decrochage.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "35c8567d-7982-4be7-8c53-ddf30c5ce4b9",
   "metadata": {},
   "source": [
    "Markdown macros\n",
    "\n",
    "$\\newcommand{\\I}{\\mathrm{i}} \n",
    "\\newcommand{\\E}{\\mathrm{e}} \n",
    "\\newcommand{\\D}{\\mathop{}\\!\\mathrm{d}}\n",
    "\\newcommand{\\bra}[1]{\\langle{#1}|}\n",
    "\\newcommand{\\ket}[1]{|{#1}\\rangle}\n",
    "\\newcommand{\\braket}[1]{\\langle{#1}\\rangle}\n",
    "\\newcommand{\\bm}[1]{\\boldsymbol{#1}}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e72f9e9b-85a0-4d52-94ac-87027a3a5393",
   "metadata": {},
   "source": [
    "# Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "653ca258-1dec-44c9-b347-4e2243d7fd9a",
   "metadata": {},
   "outputs": [],
   "source": [
    "# numpy\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f0a6d491-badc-4269-9511-af2b2f429c12",
   "metadata": {},
   "outputs": [],
   "source": [
    "# matplotlib\n",
    "import matplotlib\n",
    "import matplotlib.pylab as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "beda1165-853e-424a-906e-0419e6782bd7",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from IPython.display import Image"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "8af95e31-7c62-4b01-ad9f-f792b7fc8281",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# sympy\n",
    "import sympy as sy\n",
    "sy.init_printing(use_unicode = True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "2a12d0f6-814a-40de-9e85-687c91134bd9",
   "metadata": {},
   "outputs": [],
   "source": [
    "# figure style\n",
    "# ATTENTION: commentez l'option 'usetex' si elle ne marche pas; \n",
    "#            vous avez besoin de LaTeX pour l'utiliser\n",
    "matplotlib.rcParams['text.usetex'] = True\n",
    "#\n",
    "matplotlib.rcParams['font.family'] = 'serif'\n",
    "matplotlib.rcParams['font.weight'] = 'normal'\n",
    "matplotlib.rcParams['font.size'] = 14\n",
    "matplotlib.rcParams['axes.labelsize'] = 18\n",
    "matplotlib.rcParams['figure.figsize'] = (6.0,4.5)\n",
    "matplotlib.rcParams['lines.linewidth'] = 0.5\n",
    "matplotlib.rcParams['xtick.top'] = True\n",
    "matplotlib.rcParams['ytick.right'] = True"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5cf8600-4554-4a43-b2d8-b55661af0084",
   "metadata": {},
   "source": [
    "# Amorce de glissement: décrochage d'une échelle\n",
    "\n",
    "On étudie la chute d'une échelle; on calcule la condition de glissement au point de rotation.\n",
    "\n",
    "L'échelle est de masse $m$ et longueur $l$; le moment d'inertie (par rapport à sa base en $O$) est\n",
    "$$\n",
    "I = \\frac{ml^2}{3}.\n",
    "$$\n",
    "Le centre de masse est au milieu $l/2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "dabfe06d-7ce1-4750-a0c2-c8f004a29987",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA+gAAAPiCAYAAADsOPeCAAAJMmlDQ1BkZWZhdWx0X3JnYi5pY2MAAEiJlZVnUJNZF8fv8zzphUASQodQQ5EqJYCUEFoo0quoQOidUEVsiLgCK4qINEWQRQEXXJUia0UUC4uCAhZ0gywCyrpxFVFBWXDfGZ33HT+8/5l7z2/+c+bec8/5cAEgiINlwct7YlK6wNvJjhkYFMwE3yiMn5bC8fR0A9/VuxEArcR7ut/P+a4IEZFp/OW4uLxy+SmCdACg7GXWzEpPWeGjy0wPj//CZ1dYsFzgMt9Y4eh/eexLzr8s+pLj681dfhUKABwp+hsO/4b/c++KVDiC9NioyGymT3JUelaYIJKZttIJHpfL9BQkR8UmRH5T8P+V/B2lR2anr0RucsomQWx0TDrzfw41MjA0BF9n8cbrS48hRv9/z2dFX73kegDYcwAg+7564ZUAdO4CQPrRV09tua+UfAA67vAzBJn/eqiVDQ0IgALoQAYoAlWgCXSBETADlsAWOAAX4AF8QRDYAPggBiQCAcgCuWAHKABFYB84CKpALWgATaAVnAad4Dy4Aq6D2+AuGAaPgRBMgpdABN6BBQiCsBAZokEykBKkDulARhAbsoYcIDfIGwqCQqFoKAnKgHKhnVARVApVQXVQE/QLdA66At2EBqGH0Dg0A/0NfYQRmATTYQVYA9aH2TAHdoV94fVwNJwK58D58F64Aq6HT8Id8BX4NjwMC+GX8BwCECLCQJQRXYSNcBEPJBiJQgTIVqQQKUfqkVakG+lD7iFCZBb5gMKgaCgmShdliXJG+aH4qFTUVlQxqgp1AtWB6kXdQ42jRKjPaDJaHq2DtkDz0IHoaHQWugBdjm5Et6OvoYfRk+h3GAyGgWFhzDDOmCBMHGYzphhzGNOGuYwZxExg5rBYrAxWB2uF9cCGYdOxBdhK7EnsJewQdhL7HkfEKeGMcI64YFwSLg9XjmvGXcQN4aZwC3hxvDreAu+Bj8BvwpfgG/Dd+Dv4SfwCQYLAIlgRfAlxhB2ECkIr4RphjPCGSCSqEM2JXsRY4nZiBfEU8QZxnPiBRCVpk7ikEFIGaS/pOOky6SHpDZlM1iDbkoPJ6eS95CbyVfJT8nsxmpieGE8sQmybWLVYh9iQ2CsKnqJO4VA2UHIo5ZQzlDuUWXG8uIY4VzxMfKt4tfg58VHxOQmahKGEh0SiRLFEs8RNiWkqlqpBdaBGUPOpx6hXqRM0hKZK49L4tJ20Bto12iQdQ2fRefQ4ehH9Z/oAXSRJlTSW9JfMlqyWvCApZCAMDQaPkcAoYZxmjDA+SilIcaQipfZItUoNSc1Ly0nbSkdKF0q3SQ9Lf5RhyjjIxMvsl+mUeSKLktWW9ZLNkj0ie012Vo4uZynHlyuUOy33SB6W15b3lt8sf0y+X35OQVHBSSFFoVLhqsKsIkPRVjFOsUzxouKMEk3JWilWqUzpktILpiSTw0xgVjB7mSJleWVn5QzlOuUB5QUVloqfSp5Km8oTVYIqWzVKtUy1R1WkpqTmrpar1qL2SB2vzlaPUT+k3qc+r8HSCNDYrdGpMc2SZvFYOawW1pgmWdNGM1WzXvO+FkaLrRWvdVjrrjasbaIdo12tfUcH1jHVidU5rDO4Cr3KfFXSqvpVo7okXY5upm6L7rgeQ89NL0+vU++Vvpp+sP5+/T79zwYmBgkGDQaPDamGLoZ5ht2GfxtpG/GNqo3uryavdly9bXXX6tfGOsaRxkeMH5jQTNxNdpv0mHwyNTMVmLaazpipmYWa1ZiNsulsT3Yx+4Y52tzOfJv5efMPFqYW6RanLf6y1LWMt2y2nF7DWhO5pmHNhJWKVZhVnZXQmmkdan3UWmijbBNmU2/zzFbVNsK20XaKo8WJ45zkvLIzsBPYtdvNcy24W7iX7RF7J/tC+wEHqoOfQ5XDU0cVx2jHFkeRk4nTZqfLzmhnV+f9zqM8BR6f18QTuZi5bHHpdSW5+rhWuT5z03YTuHW7w+4u7gfcx9aqr01a2+kBPHgeBzyeeLI8Uz1/9cJ4eXpVez33NvTO9e7zofls9Gn2eedr51vi+9hP0y/Dr8ef4h/i3+Q/H2AfUBogDNQP3BJ4O0g2KDaoKxgb7B/cGDy3zmHdwXWTISYhBSEj61nrs9ff3CC7IWHDhY2UjWEbz4SiQwNCm0MXwzzC6sPmwnnhNeEiPpd/iP8ywjaiLGIm0iqyNHIqyiqqNGo62ir6QPRMjE1MecxsLDe2KvZ1nHNcbdx8vEf88filhICEtkRcYmjiuSRqUnxSb7JicnbyYIpOSkGKMNUi9WCqSOAqaEyD0tandaXTlz/F/gzNjF0Z45nWmdWZ77P8s85kS2QnZfdv0t60Z9NUjmPOT5tRm/mbe3KVc3fkjm/hbKnbCm0N39qzTXVb/rbJ7U7bT+wg7Ijf8VueQV5p3tudATu78xXyt+dP7HLa1VIgViAoGN1tubv2B9QPsT8M7Fm9p3LP58KIwltFBkXlRYvF/OJbPxr+WPHj0t6ovQMlpiVH9mH2Je0b2W+z/0SpRGlO6cQB9wMdZcyywrK3BzcevFluXF57iHAo45Cwwq2iq1Ktcl/lYlVM1XC1XXVbjXzNnpr5wxGHh47YHmmtVagtqv14NPbogzqnuo56jfryY5hjmceeN/g39P3E/qmpUbaxqPHT8aTjwhPeJ3qbzJqamuWbS1rgloyWmZMhJ+/+bP9zV6tua10bo63oFDiVcerFL6G/jJx2Pd1zhn2m9az62Zp2WnthB9SxqUPUGdMp7ArqGjzncq6n27K7/Ve9X4+fVz5ffUHyQslFwsX8i0uXci7NXU65PHsl+spEz8aex1cDr97v9eoduOZ67cZ1x+tX+zh9l25Y3Th/0+LmuVvsW523TW939Jv0t/9m8lv7gOlAxx2zO113ze92D64ZvDhkM3Tlnv296/d5928Prx0eHPEbeTAaMip8EPFg+mHCw9ePMh8tPN4+hh4rfCL+pPyp/NP637V+bxOaCi+M24/3P/N59niCP/Hyj7Q/Fifzn5Ofl08pTTVNG02fn3Gcufti3YvJlykvF2YL/pT4s+aV5quzf9n+1S8KFE2+Frxe+rv4jcyb42+N3/bMec49fZf4bmG+8L3M+xMf2B/6PgZ8nFrIWsQuVnzS+tT92fXz2FLi0tI/QiyQvpTNDAsAAAAGYktHRAD/AP8A/6C9p5MAAAAJcEhZcwAALEsAACxLAaU9lqkAAAAfdEVYdFNvZnR3YXJlAEdQTCBHaG9zdHNjcmlwdCA5LjU0LjDz2usYAAAgAElEQVR4nOzdTWhlaXrg+edM21jZGLdlahPZQw8o3IbKgF60ErohEmwaaTOEViZyM1O5VCy8GAaDIxYNeWsXgTeDYRYRyywMTUXTG6VXoVlNhTcuzSyKimmwU1RvMnpVgqEhVXjgncU9V3Gl1Mf9OOee85zz+4EISXHv0ZtFha7+ej9OVUoJAAAAoFv/XdcDAAAAAAQ6AAAA9IJABwAAgB4Q6AAAANADAh0AAAB64Le6HgAAAADdqKpqJyIOl3jKcSnl+Jbr7UbE4wWuc1ZKebHE1x0FgQ4AADBe2xGxW7+/d8djT+q328wCfeeO65wuNLqRqdwHHQAAgIiIqqqeR8TTK58+iYjPSykLR/U113kV09n31+uPcrjMoAMAABAREaWUZ1VVXZ0Bf7FMnNdO5/58ctuyeD5wSFzPTCaTp5PJZLvrcQAAAKP16srHy+xRj6qqtuvnnETEp+J8cZa498hkMnkZH/6PvD+ZTM46HhIAADAydWB/E9P96TP3F51Fr6rq5/W7+6UUTbMEM+g9MRfnEdODFd6YSQcAADatjuqre8Wv7ku/VlVVL+t3xfkKzKD3wJU4n2cmHQAA2Lj69mvfzH3qLKaz6De2SR3nuyHOV2YGvWO3xHmEmXQAAKAD9XL2+b3js33l16rj/HGI87UI9A7dEeczIh0AAOjCQofF1bdUE+cNEOgdWTDOZ0Q6AACwUfU9y+cPhtupb8F2oaqqw5h2zX4p5WST4xsigd6BJeN8RqQDAACbduMseh3nz0OcN8YhcRu2YpzPc3AcAACwETfdci0i9iLiZUR8Xs+00wAz6BvUQJxHmEkHAAA25IZbrr2M6cz5E3HeLDPoG9JQnM8zkw4AALTumluuRUQ8K6W86GI8Q2YGfQNaiPMIM+kAAMAGXHPLNVoi0FvWUpzPiHQAAGATri5lb6txRk2gt6jlOJ8R6QAAQGvqg+KeXvn0TlVVe12MZ8gEeks2FOczIh0AAGhcHedvYrrE/eqec7PoDXNIXAs2HOfzHBwHAAA0Yi7OT0spn9cf//rKw+7Xe9RpgBn0hnUY5xFm0gEAgAbMxXlExJOIi1uuvbryULPoDTKD3qCO43yemXQAAGBlVVW9jOkE4H4d5rPP70bEz+ceehbTWXTt0QAz6A3pUZxHmEkHAABWdFOcR0SUUk7i8i3XtiPi8QaHN2gCvQE9i/MZkQ4AACyljvPHcU2cz7l6y7WrJ7yzIoG+pp7G+YxIBwAAFlJV1fO4O86jlPIqIuYPhnPLtYYI9DX0PM5nRDoAAHCrqqoOY9o2+/Uy9rtcnUXvexel4JC4FSWJ83kOjgMAAL6njvOXEfHpgnEebrnWDjPoK0gY5xFm0gEAgCvm4vzJonEe4ZZrbTGDvqSkcT7PTDoAAHA1zq/G9iLPd8u1hplBX8IA4jzCTDoAAIxeVVVPYxrnr1aJ8wi3XGuDQF/QQOJ8RqQDAMAIVVW1W1XVm4h4Xn9q3dnuq89/Wu9PZwWWuC9gYHE+z3J3AAAYsHoZ+l5MZ7f3YjpZN2+2l/zVoge81dd8fMP1IqadMZtZPymlXD3xnRsI9DsMOM5nRDoAAAxUPVu+yD3KX5RSnjV8zYiIs1LKHyz42NET6LcYQZzPiHQAAICO2YN+gxHFeYQ96QAAAJ0T6NcYWZzPiHQAAIAOCfQrRhrnMyIdAACgIwJ9zsjjfEakAwAAdECg18T5JSIdAABgwwR6iPMbiHQAAIANGn2gi/NbiXQAAIANGXWgi/OFiHQAAIANGG2gi/OliHQAAICWjTLQxflKRDoAAECLRhfo4nwtIh0AAKAlowp0cd4IkQ4AANCC0QS6OG+USAcAgJGpqup5VVVvqqrSAS2pSildj6F14rw1JxGxP5lMzroeCAAA0J46yn9df/islPKiy/EM1eBn0MV5q8ykAwDAOOzNvb/b2SgGbtCBLs43QqQDAMDwzUf5485GMXCDDXRxvlEiHQAAhu1SlFdVtXfTA1ndIANdnHdCpAMAwABVVbUTETtXPi3QWzC4QBfnnRLpAAAwPNfFuEBvwaACXZz3gkgHAIBhue5QuF23W2veYAJdnPeKSAcAgOG46VA4s+gNG0Sgi/NeEukAAJBcVVW7EXHTz/QCvWHpA12c95pIBwCA3G6LcIHesNSBLs5TEOkAAJDXdfvPZ3bqE95pSNpAF+epiHQAAMjppv3nM2bRG5Qy0MV5SiIdAAASqapqkfgW6A1KF+jiPDWRDgAAeQj0DUsV6OJ8EEQ6AADksEh8b9cnvdOANIEuzgdFpAMAQI9VVbUdtx8QN88sekNSBLo4HySRDgAA/bVMdAv0hvQ+0MX5oIl0AADoJ4HegV4HujgfBZEOAAD9s1R0L3jiO3fobaCL81ER6QAA0BNVVe1ExM6STxPoDehloIvzURLpAADQD6vEtkBvQO8CXZyPmkgHAIDurRLbu/XJ76yhV4EuzgmRDgAAXVt1Nvxxo6MYod4EujhnjkgHAIAOVFW1GxGr/hy+6H3TuUEvAl2ccw2RDgAAm7fOXnL70NfUeaCLc24h0gEAYLPWieyd+gR4VtRpoItzFiDSAQBgc9adBbcPfQ2dBbo4ZwkiHQAAWlZVVRNxbR/6GjoJdHHOCkQ6AAC0q4m4tg99DRsPdHHOGkQ6AAC0p4m43q5PgmcFGw10cU4DRDoAALSjqQPe/Ky+oqqUspEvJM5p2ElE7E8mk7OuBwIAAENQz3zftg/9af3nf4iI/3LDY85KKS8aHdiIbCTQxTktEekAALAhVVXN4nG/lHLc6WAGqvUl7uKcFlnuDgAADEargS7O2QCRDgAADEJrgS7O2SCRDgAApNdKoItzOiDSAQCA1BoPdHFOh0Q6AACQVqOBLs7pAZEOAACk1Figi3N6RKQDAADpNBLo4pweEukAAEAqawe6OKfHRDoAAJDGWoEuzklApAMAACm0dh906BGRDgAA9N66M+hPIuJVQ2OBNol0AACg15rYgy7SyUKkAwAAvdXUKe4inSxEOgAA0EtN3gddpJOFSAcAAHqn0UPiRDqJiHQAAKBXGj/FXaSTiEgHAAB6o5XbrIl0EhHpAABAL7R2H3SRTiIiHQAA6FxrgR4h0klFpAMAAJ1qNdAjRDqpiHQAAKAzrQd6hEgnFZEOAAB0YiOBHiHSSUWkAwAAG7exQI8Q6aQi0gEAgI3aaKBHiHRSEekAAMDGbDzQI0Q6qYh0AABgIzoJ9AiRTioiHQAAaF1ngR4h0klFpAMAAK3qNNAjRDqpiHQAAKA1nQd6hEgnFZEOAAC0oheBHiHSSUWkAwAAjetNoEeIdFIR6QAAQKN6FegRIp1URDoAANCY3gV6hEgnFZEOAAA0opeBHiHSSUWkAwAAa+ttoEeIdFIR6QAAwFp6HegRIp1URDoAALCy3gd6hEgnFZEOAACsJEWgR4h0UhHpAADA0tIEeoRIJxWRDgAALCVVoEeIdFIR6QAAwMLSBXqESCcVkQ4AACwkZaBHiHRSEekAAMCd0gZ6hEgnFZEOAADcKnWgR4h0UhHpAADAjdIHeoRIJxWRDgAAXGsQgR4h0klFpAMAAN8zmECPEOmkItIBAIBLBhXoESKdVEQ6AABwYXCBHiHSSUWkAwAAETHQQI8Q6aQi0gEAgOEGeoRIJxWRDgAAIzfoQI8Q6aQi0gEAYMQGH+gRIp1URDoAAIzUKAI9QqSTikgHAIARGk2gR4h0UhHpAAAwMqMK9AiRTioiHQAARmR0gR4h0klFpAMAwEiMMtAjRDqpiHQAABiB0QZ6hEgnFZEOAAADN+pAjxDppCLSAQBgwEYf6BEinVREOgAADJRAr4l0EhHpAAAwQAJ9jkgnEZEOAAADI9CvEOkkItIBAGBABPo1