{ "cells": [ { "cell_type": "markdown", "id": "e2dbdbb0", "metadata": {}, "source": [ "# Incertitude type composée pour la mesure d'une distance le long du banc d'optique." ] }, { "cell_type": "markdown", "id": "7111e6ac", "metadata": {}, "source": [ "Pour exploiter la méthode d'autocollimation, on doit estimer la distance séparant l'objet de la lentille, le long d'un banc d'optique gradué.\n", "\n", "Si on note $X_{objet}$ la position de l'objet et $X_{L}$ la position de la lentille le long du banc d'optique, la distance focale $f'$ de la lentille est évaluée avec la relation : $f'=X_{L}-X_{objet}$\n", "\n", "Pour obtenir une évaluation de $f'$, on doit donc déterminer la valeur mesurée $f'_{mes}$ de $f'$ ainsi que l'incertitude type $u(f')$ à partir des valeurs mesurées directement pour $X_{objet}$ la position de l'objet obtenue avec l'incertitude type $u\\left(X_{objet}\\right)$ et $X_{L}$ la position de la lentille connue avec l'incertitude type $u\\left(X_{L}\\right)$.\n", "\n", "Dans cette situation où on évalue une grandeur à partir de la combinaison de plusieurs mesures, on dit qu'il faut réaliser la compisition des incertitudes pour obtenir l'incertitude type sur la grandeur. Pour la formule de type somme-différence qui fait le lien entre $f'$, $X_{objet}$ et $X_{L}$, la formule théorique de composition des incertitudes est la suivante :\n", "\n", "estimation de la différence :$f'=X_{objet}-X_{L}$ et estimation de l'incertitude : $u^2 (f')=u^2\\left(X_{objet}\\right)+u^2\\left(X_{L}\\right)$\n", "\n", "On se propose de la vérifier par simulation de Monte Carlo.\n" ] }, { "cell_type": "markdown", "id": "bad803d7", "metadata": {}, "source": [ "## Evaluation de la distance focale de la lentille convergente par la méthode d'autocollimation" ] }, { "cell_type": "code", "execution_count": 4, "id": "281c0e61", "metadata": {}, "outputs": [ { "data": { "image/png": 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9gCQTp789T/IcegH2rfnMrar/AB5P8uJu6Brg4XHUOSrD9EtvOealSZ6bJPT6PTSWQkdopp6TXAu8FXhVVf3vDNNX5c9rDNNz99reDhyqqtuWqOShDdNzVd1SVRuqapLea/wvVbUqPpUCHi0z6AL8EvB14CHgAPBn3fhr6O21PQU8AXyuG78E+Gzf/C3Avm7+PwAXLHdPY+73XfTeDA4Afwecs9w9DdHzYXrr6Q92l7+eoefr6B0x8m3g7cvdz7h7Bn6d3tLTQ33bXbfcPY37de57nKtZZUfL+PMDktQgl2UkqUGGuyQ1yHCXpAYZ7pLUIMNdkhpkuEtSgwx3SWrQ/wPcep1c6n/aPQAAAABJRU5ErkJggg==\n", 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\n", 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "la valeur moyenne sur les tirages réalisés est égale à 10.09704365923575\n", "l'incertitude associée à la mesure est égale à 0.31466617556376014\n", "l'incertitude théorique sur la distance focale est 0.3109126351029605\n" ] } ], "source": [ "#on commence classiquement par importer la library numpy (sous l'alias np) pour le calcul numérique\n", "#son sous module numpy.random pour effectuer les tirages aléatoires selon des lois bien controlées\n", "#et la library matplotlib.pyplot (sous l'alias pl) pour la réalisation de graphique.\n", "import numpy as np\n", "import numpy.random as rd\n", "import matplotlib.pyplot as pl\n", "\n", "#on créée une liste de 10000 tirages aléatoires uniformément répartis sur l'intervalle étudié.\n", "#ce qu'on appelle généralement simulation Monte-Carlo.\n", "N=10000 # nombre de tirage aléatoire utilisé\n", "\n", "#Liste des valeurs pour X(objet)\n", "X_O=32.0 \n", "l_XO=0.5 \n", "X_O_MC=X_O+rd.uniform(-l_XO,l_XO,N)\n", "\n", "#Liste des valeurs pour X(Lentille)\n", "X_L=42.1 \n", "l_XL=0.2 \n", "X_L_MC=X_L+rd.uniform(-l_XL,l_XL,N)\n", "\n", "#Liste des valeurs pour f'\n", "f_MC=X_L_MC-X_O_MC\n", "\n", "#on peut alors réaliser les histogrammes pour visualiser les distributions.\n", "pl.figure(1)\n", "pl.hist(X_O_MC,bins='rice')\n", "pl.figure(2)\n", "pl.hist(X_L_MC,bins='rice')\n", "pl.figure(3)\n", "pl.hist(f_MC,bins='rice')\n", "#et en demander l'affichage.