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

---
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 figs/kilogram_replica.jpg           | Bin 0 -> 1171323 bytes
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diff --git a/01_analyse_dimensionnelle.md b/01_analyse_dimensionnelle.md
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+# 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.
+
+![Différentes unités de longueur anglosaxonnes et les relations entre elles. Source:
+<https://upload.wikimedia.org/wikipedia/commons/c/c8/English_length_units_graph.png>](figs/English_length_units_graph.png){#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.
+
+![Une réplique de l'étalon international du kilogramme présentée à la
+cité des sciences et de l'industrie (Vilette), source:
+<https://upload.wikimedia.org/wikipedia/commons/a/ac/Prototype_kilogram_replica.JPG>](figs/kilogram_replica.jpg){#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.  Force et énergie.
\ No newline at end of file
diff --git a/Makefile b/Makefile
index fe6a707..eff6349 100644
--- a/Makefile
+++ b/Makefile
@@ -20,10 +20,10 @@ HTMLOPTIONS += --mathjax=MathJax.js
 
 all:  cours.pdf cours.html
 
-cours.pdf: 00_macros.md 01_lois_de_newton.md
+cours.pdf: 00_macros.md 01_analyse_dimensionnelle.md 01_lois_de_newton.md
 	pandoc -s $(OPTIONS) $(PDFOPTIONS) -o $@ $^ --metadata-file metadata.yaml
 
-cours.html: 00_macros.md 01_lois_de_newton.md
+cours.html: 00_macros.md 01_analyse_dimensionnelle.md 01_lois_de_newton.md
 	pandoc -s $(OPTIONS) $(HTMLOPTIONS) -o $@ $^ --metadata-file metadata.yaml
 
 deploy: all
diff --git a/figs/English_length_units_graph.png b/figs/English_length_units_graph.png
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-- 
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