From 27445b47bbf2b00de09cc4a5fb68df060c565830 Mon Sep 17 00:00:00 2001 From: Orestis <orestis.malaspinas@hesge.ch> Date: Wed, 26 Aug 2020 00:01:27 +0200 Subject: [PATCH] added chapter on units. work to be done on it --- 01_analyse_dimensionnelle.md | 294 ++++++++++++++++++++++++++++ Makefile | 4 +- figs/English_length_units_graph.png | Bin 0 -> 35433 bytes figs/kilogram_replica.jpg | Bin 0 -> 1171323 bytes 4 files changed, 296 insertions(+), 2 deletions(-) create mode 100644 01_analyse_dimensionnelle.md create mode 100644 figs/English_length_units_graph.png create mode 100644 figs/kilogram_replica.jpg diff --git a/01_analyse_dimensionnelle.md b/01_analyse_dimensionnelle.md new file mode 100644 index 0000000..dbbb059 --- /dev/null +++ b/01_analyse_dimensionnelle.md @@ -0,0 +1,294 @@ +# Avertissement {#avertissement .unnumbered} + +Cette tentative de polycopié contient certainement un grand nombre +d'erreurs étant donné qu'elle est développée en même temps que le cours +est donné et qu'elle en est à ses premiers mois de vie. Quand vous +trouverez des erreurs n'hésitez pas à me les communiquer et ainsi +améliorer la qualité de ce polycopié. Toute erreur trouvée concernant de +la "physique" (pas les fautes d'orthographe donc) seront récompensées +par un bonus sur la note des contrôles continus. + +# Bibliographie {#bibliographie .unnumbered} + +Il existe une multitude de bons livres de physique générale à +disposition. Ce polycopié est largement inspiré du livre de D. C. +Giancoli, *Physics: principles with applications*, 7-th edition, +Pearson, 2014. D'autres lectures possibles sont les suivantes: + +- Walker, Halliday & Resnick, Fundamentals of Physics, Wiley, 2014. + +- E. Hecht, Physique, 1999. + +Une partie de la bibliographie que je vous ai donné est en anglais. Il +existe des traductions qui sont en principe disponibles à la +bibliothèque. + +# Analyse dimensionnelle + +## Généralités + +Toues les sciences "naturelles" sont basées sur *l'observation* du monde +qui nous entoure. Mais malgré le fait qu'on ait l'impression que le +processus d'observation soit une suite simple: observation, +expérimentation, obtention de résultats, qu'on explique avec une +*théorie* (un ensemble de *lois*) cela n'est pas vraiment le cas. En +fait, de façon proche à ce qui se passe dans les arts, les sciences sont +un processus hautement créatif. En effet, lors d'une observation un +scientifique ne décrit pas tout ce qu'il voit, mais sélectionne +uniquement ce qu'il juge important pour la compréhension et +l'interprétation d'un phénomène. De plus, une fois sélectionné le +processus à observer, il convient de créer une expérience permettant de +le mesurer de façon aussi précise que possible pour pouvoir le décrire. +La mesure tient donc une place centrale dans les sciences et se complète +parfaitement avec la création de théories qui permettent l'explication +d'observations. Par ailleurs, toutes les théories ne sont pas le fruit +d'expériences (ou d'observations) mais ont souvent été le résultat de +constructions de l'esprit. Dans ce cas les expériences, viennent +confirmer (ou infirmer) les théories. En effet, une théorie physique est +supposée vraie jusqu'à ce qu'une expérience vienne l'infirmer (on ne +peut pas prouver une théorie). + +Les expériences ont donc deux fonctions principales + +* Collecter des données qui permettront la dérivation de lois physiques. +* Vérifier ou infirmer les lois physiques. + +Les lois physiques sont des outils très pratiques permettant la +prédiction *quantitative* de phénomènes (et non la "post-diction" comme +avec les expériences). Il est par exemple possible de prédire très +précisément la hauteur à laquelle il