidemment, la moyenne des écarts à la moyenne est nulle, cela par définition, alors que la moyenne des erreurs ne l'est pas en présence d'une erreur systématique\index{erreur@erreur!systématique}. -Figurent aussi dans le tableau \ref{baguettepain} les écarts et erreurs relatifs\index{ecart@écart!relatif}\index{erreur@erreur!relative} en pourcents. Il s'agit du rapport entre l'écart et la valeur de référence : la moyenne pour l'écart et \SI{60}{\centi\metre} pour l'erreur. Autant pour l'écart que pour l'erreur, on a donc : +Figurent aussi dans le tableau \ref{baguettepain} les écarts et erreurs relatifs\index{ecart@écart!relatif}\index{erreur@erreur!relative} en pourcents. Il s'agit du rapport entre l'écart et la valeur de référence~: la moyenne pour l'écart et \SI{60}{\centi\metre} pour l'erreur. Autant pour l'écart que pour l'erreur, on a donc~: \begin{equation} e=\frac{val-val_{ref}}{val_{ref}}\cdot 100 \end{equation} @@ -112,7 +112,7 @@ e=\frac{val-val_{ref}}{val_{ref}}\cdot 100 Ces indications sont importantes si on désire comparer la production de deux boulangers dont la longueur de la baguette de référence n'est pas la même. Considérons le tableau \ref{baguettepain2} qui décrit la production d'un boulanger dont la baguette de référence est de \SI{40}{\centi\metre}. On voit que la moyenne des écarts est nulle comme pour le boulanger précédent. Ce qui est normal en raison du choix de la valeur moyenne comme référence. On voit aussi que la moyenne des erreurs est la même et qu'il y a une grande systématique dans celle-ci, puisqu'elles sont pratiquement toutes positives. Cela signifie probablement, comme précédemment, que l'appareil de mesure à un biais, que la règle utilisée est trop longue. Par contre, on voit grâce à la dernière colonne donnant l'erreur relative que celle-ci est plus importante pour le second boulanger. Cela s'explique facilement. En effet, l'erreur moyenne est la même, mais la baguette de référence du second boulanger est plus courte (\SI{40}{\centi\metre}). Ainsi, malgré la différence de longueur de la baguette de référence, l'erreur relative permet de comparer les erreurs des deux boulangers. \bigskip -Imaginons maintenant qu'on s'intéresse à la production annuelle de baguettes d'un boulanger, soit des centaines de baguettes. Il devient difficile de les représenter dans un tableau, surtout si on fait des mesures d'une précision supérieures au centimètre. On peut alors réaliser des classes de mesures (on parle d'enclassement\index{enclassement@enclassement}) en mettant par exemple, toutes les baguettes entre \SI{42,5}{\centi\metre} et \SI{43,4}{\centi\metre} dans la classe des baguettes de \SI{43}{\centi\metre}. En procédant de la même manière pour les autres valeurs, on peut alors obtenir des mesures comme celles présentées dans le tableau \ref{enclassementbaguettes} où L est la longueur des baguettes et n le nombre de baguettes dans la classe associée à cette longueur, soit la fréquence d'apparition de la longueur. +Imaginons maintenant qu'on s'intéresse à la production annuelle de baguettes d'un boulanger, soit des centaines de baguettes. Il devient difficile de les représenter dans un tableau, surtout si on fait des mesures d'une précision supérieures au centimètre. On peut alors réaliser des classes de mesures (on parle d’enclassement\index{enclassement@enclassement}) en mettant par exemple, toutes les baguettes entre \SI{42,5}{\centi\metre} et \SI{43,4}{\centi\metre} dans la classe des baguettes de \SI{43}{\centi\metre}. En procédant de la même manière pour les autres valeurs, on peut alors obtenir des mesures comme celles présentées dans le tableau \ref{enclassementbaguettes} où L est la longueur des baguettes et n le nombre de baguettes dans la classe associée à cette longueur, soit la fréquence d'apparition de la longueur. \begin{table}[ht] \centering @@ -195,7 +195,7 @@ Nous en resterons ici à un niveau aussi simple que possible en admettant qu'il Imaginons donc la mesure de la longueur d'une feuille A4 à l'aide d'une règle. La présentation de cette mesure est la suivante~: \[L=L_m\pm\, I(L_m)\] où, \(L\) est la grandeur, \(L_m\) sa valeur et \(I(L_m)\) son incertitude absolue. Par exemple, une mesure pourrait donner~: -\[L=\SI{29,0}\pm\, \SI{0,2}{\centi\metre}\] +\[L=\num{29,0}\pm\, \SI{0,2}{\centi\metre}\] L'origine de l'incertitude absolue\index{incertitude@incertitude!absolue} peut être de diverses nature~: \begin{enumerate} \item une estimation de la précision suite à la lecture visuelle sur la règle selon sa graduation, la distance à laquelle ou l'angle sous lequel on la regarde, \dots, @@ -220,7 +220,7 @@ Mais qu'en est-il de son incertitude ? Pour la connaître, on va calculer l'ince \begin{equation*} s=l_1+l_2\;;\;l_1=l_{1m}\pm\, I(l_{1m})\;;\;l_2=l_{2m}\pm\, I(l_{2m}) \end{equation*} -alors, on peut calculer les extrèmums~: +alors, on peut calculer les extremums~: \begin{align*} s_{max}&=l_{1\,max}+l_{2\,max}\\ &=l_{1m}+I(l_{1m})+l_{2m}+I(l_{2m})\\ @@ -321,7 +321,7 @@ Ainsi, le résultat final est~: \smallskip L'expression du calcul en pourcent permet de se rendre compte que c'est la mesure de la largeur de la feuille qui est la plus imprécise (1,4\% contre 0,7\% pour la longueur) et que l'incertitude relative est relativement faible (2,1\%). -Par ailleurs, un problème se pose quant à l'expression du résultat dont l'incertitude absolue est arrondie : jusqu'où devons nous aller dans l'arrondi ? Il n'existe pas de réponse logique à ce problème. Mais avec l'incertitude absolue obtenue, on se rend compte que le plus grand chiffre affecté par celle-ci est celui des dizaines de centimètres. Comme l'incertitude absolue est supérieure à \SI{10}{\centi\metre\squared}, il est courant dans ce cas, d'utiliser le chiffre des dizaines arrondis à sa limite supérieure, soit \SI{20}{\centi\metre\squared}, pour écrire le résultat sous la forme~: +Par ailleurs, un problème se pose quant à l'expression du résultat dont l'incertitude absolue est arrondie~: jusqu'où devons nous aller dans l'arrondi ? Il n'existe pas de réponse logique à ce problème. Mais avec l'incertitude absolue obtenue, on se rend compte que le plus grand chiffre affecté par celle-ci est celui des dizaines de centimètres. 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