RDqJiHQAABgIgX4DkU4iIh0AAAZAoN9CpJOISAcAgOQE+h1EOomIdAAASEygL0Ckk4hIBwCApAT6gkQ6iYh0AABISKAvQaSTiEgHAIBkBPqSRDqJiHQAAEhEoK9ApJOISAcAgCQE+opEOomIdAAASECgr0Gkk4hIBwCAnhPoaxLpJCLSAQCgxwR6A0Q6iYh0AADoKYHeEJFOIiIdAAB6SKA3SKSTiEgHAICeEegNE+kkItIBAKBHBHoLRDqJiHQAAOgJgd4SkU4iIh0AAHpAoLdIpJOISAcAgI4J9JaJdBIR6QAA0CGBvgEinUREOgAAdESgb4hIJxGRDgAAHRDoGyTSSUSkAwDAhgn0DRPpJCLSAQBggwR6B0Q6iYh0AADYEIHeEZFOIiIdAAA2QKB3SKSTiEgHAICWCfSOiXQSEekAANAigd4DIp1ERDoAALREoPeESCcRkQ4AAC0Q6D0i0klEpAMAQMMEes+IdBIR6QAA0CCB3kMinUREOgAANESg95RIJxGRDgAADRDoPSbSSUSkAwDAmgR6z4l0EhHpAACwBoGegEgnEZEOAAArEuhJiHQSEekAALACgZ6ISCcRkQ4AAEsS6MmIdBIR6QAAsASBnpBIJxGRDgAACxLoSYl0EhHpAACwAIGemEgnEZEOAAB3EOjJiXQSEekAAHALgT4AIp1ERDoAANxAoA+ESCcRkQ4AANcQ6AMi0klEpAMAwBUCfWBEOomIdAAAmCPQB0ikk4hIBwCAmkAfKJFOIiIdAABCoA+aSCcRkQ4AwOgJ9IET6SQi0gEAGDWBPgIinUREOgAAoyXQR0Kkk4hIBwBglAT6iIh0EhHpAACMjkAfGZFOIiIdAIBREegjJNJJRKQDADAaAn2kRDqJiHQAAEZBoI+YSCcRkQ4AwOAJ9JET6SQi0gEAGDSBjkgnE5EOAMBgCXQiQqSTikgHAGCQBDoXRDqJiHQAAAZHoHOJSCcRkQ4AwKAIdL5HpJOISAcAYDAEOtcS6SQi0gEAGASBzo1EOomIdAAA0hPo3Eqkk4hIBwAgNYHOnUQ6iYh0AADSEugsRKSTiEgHACAlgc7CRDqJiHQAANIR6CxFpJOISAcAIBWBztJEOomIdAAA0hDorESkk4hIBwAgBYHOykQ6iYh0AAB6T6CzFpFOIiIdAIBeE+isTaSTiEgHAKC3BDqNEOkkItIBAOglgU5jRDqJiHQAAHpHoNMokU4iIh0AgF4R6DROpJOISAcAoDcEOq0Q6SQi0gEA6AWBTmtEOomIdAAAOifQaZVIJxGRDgBApwQ6rRPpJCLSAQDojEBnI0Q6iYh0AAA6IdDZGJFOIiIdAICNE+hslEgnEZEOAMBGCXQ2TqSTiEgHAGBjBDqdEOkkItIBANgIgU5nRDqJiHQAAFon0OmUSCcRkQ4AQKsEOp0T6SQi0gEAaI1ApxdEOomIdAAAWiHQ6Q2RTiIiHQCAxgl0ekWkk4hIBwCgUQKd3hHpJCLSAQBojECnl0Q6iYh0AAAaIdDpLZFOIiIdAIC1CXR6TaSTiEgHAGAtAp3eE+kkItIBAFiZQCcFkU4iIh0AgJUIdNIQ6SQi0gEAWJpAJxWRTiIiHQCApQh00hHpJCLSAQBYmEAnJZFOIiIdAICFCHTSEukkItIBALiTQCc1kU4iIh0AgFsJdNIT6SQi0gEAuJFAZxBEOomIdAAAriXQGQyRTiIiHQCA7xHoDIpIJxGRDgDAJQKdwRHpJCLSAQC4INAZJJFOIiIdAICIEOgMmEgnEZEOAIBAZ9hEOomIdACAkRPoDJ5IJxGRDgAwYgKdURDpJCLSAQBGSqAzGiKdREQ6AMAICXRGRaSTiEgHABgZgc7oiHQSEekAACMi0BklkU4iIh0AYCQEOqMl0klEpAMAjIBAZ9REOomIdACAgRPojJ5IJxGRDgAwYAIdQqSTikgHABgogQ41kU4iIh0AYIAEOswR6SQi0gEABkagwxUinUREOgDAgAh0uIZIJxGRDgAwEAIdbiDSSUSkAwAMgECHW4h0EhHpAADJCXS4g0gnEZEOAJCYQIcFiHQSEekAAEkJdFiQSCcRkQ4AkJBAhyWIdBIR6QAAyQh0WJJIJxGRDgCQiECHFYh0EhHpAABJCHRYkUgnEZEOAJCAQIc1iHQSEekAAD0n0GFNIp1ERDoAQI8JdGiASCcRkQ4A0FMCHRoi0klEpAMA9JBAhwaJdBIR6QAAPSPQoWEinUREOgBAjwh0aIFIJxGRDgDQEwIdWiLSSUSkAwD0gECHFol0EhHpAAAdE+jQMpFOIiIdAKBDVSml6zHAKEwmk5cRcdj1OIbo9PQ0Tk5O4t27dxERsbW1FQ8ePIi9vb3Y2tq6eNzZ2Vm8ffs2Tk5OLj3u4cOHsb2tSeecRMT+ZDI563ogAEB/VFU1i8f9Uspxp4MZKIEOGyTSm3d8fBzv3r2Lvb29+OSTTyJiGuyvX7+OiIjDw8PY3t6O9+/fx+vXr2N3dzcePnwYEXHxue++++7icVwQ6QDAJQK9fZa4wwZZ7t6sr7/+Os7Pz+Pw8PAiziMidnZ2Ynd3N87Pzy8e89VXX8WjR48u4nz2ubOzszg/P7+YVeeC5e4AABsm0GHDRHozTk5O4ttvv41Hjx5dWsY+s7OzExHT2fSvvvoqdnd3Lz43e/75+fnFxx999FH7g85HpAMAbJBAhw6I9PWcn5/H8fFxHBwcLPT49+/fx+7u7qXPzUf9vXv3vvf3XBDpAAAbItChIyJ9dT/72c/iwYMHce/evRsfc3b2Yev0zs7O9/aX7+7uxuPHj+PRo0fxxRdfXDsLzwWRDgCwAQIdOiTSV3NycnJpz/l15gP9ppD/5JNPYnd3V5wvRqQDALRMoEPHRPryHjx4cGk/+XVOT08v3r/rsSxMpAMAtEigQw+I9OU8evTo1r8/Pz+P9+/fR8R0r7lAb5RIBwBoiUCHnhDpzTF73jqRDgDQAoEOPSLSm/Htt99evP/xxx93OJJBE+kAAA0T6NAzIn19ZtA3RqQDADRIoEMPifTVXd1/ftut2GiESAcAaIhAh54S6atZdvZ8FvOsRaQDADRAoEOPifTlLbP//P3793F0dNT2kMZCpAMArEmgQ8+J9OXMz6Dftbz95OTEHvVmiXQAgDUIdEhApH9wfHwcL168iJ/85Cdxfn5+6e/Ozs4uLVm/Lb7Pzs7il7/8Zezu7rY21pES6QAAKxLokIRIj3j79m28ffs2zs/P4/T09NJy9ojpjPiijo+PY3d3N7a3dWQLRDoAwAoEOiQy9kifX74eEZfi+uzsLN69exePHz++9LnrHB8fx9nZWezt7bUzUCJEOgDA0gQ6JDPmSJ8tWd/e3o7Hjx9fBPrZ2Vm8fv06Hj58GJ988kk8fPgwIqYz7vPOz8/j66+/jtPT0/jiiy82O/hxEukAAEuoSildjwFYwWQyeRkRh12PY9OOj4/j5OQkHjx4EFtbWxfL3R8+fHhpP/nx8XG8ffs2dnZ24t69e3F+fn6x5/yzzz6Lra2tDv8rRuckIvYnk8n1SxoAgBSqqprF434p5bjTwQyUQIfExhrpZ2dncXp6GmdnZ7G9vX0R63c9bmdnx57z7oh0AEhOoLdPoENyY410UhLpAJCYQG+fPeiQ3Jj3pJOOPekAALcQ6DAAIp1ERDoAwA0EOgyESCcRkQ4AcA2BDgMi0klEpAMAXCHQYWBEOomIdACAOQIdBkikk4hIBwCoCXQYKJFOIiIdACAEOgyaSCcRkQ4AjJ5Ah4ET6SQi0gGAURPoMAIinUREOgAwWgIdRkKkk4hIBwBGSaDDiIh0EhHpAMDoCHQYGZFOIiIdABgVgQ4jJNJJRKQDAKMh0GGkRDqJiHQAYBQEOoyYSCcRkQ4ADJ5Ah5ET6SQi0gGAQRPogEgnE5EOAAyWQAciQqSTikgHAAZJoAMXRDqJiHQAYHAEOnCJSCcRkQ4ADIpAB75HpJOISAcABkOgA9cS6SQi0gGAQRDowI1EOomIdAAgPYEO3Eqkk4hIBwBSE+jAnUQ6iYh0ACAtgQ4sRKSTiEgHAFIS6MDCRDqJiHQAIB2BDixFpJOISAcAUhHowNJEOomIdAAgDYEOrESkk4hIBwBSEOjAykQ6iYh0AKD3BDqwFpFOIiIdAOg1gQ6sTaSTiEgHAHpLoAONEOkkItIBgF4S6EBjRDqJiHQAoHcEOtAokU4iIh0A6BWBDjROpJOISAcAekOgA60Q6SQi0gGAXhDoQGtEOomIdACgcwIdaJVIJxGRDgB0SqADrRPpJCLSAYDOCHRgI0Q6iYh0AKATAh3YGJFOIiIdANg4gQ5slEgnEZEOAGyUQAc2TqSTiEgHADZGoAOdEOkkItIBgI0Q6EBnRDqJiHQAoHUCHeiUSCcRkQ4AtEqgA50T6SQi0gGA1gh0oBdEOomIdACgFQId6A2RTiIiHQBonEAHekWkk4hIBwAaJdCB3hHpJCLSAYDGCHSgl0Q6iYh0AKARAh3oLZFOIiIdAFibQAd6TaSTiEgHANYi0IHeE+kkItIBgJUJdCAFkU4iIh0AWIlAB9IQ6SQi0gGApQl0IBWRTiIiHQBYikAH0hHpJCLSAYCFCXQgJZFOIiIdAFiIQAfSEukkItIBgDsJdCA1kU4iIh0AuJVAB9IT6SQi0gGAGwl0YBBEOomIdADgWgIdGAyRTiIiHQD4HoEODIpIJxGRDgBcItCBwRHpJCLSAYALAh0YJJFOIiIdAIgIgQ4MmEgnEZEOAAh0YNhEOomIdAAYOYEODJ5IJxGRDgAjJtCBURDpJCLSAWCkBDowGiKdREQ6AIyQQAdGRaSTiEgHgJER6MDoiHQSEekAMCICHRglkU4iIh0ARkKgA6Ml0klEpAPACAh0YNREOomIdAAYOIEOjJ5IJxGRDgADJtABQqSTikgHgIES6AA1kU4iIh0ABkigA8wR6SQi0gFgYAQ6wBUinUREOgAMiEAHuIZIJxGRDgADIdABbiDSSUSkA8AACHSAW4h0EhHpAJCcQAe4g0gnEZEOAIkJdIAFiHQSEekAkJRAB1iQSCcRkQ4ACQl0gCWIdBIR6QCQjEAHWJJIJxGRDgCJCHSAFYh0EhHpAJCEQAdYkUgnEZEOAAkIdIA1iHQSEekA0HNVKaXrMQCkN5lMXkbEYdfjgAWcRMT+ZDI5W+ciVVVtxzT69665/nEp5cbrV1W1GxHbpZTjdcYAwGZVVTWLx33fw9thBh2gAWbSSWTlmfSqqnaqqnpaVdXPI+LXEfEmIp7G9JdTs1j/aUT8uqqql1VV7Vx3jfp5u2v8NwDAIAl0gIaIdBJZKtLrMH8ZEd9ExPOI2ImIFxHxeSmlKqX8QSllv5TyaSmliojP68f8vKqqvbnrbMc04LcjwswLAFxhiTtAwyx3J5Fbl7vXQf20fouIOIuIZxHx+rYl7HPPfxrToP+0lHJSR/5hRJyVUv6gif8AADbHEvf2mUEHaJiZdBK5cSa93ic+W8IeMZ0xv19KebVInEdElFJe1M97U1XV8/jwiys/1AHANQQ6QAtEOol8L9KrqjqMD/vEzyLiSSnl2aJhPq+U8qy+xtO5T5+sN2QAGCaBDtASkU4iF5Fex/nLmO4TP4vpMsZ1/3989flm0AHgGgIdoEUinUR2//Zv//b/jmmcR3yI8yZmu1/PvX/W0DUBYHAEOkDLRDoZvHv3Lt68efM/zH3qWVMhXUo5jYjT+kOz5wBwA4EOsAEinT57//59HB0dXXz8h3/4h2dffvnl61uesopZoJs9B4AbCHSADRHp9NXR0VGcn59HRMTW1lb86Z/+6XYscZ/0Bc0OmDODDgA3EOgAGyTS6Zvj4+N4//79xce7u7uxtbUVccst2NZg/zkA3EKgA2yYSKcvzs/P4+TkQy9vbW3FZ599Nv+QJiN9O8yeA8CtBDpAB0Q6ffCzn/3sYml7xKXZ83lNRfpZ2H8OALcS6AAdEel06ersecQ00G+wdqSXUj4vpbxY9fkAMAYCHaBDIp2u/PKXv7w0e37v3r3Y3r61v9vYkw4AzBHoAB0T6XTh9PT00scPHjxY5GkiHQBaJNABekCks2nv3r279PHOzs6iTxXpANASgQ7QEyKdTbk6e761tRX37t1b5hIiHQBaINABekSkswnz9z2PiPj4449XuYxIB4CGCXSAnhHptO2777679PGSs+fzRDoANEigA/SQSCeD8/PzePHixe6Pf/zjX1dVVaqq+nVVVW9uePt5/ZjnXY8bAPrqt7oeAADXm0wmTyaTSUTEYcdDYeDuuL3arWb3Tv/Nb37zX3/xi1/8h9/85jf/PCIeX3nYaUS8jojj+k8A4BpVKaXrMQBwi8lk8jJEOg06Pj6Ot2/fXnz8ox/9aJlT3G9zEhH7P/7xj38eEbMLnkXE/VLKWRNfAIDuVFU1i8f9Uspxp4MZKEvcAXrOcnea9tFHH136+OqhcWvYjYg38SHOIyKOxTkALEagAyQg0mnS1UPhrh4at47T09PdK586aeziADBwAh0gCZFOU3Z2dmJra+vi47Oz5ia4r95jPab7zgGABQh0gEREOk158ODBxfvXRPXK5pfL/87v/M7/9+WXXzZ3cQAYOIEOkIxIpwkPHz68eP/8/DxOTtZfiX5+fn4p9u/fv/9b4T7pALAwgQ6QkEhnXdvb25ciff5U91X97Gc/u/Txxx9/HFEfHCfSAeBuAh0gKZHOuj777LOLA+POzs7i66+/Xvla79+/j5OTk0t72+du3SbSAWABAh0gMZHOOra2tuKLL764iPSTk5OVIv3du3fx1VdfxYMHDy72tm9tbV09LV6kA8AdBDpAciKddcwifTbbfXJyEn/1V3+10J7009PT+MlPfhJHR0ext7cXjx49ik8++SQiLs2ezxPpAHCLqpTS9RgAaMBkMnkZEYddj4O8Tk5O4u3btxe3Xdva2oqdnZ3Y3r7c02dnZ/H+/fv47rvv4sGDB/Hw4cNLjzk+Po6PPvro0h73q18qIvYnk0lz93cDoHVVVc3icb+U4jaaLRDoAAMi0mnCu3fv4ttvv43379/Ht2KmvZEAACAASURBVN9+G+fn5xExDfaPP/44tre34969e/HgwYNLe86XJNIBkhHo7RPoAAMj0klEpAMkItDbZw86wMDYk04i9qQDwByBDjBAIp1ERDoA1AQ6wECJdBIR6QAQAh1g0EQ6iYh0AEZPoAMMnEgnEZEOwKgJdIAREOkkItIBGC2BDjASIp1ERDoAoyTQAUZEpJOISAdgdAQ6wMiIdBIR6QCMikAHGCGRTiIiHYDREOgAIyXSSUSkAzAKAh1gxEQ6iYh0AAZPoAOMnEgnEZEOwKAJdABEOpmIdAAGqyqldD0GAHpiMpm8jIjDrscxJu/evYtvv/023r9/HxER9+7di48//jh2dnZia2ur49H12klE7E8mk7OuBwIwFlVVzeJxv5Ry3OlgBkqgA3CJSN+M4+PjePv27a2PefjwYXz22WdC/WYiHWCDBHr7BDoA3yPS2/P+/fs4Ojq6mDG/y/b2djx69Ch2dnZaHllaIh1gQwR6+wQ6ANcS6c179+5dvH79eqXnPnr0KHZ3dxse0WCIdIANEOjtc0gcANdycFyzzs/P4+joaOXnHx8fx9mZ/ryBg+MAGASBDsCNRHpzjo6O4vz8fOXnn5+fx9dff93giAZHpAOQnkAH4FYifX2np6fx7t27Rq5zcnLSwIgGS6QDkJpAB+BOIn09p6envbzWQIl0ANIS6AAsRKSvronZ8xmBvhCRDkBKAh2AhYn01SxyuNveD38Yj/7Vv7rzcefn5wvfom3kRDoA6Qh0AJYi0tvx8P792P0X/yJ2fvCDOx/73XffbWBEgyDSAUhFoAOwNJHerY8++qjrIWQi0gFIQ6ADsBKRvrh79+41dq2tra1GrzcSIh2AFAQ6ACsT6YvZ2dlp7FoPHjxo7FojI9IB6D2BDsBaRPrdmgz0Jq81QiIdgF4T6ACsTaTfbmdnJx4+fLj2dT755JP45JNPGhjRqIl0AHpLoAPQCJF+u88++yy2t1dvwq2trTg4OGhwRKMm0gHoJYEOQGNE+s22trbi0aNHsbW1tdLzDw4OVn4u1xLpAPSOQAegUSL9Zjs7O3F4eLjUPvJ79+7F4eGhpe3tEOkA9IpAB6BxIv1m29vb8aMf/Sj29vZunRHf2tqKhw8fxuHhoduqtUukA9AbVSml6zEAMFCTyeRlRBx2PY4+e//+fZyensbDOtb/r//23+LeH/2RKN+8k4jYn0wmZ10PBKCvqqqaxeN+KeW408EMlEAHoFUifUEnJ9M//+W/jPi93+t2LOMl0gFuIdDbZ4k7AK2y3J1ELHcHoFMCHYDWiXQSEekAdEagA7ARIp1ERDoAnRDoAGyMSCcRkQ7Axgl0ADZKpJOISAdgowQ6ABsn0klEpAOwMQIdgE6IdBIR6QBshEAHoDMinUREOgCtE+gAdEqkk4hIB6BVAh2Azol0EhHpALRGoAPQCyKdREQ6AK0Q6AD0hkgnEZEOQOMEOgC9ItJJRKQD0CiBDkDviHQSEekANEagA9BLIp1ERDoAjRDoAPSWSCcRkQ7A2gQ6AL0m0klEpAOwFoEOQO+JdBIR6QCsTKADkIJIJxGRDsBKBDoAaYh0EhHpACxNoAOQikgnEZEOwFIEOgDpiHQSEekALEygA5CSSCcRkQ7AQgQ6AGmJdBIR6QDcSaADkJpIJxGRDsCtBDoA6Yl0EhHpANxIoAMwCCKdREQ6ANcS6AAMhkgnEZEOwPcIdAAGRaSTiEgH4BKBDsDgiHQSEekAXBDoAAzSZDJ58g//8A//T9fjgAWIdAAiQqADMFBVVb3867/+6x+enJx0PRRYhEgHQKADMDxVVb2MiMOIiK+//vpVWO5ODiIdYOQEOgCDMh/nEfGqlPLEnnQSEekAIybQARiM6+J89ncinUREOsBICXQABuG2OJ8R6SQi0gFGSKADkN4icT4j0klEpAOMjEAHILVl4nxGpJOISAcYEYEOQFqrxPmMSCcRkQ4wEgIdgJTWifMZkU4iIh1gBAQ6AOk0EeczIp1ERDrAwAl0AFJpMs5nRDqJiHSAARPoAKTRRpzPiHQSEekAAyXQAUihzTifEekkItIBBkigA9B7m4jzGZFOIiIdYGAEOgC9tsk4nxHpJCLSAQZEoAPQW13E+YxIJxGRDjAQAh2AXuoyzmdEOomIdIABEOgA9E4f4nxGpJOISAdITqAD0Ct9ivMZkU4iIh0gMYEOQG/0Mc5nRDqJiHSApAQ6AL3Q5zifEekkItIBEhLoAHQuQ5zPiHQSEekAyQh0ADqVKc5nRDqJiHSARAQ6AJ3JGOczIp1ERDpAEgIdgE5kjvMZkU4iIh0gAYEOwMYNIc5nRDqJiHSAnhPoAGzUkOJ8RqSTiEgH6DGBDsDGDDHOZ0Q6iYh0gJ4S6ABsxJDjfEakk4hIB+ghgQ5A68YQ5zMinUREOkDPCHQAWjWmOJ8R6SQi0gF6RKAD0JoxxvmMSCcRkQ7QEwIdgFaMOc5nRDqJiHSAHhDoADROnH8g0klEpAN0TKAD0Chx/n0inUREOkCHBDoAjRHnNxPpJCLSAToi0AFohDi/m0gnEZEO0AGBDsDaxPniRDqJiHSADRPoAKxFnC9PpJOISAfYIIEOwMrE+epEOomIdIANEegArEScr0+kk4hIB9gAgQ7A0sR5c0Q6iYh0gJYJdACWIs6bV0c6ZCDSAVok0AFYmDgHQqQDtEagA7AQcb4h//iPf9P1EGABIh2gBQIdgDuJ8w361a/+t7AnnRxEOkDDBDoAtxLnm+fgOBIR6QANEugA3Eicd0ekk4hIB2iIQAfgWuK8eyKdREQ6QAMEOgDfI877Q6STiEgHWJNAB+AScd4/Ip1ERDrAGgQ6ABfEeX+JdBIR6QArEugARIQ4z0Ckk4hIB1iBQAdAnCci0klEpAMsSaADjJw4z0ekk4hIB1iCQAcYMXGel0gnEZEOsCCBDjBS4jw/kU4iIh1gAQIdYITE+XCIdBIR6QB