\n", "pl.show()\n", "\n", "#on évalue alors la distance focale pour cette expérience\n", "moyenne_f=np.average(f_MC)\n", "print(\"la valeur moyenne sur les tirages réalisés est égale à \",moyenne_f)\n", "#on évalue alors l'inceritude type \n", "u_f=np.std(f_MC,ddof=1)\n", "#on affiche l'incertitude associée\n", "print(\"l'incertitude associée à la mesure est égale à \",u_f)\n", "\n", "#on compare alors l'incertitude à la valeur numérique théorique. \n", "#Evaluation de type B de l'incertitude\n", "u_XO=l_XO/np.sqrt(3)\n", "u_XL=l_XL/np.sqrt(3)\n", "u_f_theo=np.sqrt(u_XO**2+u_XL**2)\n", "print(\"l'incertitude théorique sur la distance focale est \",u_f_theo)" ] }, { "cell_type": "markdown", "id": "7637365f", "metadata": {}, "source": [ "**Conclusion : On retiendra qu'on peut utiliser la formule de propagation des incertitude pour évaluer l'incertitude associée à une grandeur s'exprimant à l'aide d'une relation de type somme-différence en fonction des paramètres expérimentaux directement mesurés**\n", "\n", "**Pour Y s'exprimant par la formule générale de type somme-différence $Y=a*X_{1}+b*X_{2}$, l'incertitude type composée est exprimée par la relation $u^2\\left(Y\\right)=a^2*u^2\\left(X_{1}\\right)+b^2*u^2\\left(X_{2}\\right)$**\n" ] }, { "cell_type": "markdown", "id": "488ced46", "metadata": {}, "source": [ "## Evaluation de la distance focale de la lentille divergente par la méthode d'autocollimation\n" ] }, { "cell_type": "markdown", "id": "3f9a28e3", "metadata": {}, "source": [ "Pour la lentille divergente, on a déjà mis en place la méthode d'autocolimation pour le doublet ce qui donne." ] }, { "cell_type": "code", "execution_count": 17, "id": "f1f00e5f", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "la distance focale obtenue pour le doublet est alors 22.0\n", "l'incertitude associée est 0.3109126351029605\n" ] } ], "source": [ "#pour l'objet\n", "X_Od=32.0 #position de l'objet\n", "l_XOd=0.5 #demi largeur d'intervalle\n", "u_XOd=l_XOd/np.sqrt(3)#incertitude type \n", "\n", "#pour le doublet de lentille\n", "X_Ld=54.0 #poisition du doublet\n", "l_Ld=0.2 #demi largeur d'intervalle\n", "u_Ld=l_Ld/np.sqrt(3) #incertitude type \n", "\n", "#pour la distance focale\n", "f_d=X_Ld-X_Od\n", "u_f_d=np.sqrt(u_XOd**2+u_Ld**2)\n", "\n", "print (\"la distance focale obtenue pour le doublet est alors\", f_d)\n", "print(\"l'incertitude associée est\", u_f_d)\n", "\n" ] }, { "cell_type": "markdown", "id": "4500fca2", "metadata": {}, "source": [ "On a donc obtenue une estimation de la distance focale de la lentille convergente et de la distance focale du doublet, on peut donc obtenir une estimation de la distance focale de la lentille divergente en composant les incertitudes à l'aide de la formule $V_{doublet}=V_{conv}+V_{div}$ ce qui se traduit par $f'_{div}=\\frac{f'_{doublet}*f'_{conv}}{f'_{conv}-f'_{doublet}}$ qui n'est pas une formule simple. \n", "On va devoir composer les incertitudes avec la méthode Monte Carlo.\n" ] }, { "cell_type": "code", "execution_count": 23, "id": "a3a0472e", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "la valeur moyenne sur les tirages réalisés est égale à -18.67597685318012\n", "l'incertitude associée à la mesure est égale à 0.8864607127420258\n" ] } ], "source": [ "#on a déjà créer les listes de tirages aléatoires pour la lentille convergente.\n", "#on crée maintenant les listes des positionspour le doublet\n", "X_Ld_MC=X_Ld+rd.uniform(-l_Ld,l_Ld,N)\n", "\n", "#Liste des valeurs pour la distance focale de la lentille divergente\n", "f_div_MC=(X_Ld_MC-X_O_MC)*(X_L_MC-X_O_MC)/((X_L_MC-X_O_MC)-(X_Ld_MC-X_O_MC))\n", "\n", "pl.hist(f_div_MC,bins='rice')\n", "pl.show()\n", "\n", "#on évalue alors la distance focale pour cette expérience\n", "moyenne_f_div=np.average(f_div_MC)\n", "print(\"la valeur moyenne sur les tirages réalisés est égale à \",moyenne_f_div)\n", "#on évalue alors l'inceritude type \n", "u_f_div=np.std(f_div_MC,ddof=1)\n", "#on affiche l'incertitude associée\n", "print(\"l'incertitude associée à la mesure est égale à \",u_f_div)\n" ] }, { "cell_type": "markdown", "id": "1e027311", "metadata": {}, "source": [ "On pet observer qu'on obtient bien une distance focale négative ce qui est logique puisque la lentille est divergente. On peut noter aussi l'allure de la distribution des valeurs aléatoire qui n'est plus du tout uniforme mais se centre de plus en plus sur la valeur moyenne. On peut noter enfin que l'incertitude sur cette valeur est grande." ] }, { "cell_type": "code", "execution_count": null, "id": "4e4173bc", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.10" } }, "nbformat": 4, "nbformat_minor": 5 }