faut lancer un satellite pour qu'il +se retrouve en orbite géostationnaire (et donc connaître la quantité de +carburant nécessaire par exemple) grâce aux *lois de Newton*. Ce qui +serait certainement beaucoup plus difficile à déterminer +expérimentalement, s'il fallait faire des dizaines d'essais jusqu'à ce +que ça marche. + +Par ailleurs, beaucoup de "lois" ont des capacités prédictives mais ne +sont pas complètement générales. Par exemple, bien que les lois de +Newton marchent très très bien pour notre vie de tous les jours, +certaines applications d'usage quotidien ne fonctionneraient pas si on +s'en tenait là. En effet, le GPS requiert l'extension des lois de Newton +à la relativité générale pour pouvoir fonctionner correctement. En fait +la gravitation Newtonienne est une approximation de la relativité +générale. + +Ces approximations sont souvent le résultat de simplifications faites +dans la représentation dont ont se fait de processus physiques: les +*modèles*. Un modèle est une vision de l'esprit qui permet de réunir +plusieurs situations qui à première vue peuvent paraître non-semblables +ou à simplifier un problème afin de pouvoir le résoudre plus simplement. +Par exemple un liquide est composé d'atomes qui se déplacent. Il serait +possible (mais complètement infaisable et inutile dans presque tous les +cas) d'étudier chaque atome individuellement pour avoir une description +très détaillée du mouvement d'un fluide. Néanmoins, il est beaucoup plus +simple de faire *l'hypothèse* qu'un fluide peut être considéré comme un +objet continu. + +Les modèles permettent également un traduction en équations mathématiques +d'un phénomène naturel. Ces équations peuvent ensuite d'être résolues +à la main ou par ordinateur, chose que nous ferons durant ce cours. +Mais avant d'aller plus loin, nous allons discuter des dimensions des différentes grandeurs +physiques et leurs dimensions. + +## Unités, Système International + +Toute mesure doit être effectuée par rapport à un "standard" ou unités. +Cela n'a aucun sens de dire qu'un éléphant pèse 36, si nous ne disons +pas 36 en quelles unités. Pour chaque grandeur de multiple standards ont +été créé au cours des années qui sont devenus de plus en plus précis +avec les avancées technologiques (combien exactement mesure +$1{\mathrm{m}}$, la durée d'une seconde, etc). Dans cette section nous +allons discuter les unités des grandeurs de base de la physique. Nous +verrons en particulier le *Système International* (ou SI). + +Ne pas se mettre d'accord sur un système d'unités peut avoir des +conséquences catastrophiques. Un exemple récent est la perte du +satellite Mars Climate Orbiter (le coût de la mission était de 330 +millions de dollars) qui a explosé alors qu'il essayait de se mettre en +orbite autour de Mars, car une partie du code informatique récoltant les +donnée de la sonde donnait des résultats en unités non-SI alors que la +NASA travaillait en SI. Il y a donc eu une très grossière erreur dans le +calcul de la trajectoire à adopter pour la mise en orbite et le +satellite s'est écrasé (plus de détails ici par exemple +<https://fr.wikipedia.org/wiki/Mars_Climate_Orbiter>). + +Les unités sont définies par rapport à des grandeurs "facilement" +mesurables avec une grade précision et qui ne changent pas (ou très très +très peu) au cours du temps. + +### Longueur + +Le standard international fût établi par la France dans les années 1790. +Pour les unités de longueur est le mètre (abrégé ${\mathrm{m}}$). A +l'origine le mètre était 1/10'000'000 de la distance entre l'équateur et +un des pôles. A partir de cette mesure un étalon en platine fût forgé +(c'est quand même plus pratique à utiliser). Puis, en 1889, le mètre a +été défini comme la distance entre deux très fines encoche sur une barre +d'un alliage platine-iridium. Comme cette façon de définir le mètre +n'était pas suffisamment précise pour beaucoup d'applications, en 1960 +le mètre devint $1'650'763.73$ longueur d'onde d'une lumière émise par +le gaz krypton-86. En 1983, fût redéfini comme la distance parcourue par +la lumière en 1/299'792'458 secondes. + +Il existe d'autres unités de longueur, par exemple les britanniques +utilisent le pouce ou inch (1$\mathrm{in.