3EOgAIyPOh0ekk4hIB7iFQAcYEXE+XCKdREQ6wA0EOsBIiPPhE+kkItIBriHQAUZAnI+HSCcRkQ5whUAHGDhxPj4inUREOsAcgQ4wYOJ8vEQ6iYh0gJpABxgocY5IJxGRDhACHWCQxDkzIp1ERDowegIdYGDEOVeJdBIR6cCoCXSAARHn3ESkk4hIB0ZLoAMMhDjnLiKdREQ6MEoCHWAAxDmLEukkItKB0RHoAMmJc5Yl0klEpAOjItABEhPnrEqkk4hIB0ZDoAMkJc5Zl0gnEZEOjIJAB0hInNMUkU4iIh0YPIEOkIw4p2kinUREOjBoAh0gEXFOW0Q6iYh0YLAEOkAS4py2iXQSEenAIAl0gATEOZsi0klEpAODI9ABek6cs2kinUREOjAoAh1gg6qq2q6q6qdVVb2pqmp3gceLczoh0klEpAODIdABNutpRDyOiL2IOL4t0sU5XRPpJCLSgUEQ6AAbUlXVTkwDfeb344ZIF+f0hUgnEZEOpCfQATbn+TWf+16ki3P6RqSTiEgHUhPoABtQVdVeTJe2X+ci0sU5fSXSSUSkA2kJdIDNuG72fN7vR8TbEOf0mEgnEZEOLbj3z/5ZPLx/v+thDJpAB2hZVVVPY/rD4l1+p/7zP4lz+kqkk4hIh4Z98W//bez98IfxP/2bf/M/dj2WoRLoAC2qqmo7Lh8Mt4h/t8gt2KArIp1ERDo0ZHJwsLf1278dERHb//Sf/lHHwxksgQ7QrqcRsewPhjee7g59IdJJRKRDM/Zm7/ze1ta/7nIgQybQAVpyzW3VliHS6T2RTiIiHdZ38TPJb/+Tf3JvcnDgZ5QWCHSA9rxc8/kind4T6SQi0mFFk4OD7ZibQa9d/ZgGCHSAFtS3VWvihUuk03sinUREOqzmup9p/GzSAoEO0I51Z8/n/X7cfA916AWRTiIiHZZ3XYz72aQFAh2gYVVVPY+InQYveRzChwREOomIdFjOtasCJwcHlrk3TKADNKi+rdphQ5c7jYjPSyn7pZTThq4JrRLpJCLSYQGTg4OduHk5u0BvmEAHaNbzWP62aledRcSLUsr9UsrrBsYEGyXSSUSkw91ui3CB3jCBDtCQ+iC3dWfPX0XEp6WUZw0MCToj0klEpMPtbjsMbrc+4Z2GCHSA5jxf47nHEbFfSnliOTtDIdJJRKTDze46DM4seoMEOkADqqp6HKu9QJ1GxJN6n/lxw8OCzol0EhHpcMXk4GA37t6653ZrDRLoAGuqD4ZbZfb8RUyXs4sXBk2kk4hIh8sWmXxwu7UGCXSA9T2N5W6r9joi7pdSnpVSzloaE/SKSCcRkQ4fLDI7vlOf9E4DBDrAGqqq2omIJws+/CSm+8w/t8+cMRLpJCLSYWrR2XH70Bsi0AHW8zQifv+Ox5xFxLNSyqf2mTN2Ip1ERDqjNjk4WCa67UNviEAHWFFVVXtx923VXsR0OfuLDQwJUhDpJCLSGbNlAt0+9IYIdIDVPb3l747DPnO4kUgnEZHOWC0T6Nv1ie+sSaADrKCqqsO4/oXrNKb7zPftM4fbiXQSEemMyuTgYDuWX7ZuH3oDBDrAkm64rdpZRLwopdy3zxwWJ9JJRKQzJqvEtkBvgEAHWN5hRMz/gPYq6uXsHY0HUhPpJCLSGYtVlqsL9AYIdIDlzfaUH0fEp6WUJ/aZw3pEOomIdMZgpUPfljz5nWsIdIAllVJelVKqep/5SdfjgaEQ6SQi0hmsycHBTkTsrPh0gb4mgQ4A9IZIJxGRzlCtE9kCfU0CHQDoFZFOIiKdIVonsnfrE+BZkUAHAHpHpJOISGdo1p0FN4u+BoEOAPSSSCcRkc4gTA4OduPynWpWIdDXINABgN4S6SQi0hmCJuJaoK9BoAMAvSbSSUSkk10Tcb1TnwTPCgQ6ANB7Ip1ERDqZNTX7LdBXJNABgBREOomIdLJ6ERHHt7xFRMRv/vEf//6Wx7yOiJONjnpAqlJK12MAWFlVVdsR8TQiHsf0t7WnEfGqlPJiwefv1s99XUrxYkJnJgcHsxfk/cnR0fGtDx65yWTyMiIOux4HLOAkIvYnk8lZ1wOBJsxeq/7uV7/6i7/5xS/+suvxDJEZdCCtOs7f1B9+WkqpYvrD0POqqp4v8Pyd+vlPI+LOxwP9YCadRMykA0sR6EBmLyPiuJTyrJQym504rf9cZHbtcXy4lcjpbQ8E+kWkk4hIBxYm0IGUqqrai4jdUsqzGx6yXT/mNrtz71veDsmIdBIR6cBCBDqQ1fOYHmRy1XyU3zUrPv9Ye34hIZFOIiIduJNAB9KpD3bbKaW8uvL5nfgwK35aSrkx0OtrbC/yWKDfRDqJiHTgVgIdyOhxXP/D+Py+89d3XMPsOQyISCcRkQ7cSKADWV0X4I/n3r/rB3X7z2FgRDqJiHTgWgIdSKc+tf1SVFdVNbsPekTEyQJL1s2gwwCJdBIR6cD3CHRgKOaD+9Yfzu0/h2ET6SQi0q9RVdXzqqoWuV3qKtferarqeRvXhiYIdGAo5pe3238OIyfSSUSkz6mq6mVMX6fvei1fSb0Cb7uqqp9XVeV/c3pHoAPp1cvbZy+yJ6WUszueYv85jIBIJ5FUkV5V1U5VVU/rme7n9V1Umrjuy5j+b7G/wGv5Op7Vf74R6fSNQAeGYD64F5kRN4MOIyHSSaT3kV5V1Xa9PPyb+HDuS0TEN+suSa/j/HFEPGk5zqO+/n5Mf7n/ss2vBcsS6MAQLBzo9p/D+Ih0EultpNczzW9iekvT/VLKk/rQ1mcR8SIiXtYr2la59mF93RdXD4FtSx3pTyLisT3p9IlAB4bgYka8lHLXjLjZcxghkU4ivYv0Os6/ienYPr/mtXb2b2vp0K2Xxz+PiONSyou1Brqk+r/jRUQ8rX+BD50T6MCQLDIbbv85jJRIJ5HeRPrczPl2TGe4v/fL7Xo12llE7Kwwi/6yvvaTNcd5WFXVmxWW2r+I6dh/us7Xh6YIdGBIFgl0M+gwYiKdRPoS6bOD207r5ew3mf3Se++Wx1xSx/ReRLxaZ8tZPQs/O/19qUCvl7q/iukvFyx1p3MCHRiC2Yv6rafIVlX1NOw/h9ET6STSaaTXs+GzGfHb4nzeMkvFn9Z/rru0ff6XAqu8ts++/qFT3emaQAeGYHav1J2bXljrvWXzvxk3ew4jJtJJpJNIr19PZyecH5dSFr0v+UKBXs+e79TXXvcX5mttX5ubRd+OD780gE4IdGAIZvvHIq45oKaeOf9pXH7Rtv8cRk6kk0gXkT6/6mzROF/GbCl6E9duYvvabBxr3S4O1iXQgfTm7md6HNPlaW+qqnpev30T0xfu2f1OZ8ygAyKdTDYW6fWe7tlM8mkpZZF/IxfjumuZeH392az3WoFeX2u2xe1s1du01YffnUbE9qq3i4MmCHRgEEopJ6WU/Yj4ND7E91lMbwczi/PZC7j958AFkU4im4r0+WXejS5tr13Mnte/ZF9Hk4e/zp4v0OmMQAcGpQ71F6WUZ/Wf150q28ZSPSAxkU4irUZ6Pfs9v8z7zn8X9Sz2hQWiexbATWw3a/L2qbPnC3Q6U5VSuh4DwMqqqtqLBWbE66Xusx8g7ptBZ9MmBwe/jsvbLFZxFhGfTo6O/P+3JZPJ5GXYg0oOJxGxP5lM1p2BvqQ+t2V2nstpKeX+As/Zi+m90iMiopRS3fLYnYj4pv5w/7r7qi/jyuv7p6sucW9jbEM0OTgoERF/96tf/cXf/OIXf9n1eIbIDDqQUlVV21VV/TymPxB8c/W391ceOzspNiLihTinI03Mdrn9T8vMpJNIWzPp87+gWnTF2TK3Obt47DIBXFXV07nzZS7e4vItVh9f85hbb8E6r/75YPYLj4Xv5w5NEuhAVk/j8rK2a1+A66V6FwfdxPr3WoVVNbGU89TseftEOok0Gun1LUnnX08XDej51+O7vkfNrr/s98S9+uvMv80vRT+95u93Y/lfbM7GtcyeemiMQAeyurrn7KYX+qcx/WFgdmBco0sBYQlNLJW03HJDRDqJNBnpl/ZeLzHDvcw+8Nljl3o9LqXsX32Ly98TX133mBWWvAt0OiXQgaxOY/oi+iym+8S+90JfL21/GtMfAlZ5kYYmNRHX/j+8QSKdRJqK9Pll3SdVVe0t8HYYl2epF13i3sT3syZPcL/KliI6IdCBrF7HdGb8+IY4fx4RL2P6g4I4p3OTo6Mmfnh0B4INE+kkslakX7k3+cX1Fnh7eeVSG1np09T9z69xMf768DvYKIEOpFQvu3sWEW/qQ2Bmv8l/Wp/oehjT/eZrnegKDVsnsE8mR0e2aHRApJPIOpF+dUn3s4jYX+Bt/jX21ruqXAnedc/TaHP2HDoj0IG0SimvIuLT+sOn9dt2TMP8fn0vdEFDn6zzyyI/gHZIpJPIqpF+KdBLKS9KKcd3vcXlQ+WW+SXkuoHe5P3Pb7LwCfDQFIEOpFZKOa1DfHYYzLNSyithTk+tE9lWgnRMpJPIKpE+H7wLfa+qT32f/xqb3IaziRl0gc7GCXQA2JDJ0dFJLHly8dxz7T/vAZFOIstG+nyMLvoLwUu3OdvUlrIW959HrD+zD2sR6ACwWauEtuXtPSLSSWSZSF8l0OdnsYc2ew6dEOgAsFmrzPRY3t4zIp1EVlnufucs8jWnvm/y30Ob+88ta6dTAh0ANmuV2R7L23tIpJPIUpG+4JLxw7n3X912ensLNjWDbrk7GyfQAWCDJkdHp7HcjM9ZvXedHhLpJLJopC8apfP7z1+sMJ517tfe1v7zqwQ6GyfQAWDzlpnxsb+y50Q6iSwS6Yssb38cHyL5xaKz5/Vt2ebHsoqFZs+rqtquYx5SEegAsHnLzPgI9AREOoncFOnL3GFitrz9LFabPV/HovvPn8flmG/6+tAKgQ4Am2cGfYBEOolcF+kLxWg9ez4L3yellGVvHTn7nrbq7PadM+j1zPnjWO38jtn/Jmcr/LfB2gQ6AGzY5OjoLBYL79N6zzpJiHQSuRrps0C/a2/48/rPV6WUVQJ49j1t6UCvqmo7Ftt//rQe3yqBPbu+2XM6IdABoBuL/PBn9jwhkU4i85H+eu5z16qq6nlMA/YkIp6t+DVngb7KHvQ7Z/yrqtqL6Sz7qkvvBTqdEugA0I1F4lugJyXSSWQ3It58+eWXp1F/z6mq6nvxXC9tfxrTcN1fY/n3xfe1677OberD6G78uvX1fhoRn68yvnqGfjYmgU4nBDoAdGBydHQcdx/KJNATE+kkshsRb37wgx/87/XHT+f/so7zl7F+nM/usT57/iqHuM3+TV1aIl/PnL+JiGdr3Hpt/hcGvv/SCYEOAN257QfAk3qvOomJdBLZ/bM/+7N//7u/+7v/a0TsVVX1pqqq51VV/TSmcf6qlPJpQwen3bmc/iallGcx/Te1U1XVT6+Mcb+Uss6/t9kvDF47II6uCHQA6M5tszxmbwZCpJPI7p//+Z//z3/8x3/8r+NDRL+OiPt1GDdl9v1tlRn0KKU8iYhP48N+9lellPtrzJzPzMbj+y+dEegA0J3bfgj0A+KAiHQS2f2TP/mT//jll1++LqU8K6U0Pptcn/5+GhHb9fL5Va5xUo/vWSll7e+X9a3ZdmN6Orx/q3RGoANARyZHRyfxYQbo6t8J9IER6SRy3X3SmzaboV9pFr0Fh/Wf/o3SKYEOAN26LsTF+UCJdBJpO9JfxPSwuMN69rprAp1eEOgA0K3r9kwK9AET6STSWqTXy+Zn/w4Ob3ts26qqOozpPdZf1bdyg84IdADo1utrPifQB06kk0ibkf4sptt8Dut7kG9c/XWfxnQ2v8mD8GAlAh0AOlTfSm1+Fv2s3pvOwIl0EmlzufuTmM5eP2/h2ot4GtN7qj9xazX6QKADQPfmZ8yvm1FnoEQ6ibQS6fUJ7C9iOou+0QPjqqrajWmgv6pPlofOCXQA6N7JDe8zAiKdRNqK9Gcx/UXlyw0vdX8Z0++5g17aXlXVblVVe1VVHVZV9fzK25srb99UVVVueptd8/89P//vu/xvGrKqlHL3owCAVk0ODn4dEVVE7E6OjhxSNEKTyeRldHxYFizoJCL2J5NJY0vC6zB/E9P7kO83dd1bvt7LiHgcEfezL22vVwJsx/QXKNsxXbI//3Fjvnz0KCIi/o///J//6v/8+7//X5q8NlMCHQB6YnJwsCPOx02kk0ibkX5SSnnS1HWv+TrPYxrnn5dSer9qqcEAv3rmSVzzccQtB5V++ejRm4iIv/vVr/7ib37xi79c4muzIIEOANAjIp1E2or0lxFxXEppfOtHHbvPYxrnvZs5r+8Jv3vlbdEI///bu5vYOPPEvvO/J5lxa8frgdkBbAjZYG02ehOMscAGFPYiIPACFHLpOgXUrY9L3Xpupo7P3CT4MJ45mfQlQQM5iLAPU/JJPBnTe3ETBjJeZYFFE5PDWBNsdjgziwSacYJnD89T4sMSX6qK9fY89fkAgsRivTzdLbL5rf/bWfPrvPX7aepZCXN5I6IcDKpEoC+SQAcAWDMinQ6Ze6RviubNiJ0ku6lHxXdzc4yf5f0IP0tytqzz2wX64gl0AIA1JNLpEJE+gWaX+p1cxPj2DXc/bf9al6n4An3x7OIOALCG7O5OhyzynPROKopiqyiKvaIoDoui+LLZAf1V6un1+7kc52epv9afJnlUVVVRVdWDqqqeVFV1tC5xznIIdACANSXS6ZCNj/SiKLaLojgoiuJFkp8leZE6xndadztPcpz67PdHST6squqjJsafN+fCs8EEOgDAGhPpdMjGRXpzvvizoii+TPJV6hHyvdZdTlPH+OPUR7p9WFXV46qqnlZVdbKOG9WxWgIdAGDNiXQ6pNeR3kxd32+mrv8s9bT1g1yMko9GyJ+kDvIHTYwfL2sjN7pNoAMAdIBIp0N6FelFUew0U9dfpZ66PtrAcfTPd5ZmlLw1Qn4kyJmFQAcA6AiRTod0OtKb9eTPiqL4KsmXqaeu77bucpJ6U7cHzRryp1VVHa/iWukXgQ4A0CEinQ7pVKS3pq+P1pMf5GK39fPUX3dPUm/s9qjZ1M0O68yVQAcA6BiRToesfaQ3U9gPU0f5YS6vJ3+e+uizD1vHntnYjYUR6AAAHSTS6ZC1i/RmtPygNYW9vab8OMmTJsqfOvqMZSqqqlr1NQDAWioHg2e5+KHtNMnzcjh8b41hORjsp16buJuLH/BOk5yUw+HTa577uscclcPhnaOrHAx2Ux/1s5t6iuZoZ+Gn5XBo9KdHyrIcbVgF6+40yaOyLFf2Pagoir3U3xv3xj51lvp7pM3dblAOBlWS/PWPf/xHf/mjH/3xqq+njwQ6AFyhifOt1JsAJfVROjtJnowCuhwMdlJPhzxPHdbHze1bSV6kjuP6B9ImiluPGQV8+zGj1zgph8NHM153+7WPU7+pcNp87iD1msoHzescNg97UA6HfiDtMJFOhyw90oui2E799bGXizXlI0dJjo2ST0agL55AB4AxzejziyQftcL6Weq4PU/yUeof8l6kHpG+alR9O/V6xqSO5KdNnL9IHfnv/TB41WOmvO6d1JG/ldYbCWP3eZY6zrdysc7ywSji6S6RTocsJdKLohhF+e7Yp05zEeZmFE1BoC+eNegA8L691CPiV/3gtpU6gl4keXxVnCdJMyI9GpXea8X546vivPWY0WuO/0B5o7E4f37dNPkm+nfS2gRJnPeDNel0yELXpDc7sY82fBt9Lx1t+PagqqoHNntjXQl0AHjfXt4PnfYPks/Smjp+g1Ggt0fbJ33Mzo33amlG3kdxfu2695b2mwqmdfaISKdD5h7pY2E+msp+kssbvnlDkrUm0AGgpRwM9pKcXbEmux3MxxNu5NYeBT+5brT9hteZ1GEu3kB4PsH926NGfljtGZFOh9w50sd2Yx8P80fNeeW+HugMgQ4Al+1kbFS52XitHc63rg1vRrXbbg3nscdMFM6t3eCT5PS6m2fs2gAAIABJREFU6fNjLr1xMMnr0C0inQ6ZKdKbMH+Wet+OZ7k6zH1/o3MEOgC8b3ykux20xxPueD4+4j7JY9qvM+nI9rPWn28NsrE3G6w/7zGRTodMHOljYX6Qi9lDR0k+EuZ0nUAHgJZyOLxqnXg7ticN2oU/phk9b/9AO+3ouTjvOZFOh9wY6UVRbN8S5k+cX04fCHQAuN0sU8KX8ZhLsT3DyL5A3wAinQ55L9KbMD+MMGdDCHQAuMEsU8JnfMxOLtZQThrbe60/L/KNAzpOpNMhO0le/eEf/uHvtUbM95vPjY5KE+b0lkAHgJsta/S8Hdu37vZeDgbj56RP+8ZBJtxQjp4Q6XTF6enpzt/8zd/8X6lHzJPLYf5UmNNnAh0Abras9edTBfoV7N7OrUQ66+zNmzf5/PPP8/Lly/zyl7/8IEm+/vWv/+tchPn5jU8APfC1VV8AAKy5hY+g3za9vTl+7bwcDq/74fSmz7VZf07KsnxSlmVyMW0YVurt27c5OTnJ6enFt6Xt7e3s7u7m/v37//MKLw2Wzgg6AFxjWevPczmUrho9f5bLI+zjJo1tI+gkMZLO+vjiiy/yve99712cb21t5ZNPPsmnn36a+/fvJzOekw5dJdAB4HqzHEl21/Xnl6KpCf69KZ7rStafM06ks0pnZ2f5/ve/n5OTk7x9+zZJ8vDhw+zv72dnZ2f87iKdjSHQAeB6yzjLfDcXxwYdXzFVfa+5/fK09zqwp1mPedj6szgniUhn+c7Pz3N8fJzPP/885+f1t7Bvfetb+eyzz7K7u5t79+5d91CRzkYQ6ABwvaWfZX7F5w9yfUCNpsOP7+h+STkYHObyKL3157wj0lmG0Trz73//+3n9+nWSejr7p59+mr29vWxtTdTdIp3eE+gAcIVZpoTPuP789Jo/pxwMDpKc3fDaT9OMopeDwZUbfjVxvpOkPQJvBJ1LRDqL9Pr16xwdHeWLL75Ikty7dy+7u7v57LPPsr29fcuj3yPS6TWBDgBXa09Vn/TYs1lG3NtT1d/9wNkE90GSx9c9sJkO/6h5/LP22ejlYLBXDgZfNR8+zsUu8dafcyWRzry9ffs2x8fHOT4+fjedfWdnJ9/+9rfz8OHDuzy1SKe3BDoAXK09rDNp0LYfM1HUN5H9OHVkH5SDwbNyMHiRemf3R7cdn9aM0n+UOqwOy8GgKgeDn6V+s+BJORw+yeU3G8Q51xLpzMvr16/zve9979109vv372d/fz+ffPLJTevMpyHS6aWiqqpVXwMArKVmivl5ORxOHCytaemTjrqPHjfarX079VnoUz3+lud+lno0Pkmel8Ph03k9N/1UluVhnJO+8b744ovcu3fvql3Vr3XVmeYPHz7M7u6NW2XcxWmSR2VZTrNpJjMqB4MqSf76xz/+o7/80Y/+eNXX00cCHQB6rhwMvszFKPojU9yZhEjfbKenp3n58mWSZHd3d6Ip6WdnZ3n58uW76ez379/PYDAYnWe+SCJ9SQT64pniDgA95vxzZmW6++Z68+bNuzhPkpOTk7x58+ba+49GzdtHp43ONF9CnCemu9MjAh0A+q19vNrcps2zGUT65hlt7Dbu+Pg4b9++fe/2s7OzSzu0b21tZX9/f5FT2q8j0ukFgQ4AHVMOBvvNZnKT/CDaDnSj50xNpG+Wk5OTd6Pgbefn5xkOh+/dd4Wj5lcR6XSeQAeADmnONT9Mvenb4S333c3F0W8n02x2B20ifTO8fv360uZuV33+iy++yPn5+Xuj5p9++ml2d3fntUP7XYh0Ok2gA0BHNCPm7U27rv0BtBwMdpK8aD48zQ3nqcMkRHq/vX379r0R8qucnJzkT//0T9+tSd/Z2cn+/n62t7dveeRSiXQ6S6ADQDedJ7nyuLRm5PxV6oA/yQTnqcMkRHp/DYfDK9eYX+XXv/51Pvjgg3zyySfzPNd83kQ6neSYNQDokNaZ5sep4/us9ent1GvOd5vbn5vWziI4gq1f2keqTWp7ezuffvrpgq5orhzBNkeOWVs8I+gA0CHlcPg0yUepf+jcTh3ro187ze