}$ correspond à +0.0254${\mathrm{m}}$). Dans ce cours nous nous concentrerons +principalement sur le système SI. + +A titre de comparaison la @fig:imperial montre les relations entre les +différentes unités de longueur qui existaient dans l'empire britannique. +On y voit un très grand nombre d'unités différentes reliées entre elles par des relations +plus ou moins compliquées. La page wikipedia +<https://fr.wikipedia.org/wiki/Unit%C3%A9s_de_mesure_anglo-saxonnes> contient les unités anglosaxonnes +pour des surfaces et des volumes également. + +{#fig:imperial width=50%} + +### Temps + +La mesure du temps en SI est donnée en secondes (abrégée +${\mathrm{s}}$). Une seconde a longtemps été définie comme étant +$1/(3600\cdot 24)=1/86'000$-ème de journée solaire. La vitesse de +rotation de la terre se ralentissant légèrement d'année en année, il a +été nécessaire de raffiner de plus en plus cette définition. A présent +une seconde correspond à un processus atomique. Il s'agit du temps +nécessaire à 9'192'631'770 de la transitions entre deux états de l'atome +de césium 133. + +### Masse + +Le kilogramme (abrégé ${\mathrm{kg}}$) est la masse d'un étalon +international du kilogramme. En 1795, le kilogramme était la d'un +décimètre cube d'eau à une température de $4^\circ{\mathrm{C}}$. Puis il +a été remplacé par un étalon en platine iridié (voir Fig. {@fig:kg}). +Il s'agit de la seule unité utilisant encore un étalon, aucune "grandeur +naturelle" n'ayant pu être utilisée pour définir le kilogramme +autrement. Des copies de cet étalon ont été fabriquée et envoyées à +chaque état qui en ont fait d'autres copies officielles pour contrôler +les balances utilisées un peu partout sur les territoires. + +{#fig:kg width=50%} + +### Température + +La température mesure le degré d'échauffement d'un corps. En SI l'unité +de la température est le degré Kelvin (abrégé ${\mathrm{K}}$). Il est +défini comme le $1/273.16$-ème de la température du point triple de +l'eau (la température où les trois phases, solides, liquides, gazeuse, +de l'eau peuvent coexister en équilibre thermodynamique). Cette +température se trouve par définition à $273.16^\circ{\mathrm{K}}$ ou +encore à $0.01^\circ{\mathrm{C}}$. Nous verrons un peu plus de détails +sur la définition du zéro degrés Kelvin (ou zéro absolu) dans la suite +du cours. + +### Courant + +L'intensité du courant électrique est mesurée en Ampères (abrégé +${\mathrm{A}}$). Un ampère est l'intensité de courant constant qui +circulerait dans deux fils conducteurs infinis placés dans le vide, de +section négligeable, et placés à une distance d'un mètre l'un de l'autre +qui produirait une force de $2\cdot 10^{-7}$ Newton par longueur de +mètre. Les autres unités très utiles en électricité, l'ohm (abrégé +$\Omega$) et le volt (abrégé ${\mathrm{V}}$) se déduisent ensuite par +les fameuse formules $U=R\cdot I$ +($[{\mathrm{V}}]=[\Omega]\cdot [{\mathrm{A}}]$) et $P=U\cdot I$ +($[{\mathrm{W}}]=[{\mathrm{V}}]\cdot [{\mathrm{A}}]$). + +## Ordre de grandeur + +Souvent nous pouvons vouloir qu'une estimation rapide d'une quantité ou +simplement vouloir rapidement avoir une idée de comment "marche" un +processus. Pour ce faire plutôt que d'entrer complètement dans tous les +détails compliqués des calculs il peut être beaucoup plus simple de +fonctionner avec des ordres de grandeurs de nos quantité (en gros on +arrondit tout à l'entier ou même à la puissance de 10)[^1]. On a donc un +résultat précis "à la puissance de 10 près". + +Exercice (Volume d'un lac) # + +Calculez le volume du lac Léman sachant qu'il fait environ +$70{\mathrm{km}}$ de long pour $10{\mathrm{km}}$ de large et +$100{\mathrm{m}}$ de profondeur.[^2] + +Exercice (Hauteur d'un bâtiment) # + +Je souhaite estimer la hauteur d'un bâtiment. Supposons que mes yeux +soient à une hauteur de $1.5{\mathrm{m}}$ du sol. La seule information +connue est que quand je me place à une distance d'un écartement de bras +d'un arbre (mesurant environ $3{\mathrm{m}}$ de haut et se trouvant à +$20$ pas du bâtiment) placé entre moi et le bâtiment, l'arbre cache tout +juste le haut du bâtiment. + +Exercice (Épaisseur d'une feuille de papier) # + +Vous avez à disposition une règle (précise au millimètre) et un livre. +Estimez aussi précisément que possible et avec un minimum d'effort +l'épaisseur d'une feuille du livre. + +## Analyse dimensionnelle + +Lorsque nous parlons de dimensions d'une quantité, nous nous référons +souvent au type des unités de la quantité. Une longueur sera représentée +par $[L]$[^3], un temps par $[T]$, une masse par $[M]$, etc. Cette +notation se généralise pour toute quantité dont les quantités sont des +combinaisons (multiplication ou division) de ces unités de base. Ainsi, +une surface sera $[L^2]$, une fréquence $[1/T]$, une vitesse $[L/T]$, +une énergie $[M\cdot L^2/T^2]$, une force $[M L/T^2]$, etc. + +Exercice (Quantité de grandeur de base) # + +Écrivez les 5 types d'unités fondamentales nécessaires à la dérivation +de toutes les autres. + +L'analyse dimensionnelle peut se révéler particulièrement utile pour +vérifier si des relations font du sens ou pas. Les lois physiques +mettent en relation différentes quantités qui doivent être consistante +également du point de vue des unités. On ne peut naturellement pas +additionner des quantités qui n'ont pas les mêmes unités. Cela +reviendrait à ajouter des éléphants à des lettres, le résultat serait +alors peu clair. + +Si nous prenons comme exemple la relation +$$ +s=x_0+\frac{1}{2}v_0 t^2, +$$ +qui décrirait la position d'un objet en mouvement rectiligne uniforme, +$s$, qui partirait d'une position $x_0$, aurait une vitesse $v_0$ après +un temps $t$. Si nous effectuons l'analyse dimensionnelle de cette +relation nous avons +$$ +\begin{aligned} + [L]&\stackrel{?}{=}[L]+[L/T]\cdot [T^2],\nonumber\\ + &\neq[L]+[L\cdot T]. +\end{aligned} +$$ +On constate donc que cette équation +est certainement fausse. On note aussi que le $1/2$ n'ayant pas d'unités +a été simplement ignoré dans la relation ci-dessus, car il n'est pas +porteur d'unités. + +Si le résultat de l'analyse dimensionnelle se révèle incohérent, nous +sommes certains que l'équation est fausse. L'inverse est cependant faux. +En effet, une analyse dimensionnelle d'une équation cohérente ne permet +pas d'être sûr que l'équation en elle-même est correcte. Par exemple +tous les facteurs numériques peuvent être complètement faux. Ou alors +certaines quantités peuvent avoir les bonnes unités mais n'avoir aucun +sens physique dans les cas étudiés. + +Exercice (Analyse dimensionnelle) # + +Essayez de deviner les relations entre les quantités suivantes à partir +de leurs dimensions + +1. Distance et vitesse. + +2. Accélération et vitesse. + +3. Distance et accélération. + +4. Énergie et vitesse. + +5. 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