0PyuHwI3HOohhJ74/z8/O8evVq6sednZ29W4O+5oyk0ylG0AEAmImR9O77/PPPc3Z2dvsdr/Hpp5+u2/rz6xhJnwMj6ItnBB0AgJkYSe+2L7744k5xnlx/PvoaMpJOJwh0AABmJtK76c2bN/mrv/qrOz/P27dvc3x8PIcrWgqRztr72qovAACAbivL8klZlonp7p3xgx/8IL/+9a/n8lyj9egPHz6cy/Mt2CjSTXdfU+VgcJAbjhFNclwOh6dTPN92JvvedF4Oh88nfd5FEegAANyZSO+Ok5OT/PSnP131ZaySSF9vu9f8eWQ7yeMpnm879X/znVwf/mepT0ZZOZvEAQAwNzaOW29v3rzJ0dH8ViTcv38/Ozs7+YM/+IN1PQ/9JjaOm9IqNokrB4MvU8d120flcDj1BgrlYLCb5EXqUD9P8nTdTjyxBh0AgLmxJn29/eAHP5jL84zOQd/f38/Ozk4X4zyxJn3tlYPBVur/TuNTz2d6E7AcDk9Sj5SfJ3m0bnGeCHQAAOZMpK+neUxtH4V5h45Xu41IX2+7Sc7K4fBp6qge2bvDc25lynXsyyTQAQCYO5G+XkYbuc3qW9/6Vt/CvE2kr6+dXKwNb38/2S4Hg1kjfSfJ2h49INABAFgIkb4e3r59m+FwONNjd3Z28tlnn2Vvb6+PYd4m0tfTbuq9ApL3v5dMHejlYLCTZKuZ6r6WBDoAAAsj0lfvhz/8YX7+859P9ZhRmH/yySfZ2tqYZhXpa6S1/vwkSZpN4dphvdccoTaN3azJbu3XEegAACyUSF+d169fTzW1fUPDvE2kr4/R+vP2bu3jU9On3SxuJxcj8mtJoAMAsHAiffmmmdouzC8R6euhvf48SdLsun6XzeKMoAMAQCLSl204HObt27c33keYX0ukr157/XnbTJvFdWH9eSLQAQBYIpG+HK9fv87r16+v/bwwn4hIX5Hx9edjZt0sbu1HzxOBDgDAkon0xbpparswn5pIX42r1p8nudNmcWu//jwR6AAArIBIX5wf/vCHl6a237t3Lw8fPhTmsxPpy3fd6PnILJvFGUEHAIDriPTFaO/a/vDhw3z729/O7u6uML8bkb5c160/TzL9ZnFdWX+eCHQAAFZIpM/fJ598kocPH2Z/fz+7u7u5d+/eqi+pL0T6Etyy/rxtfLO4m0bROzF6ngh0AABWTKTP187OTnZ3d3P//v1VX0ofifTFu3b9+ZhpNovrxPrzRKADALAGRDodItIXa5LR86s2i9u9YbM4I+gAADANkU6HiPTFuXH9+ZhbN4vr0vrzRKADALBGRDodItLnbIr15/X9398s7qp16J0ZPU8EOgAAa0ak0yEifb4mXX/e1v5esXXFZnGdWX+eCHQAANaQSKdDRPr8TDx63nLbZnFG0AEA4K5EOh0i0udjmvXnSW7eLK5r688TgQ4AwBoT6XSISL+Dadefj7lus7hOjZ4nAh0AgDUn0ukQkT67WdafJ7lxs7hOrT9PBDoAAB0g0ukQkT6bWUfPR67aLM4IOgAALIJIp0NE+vSmXn8+Zvx7w0E6tv48EegAAHSISKdDRPqE7rj+vH6O9zeL277L862KQAcAoFNEOh0i0icz8/rzMeObxXVq/Xki0AEA6CCRToeI9NvtJ7lrnF+1WZwRdAAAWAaRToeI9GuUg8FB6hH07Waq+129+57QtfXnSfK1VV8AAADMqizLJ2VZJhfHKsG6GkX6o7Isz2+9d0+Vg8FOkr3mw73Ua8XT/P5lORgcpx4FP55xyvtR6g3iOhfniRF0AAA6zkg6HWIkPXmWOqAPchHnI9vN7c9Sj6pPrbVZXCcDvaiqatXXAAAAd1aW5WGMpNMNp0k6N5JeDgZVkvz1j3/8R3/5ox/98aqvp4+MoAMA0AtG0ukQI+lcSaADANAbIp0OEem8R6ADANArIp0OEelcItABAOgdkU6HiHTeEegAAPSSSKdDRDpJBDoAAD0m0ukQkY5ABwCg30Q6HSLSN5xABwCg90Q6HSLSN5hABwBgI4h0OkSkbyiBDgDAxhDpdIhI30ACHQCAjSLS6RCRvmEEOgAAG0ek0yEifYMIdAAANpJIp0NE+oYQ6AAAbCyRToeI9A0g0AEA2GginQ4R6T0n0AEA2HginQ4R6T0m0AEAICKdThHpPSXQAQCgIdLpEJHeQwIdAABaRDodItJ7RqADAMAYkU6HiPQeEegAAHAFkU6HiPSeEOgAAHANkU6HiPQeEOgAAHADkU6HiPSOE+gAAHALkU6HiPQOE+gAADABkU6HiPSOEugAADAhkU6HiPQOEugAADAFkU6HiPSOEegAADAlkU6HiPQOEegAADADkU6HiPSOEOgAADAjkU6HiPQOEOgAAHAHIp0OEelrTqADAMAdiXQ6RKSvMYEOAABzINLpEJG+pgQ6AADMiUinQ0T6GhLoAAAwRyKdDhHpa0agAwDAnIl0OkSkrxGBDgAACyDS6RCRviYEOgAALIhIp0NE+hoQ6AAAsEAinQ4R6Ssm0AEAYMFEOh0i0ldIoAMAwBKIdDpEpK+IQAcAgCUR6XSISF8BgQ4AAEsk0ukQkb5kAh0AAJZMpNMhIn2JBDoAAKyASKdDdpK8WvVFbAKBDgAAKyLS6ZCdVV/AJhDoAACwQiKdrvngG9/471Z9DX0l0AEAYMVEOl3yzx48+N+tSV8MgQ4AAGtApNMVv/HBB/9DbBy3EAIdAADWhEinQ+zuvgACHQAA1ohIp0NE+pwVVVWt+hoAAIAxZVkeJtlf9XXAO6en9e8ff5x885uXPpPkUVmW5yu4qoUoimIrycEinruqqqfXvq5ABwCA9STSWSvXB3rSs0gvimI/yeEtdzuZ4Km2m18jR1VVPbn2dQU6AACsL5HO2rg50JMeRXpRFC+S7LVuOk+99OSkqqpJwjxFUewkeZVktATgNMmjqqqu/fdjDToAAKwxa9LpkF6sSW+mt7fj/HlVVR9WVfV0ijjfzuU4P8stcZ4IdAAAWHsinQ7pQ6Tvtv786KY141dpAv9FLuL8PMnj2+I8EegAANAJIp0O6XqkjwL9yaQj5mNepf53kNRx/qiqqtNJHijQAQCgI0Q6HdLlSN9LclxV1dRfa0VRHOYizpPk6aRxngh0AADoFJFOh3Qu0oui2E09NX2qae3NY8c3dHwybeQLdAAA6BiRTod0LdJ3U4+en03zoOZYtnacH80yAi/QAQCgg0Q6HdKlSN9LcjzNA644M/3Gs85vItABAKCjRDod0pVI36qqauJAb846b8f56axxngh0AADoNJFOh6x9pFdV9eGk923i/FXrptMkj+7y+gIdAAA6TqTTIWsf6ZO45qzzR5OcdX4TgQ4AAD0g0umQTkd6E+evkmw3N80lzhOBDgAAvSHS6ZAuR/qLXD7r/PE0Z53fRKADAECPiHQ6pHOR3px1vtu66UlVVSfzen6BDgAAPSPS6ZDORHpRFM9y+azzp7OcdX4TgQ4AAD0k0umQtY/05qzzg9ZNR1VVPZ/36wh0AADoKZFOh6xtpBdFsZfLZ50f3+Ws85sIdAAA6DGRToesXaQ3Z5234/w0yULiPBHoAADQeyKdDlmbSG/i/FUuzjo/zRTHqRVFcdAcyTYxgQ4AABtApNMhK4/0JqwPcxHn56l3bJ80zreTPJv2bHSBDgAAG0Kk0yEri/Qmzl/l4qzz89Qj59Ocdb6X5Hja1xboAACwQcqyfJL/+B9P8pOfJD/5SfKf/tOqLwmus6pIf5aLOE/qkfNp4jypA33q89GLqqqmfQwAANBB5WCwk+RFkuTDD3+Z3/iN/yW/+lXyX/5Lsr2dfOMbq71A1ttp06gff5x885tLfeUkj8qynGq6+CyKojjM5bPOn0x71nmzdv3LJB+a4g4AALynifPRhlcPyn/zb/558Tu/86/zwQfJf/2vyX/4Dyu+QrjWUkbSi6I4yOU4fz5tnDf2Ux/FNvUbCgIdAAA2w4vUcX5UDofnSVL9u3/36/z0p8l/+2/1KDqsr4VGelEU+6mnto8cVVX1dIbn2Ukd6NNOiU8i0AEAoPfKwWA/yXbzYXtd7MVo4QcfLPOSYBYLifQrzjo/qapq1rPOR88z9QZxiUAHAIBN8C7Ey+GwHeij6bvn+Sf/5C+We0kwk7lGeuus85HTJI9nfK7D1Nd3WlXV2SzPIdABAKDHysFgKxc7Ul+adlsOh0+SPCiHww/L7373X8URbHTDXCK9dZza6HnOUh+nNtXa8aIotoqieJGLN8JmGj1PBDoAAPTdbuvP762LLYfDd7c5J50OuVOkXxHn50keTxPnRVFsF0XxLMlXqY9VG5k50L826wMBAIBOaJ/nfOu027Isn5RlmVzezRrW0SjSZzmC7UUuf22cJNkrimLvmvuPv+52LvZ1aDubdXp7ItABAKDv2hEy0c7SIp0OmTrSi6LYzeWZJcnlEfC7mHn0PDHFHQAA+u5diIxtEHcj093pkKmmu1dVdVJVVbGgX1MfzdYm0AEAoKfKwWCq6e3vPV6k0x0LPSd9WQQ6AAD0V3uN7ETT28eJdDqk85FeVFW16msAAADmoBwMDnKxK3VSB8toivtJ3o/0k0mnvZdleRhr0jfbafPX5+OPk29+c7XXcrPTJLNsHLdyRtABAKA/dlNHefvXyNYVn7tqF+orGUmnQ3ZSH6HWOUbQAQCgp8rB4Ge5GFH/sBwO7zyiaCR9g3VnBD1Jjpo3lTrFCDoAAPRQORhs5yLOz+cR54mRdDqhk3GeCHQAAOirqc8/n5RIZ411Ns4TgQ4AAH21sEBPRDprqdNxngh0AADoq/YGcFOfgT4Jkc4a6XycJwIdAAD6aqEj6CMinTXQizhPBDoAAPTVuxH0cjhcWKAnIp2V6k2cJwIdAAB6pxwMdlsfnizlNUU6y9erOE8EOgAA9FF7evtC1p9fRaSzRL2L80SgAwBAHy18g7jriHSWoJdxngh0AADoo6VsEHcdkc4C9TbOE4EOAAB99C7Qy+FwKWvQx4l05u0Xv/jFX/Q5zhOBDgAAvVIOBisdPW8T6czL6elp/uRP/uR/K4pia9XXskgCHQAA+mVtAj0R6dzdL37xi794+fJlkmwledXnSBfoAADQLyvbIO46Ip07OPrud7/7r5KMprbvJHm2wutZKIEOAAD9slYj6CMinRm82xCuqqqjJM+b2/eLouhlpAt0AADol93RH1a1Qdx1RDpTeG+39qqqniYZ/Z0+KIpib/mXtVgCHQAAemKdNoi7jkhnAjcdpfY4F3+3D4ui2Lnmfp0k0AEAoD/asbJWo+dtIp0b3HjOeVVV56nXo5+n3jTuxbIubBkEOgAA9Ed7g7i1HEEfEelc4cY4H6mq6jT1SHqSbBdFsb/Yy1oegQ4AAP3RiRH0EZFOy0RxPlJV1Ula69EXc0nLJ9ABAKA/RhvEHZfD4flKr2RCIp1MGecto13dt/uyq7tABwCAHhjbIG7tR8/bRPpGmzXOR6Poo783+0VRbM3vslZDoAMAQIeUg8GzcjD4WTkYfFUOBu0156PR8/Mkxyu4tDsR6Rtp5jhvGY2ib6UHU90FOgAAdEQ5GOynjpCt1BvCtTfHGo2gH3Vlevs4kb5R5hHnqarqLBeRflAUxfZN9193Ah0AALpjPD6Ok6QcDLaS7CVpx0onifSNMJc4b3meeuZI0vFR9K/d9Mnm3Ydptqw/adYBTKwoir1c3m3yKqdVVXVumg4AAMzZWfP7SeqR8tFRaqMoedLV0fO2siyflGWZTNcidMO84zxVVZ0XRXGU+utgL/U56Z1UVFV1/SfreB59Uexee8f6G8VZkqNpQ7rZbW8/9TSdq5wmeS7QAQAgKQeDL1OPFh41v++m/nn6aTkc9mrkuSzLw4j09XHavB/08cfJN785yzPMPc5HiqLYSfJl8+GjaQeO18WNgf7enYviqi+QB81B8Xe7kKL4Mhcj6SeZIfYBAGATNGvRd1MPcp2mHk0/u/lR3STS18jdAn1hcT5SFMVXqZeBPK+q6ukiX2tRpg30rSQ/G7v5o2Zh/t0u5CLQn1ZV1el1MwAAwPyI9DUxe6AvPM6Td7OzD5KcVVX10aJfbxGm2iSuqqrRVJq2O3+hFEVxkDrOn4hzAACgzcZxnbaUOG+MprVvN1PeO2eWXdzH5/Lv3eUCmn9xz1LHuS86AADgPSK9k5YZ52nWnY82SbxpD7W1NXWgN+vC21Pat5vN5KbWxPmriHMAAOAWIr1TlhrnLaN9zO40kLwqs56DPr5529TvTjTr2V+k3gyud19kRVHsFkXxbNY3LwAAFqkcDHbKweCgHAw6fWYwm0ekd8Kq4jy5mPG90xwb3ilTbRL37kH1P+hXrZvOU28WN9GZi02cv0p9vnlnz6hra2YD7KZeS38pyquqKlZyUQAAjXIw2M7Fzyq7qXc6HnlUDoedPJKIzWXjuBWYbJO4VcZ5kqQoilHkdu64ta/N8qCqqs6KojjNxbFoW6mjdNJ3sg6TnHc5zps3KdpBft057gAAS1cOBlu5HOSd3DAJrlOW5ZOyLBORvjb+7u/+7q+Ojo7WofHOU/fZTt7fQ22tzRTojeNc/kY/UaA3Z6lvJ3l0h9deumbU3//kAIC1VQ4Go4GD0c8r0GsifX2cnp7m5cuX/+LP/uzPdqqqOl315aT+Hti5QdS7BPpR6t3XR3aLoti+6Uz05ji1vUwxHX6ViqLYzeUoBwBYG8209XaQd+6HUbgrkb56TZyPPjxM8mCFl5NcbGreuTXoMwd6VVXnRVEc5/J66/0kT6+6f1EU+6kPjX+0rnE+to7c/+QAgLXSTFtvB3nnfviERRDpqzMW50m9OdtBVVXPV3VNuThqrXM9d5cR9KSe5t4O9L1cEehN+B4mebwG0x3eaaat+58cALC2ysGg/bOKJXZwDZG+fFfE+cheklUG+mgEvXOzoO8U6FVVHRdFMVqAn9Rnou+2d8obO+t8/Hi2pWuOPfM/OQBgLTXryHdbv4AJifTluSHOk3oUfWuFM6evXXa97mY6Zu3SExTFs9RT10eORruzt45TO6mq6sqp78vQrH1f5f/kVvnuEQCw5v7Fxx//0//pd3/3n/3Ob/3W7339H/7De8t+/Z/8/Oev/7+3b3+57NeFRfnHv/d7v/9bv/3bv7vq6+idn/40SfJ/J/m318f5yMqOOGv2EnuVdO/I63kE+k6SL1s3nVdV9eG6nHV+xfUBAKyVg3/5L3Pv619f9WUATOTk3//7fPHVV7fdTaDP4K5r0FNV1en4mejNhnA7zedXeg5ec33PU09zWdUmAZ06ew8AWK7/46uv/vH/+vu//z/+9x988I1VvP7/+5//8y9/9fd///ereG1YpN/+R//oN7/xm7+59FkpffWrX/4yv3r7Nv/nmzervpTeuvMIevJuCnn7yLXz1PP+12rH9lUdm9a1d20AgNVY4bFpj8rh0IACvVSW5WGsSb+zW9acX+Wjm47gXqQuj6DPK9C3k7TnOJynjvO12bF9XDMFvx3rC9swrmt/KQCA9dBsGNcO9kUR6PSaSL+bGeL8rKqqjxZ1PbdpNgZ/kXSvxeYV6FtJfta66fkqN4WbRfMmQzvY53bkWtf+UgAA62mBR64JdHpPpM9mhjhPVtyDrY3MT6uqerCq65jFvAL93TsUjZVtCDAvzeZyo//57d1y9xsJdABg3prp8PMaXBDobASRPp0Z4/wsyYNVLnVuBfpJVVWPVnUds5hXoF86aq2PQdpavz71O9Z9/PcBAKyX1vnps6xfF+hsDJE+mRnjPFmDwdqiKF6l/j7YvZndcwr0L3MRrZ17l2JaY+vX93LLO9YCHQBYtnIwmGZwQaCzUUT6ze4Q52sRxEVRfJW60dbieqYxj3PQO7/+/K7G1q/vZewda4EOAKxSORjcNrgg0Nk4Iv1aR9/5zneOkhxm8pnD56k78PniLmtyRVGMInflo/nTmkeg9279+V211q/vpt6YYKPesAAA1tvY+vWtJE/K4XBtjsaFZRHp7zkqy/LJ6IPxpczXOEnyZFVHqo1rH7GWei382p4sdpV5BHrv158DAAD9JNLfuRTnI2PLe0cj6uepw/xkXcJ8pNWnKz3qbVbzCPSNWn8OAAD0i0i/Os67qMvrz5PkH9zlwc27Ke11CZ2aPgAAANDE6dGqr2NF+hTnO7nYY+N4ldcyqzsFeuqpDm0bvfYcAADopg2N9N7EeWPUp+ddW3s+MtdA3/TN4QAAgO7asEjvW5wn9SkVSUdHz5Mp1qC3jhIbTRnYzsW/gJH2tvongh0AAOiaDViT3rs4b3r1q+bDx1VVdTLSpwn0SbbYbztP8lFVVY7sAAAAOqXHkd67OE+SoigOkjxLun2y2J13cQcAAOijHkZ6L+M8ubR7+3FVVY9XfT2zuusadAAAgF7q2Zr0Psf5s1wsxe70fy8j6AAAADfowUh6n+N8K/Xa860kR1VVdfqf0wg6AADADTo+kt7bOG8cpI7z5PKm5Z0k0AEAAG7R0UjvdZw3O7ePNjJ/XlXV2SqvZx4EOgAAwAQ6Fum9jvPGKM7P04PR80SgAwAATKwjkd77OC+KYicX+wIc9eV4b5vEAQAATGmNN47bhDjfSvIqyU7q0fOP+hLoRtABAACmtKYj6b2P88aL1HGeJE/6EueJQAcAAJjJmkX6RsR5c+b5bvPh86qqjld5PfNmijsAAMAdrMF0902J8/0kh82HnT/z/CoCHQAA4I5WGOmbEuc7Sb5sPjxN8qhPU9tHBDoAAMAcrCDSNyXOt1PH+VZ6tincOGvQAQAA5mCZa9JPT0/zne9852wZr7VKzcj5q9RxnvR05HxEoAMAAMzJMiL9b//2b/+fly9fJsmzoigOm2PHeqcoir3Ucb7d3PSkqqrTFV6d35hMAAAGCUlEQVTSwgl0AACAOVpwpB/9+Z//+T9NctJ8vJ/kVd8ivdmt/UUuprU/qqpqXXbMXxiBDgAAMGcLivSjsiyfVFV1XlXVo9bz7yT5qpkO3mlFUWwVRXGY5KC5abQh3MkND+sNm8QBAAAsyBw3jrtyQ7ixo8fOkzzt6khzMwvgVeo3HJJ6lsDjPq85H2cEHQAAYEHmNJJ+7W7tTYw/SB3nW0kOm3Xp21fdf10VRXGQ5KtcxPnzqqp6vSHcVYygAwAALNgdRtInOkqtCfIXuQjcJHmeOnTXNnKbGQAHudgILqk3g+vkLIC7EugAAABLMEOkT33OebO52n4ujiU7Tz2Cv1ahXhTFbpJnufyGwnHqKfq9Pz7uOgIdAABgSaaI9KnjfKRZy32Qi43WkjVZn95sZPcsyW7r5pPUbyBsxEZwNxHoAAAASzRBpM8c523NtPeDsdc6Tz1SfZrkeBmj6k2U76WO8vaI+Vnq6ewbH+YjAh0AAGDJboj0ucR5WxPqh7k8aj1ynHoE+2SeU8uLohgF+W4ury9P6jB/vurR/HUk0AEAAFbgikife5y3NaG+1/y66sz009Sxnub386qqTid4zvFfu7lYAz9ylmbkvqqq41n/GfpOoAMAAKxIK9IXGufjmnXqo1HuvVvufp463tu2cnXkt42C//i20Kcm0AEAAFaoLMv9sixXNt27ifXR+vCdTBbf40YRP/r9eJN3Y5+VQAcAAOA9renro2gfd5IkNnmbH4EOAAAAa+AfrPoCAAAAgORrq74AAACArmjO9N7J+0eHnZjqzV0ZQQcAALhBURS7RVEcFkXxsySv8v6u5ztJXhVFUTX3u+q8cbiVNegAAABXaEL7IPUO56Pjwq7cbb3ZCf2g+ZUkz5M8r6rqfBnXSj8IdAAAgJYmtg9Tj5SfJ3l6XZhf8did1KPsW6mPG3sk0pmUQAcAAGg0o+YvUgf2SZLH0wZ28xyvmg9Pq6p6MN+rpK+sQQcAAEhSFMV+Lka/j6qqmmn0u9ks7nnz4U5RFM/meJn0mBF0AABg4zURPVo/flRV1ZM7Pt92kq9aN31UVdXZXZ6T/jOCDgAAbLRm5HwU56d3jfMkaWL8uHXT/l2fk/4T6AAAwMZq4vyw+fA8yeM5Pn17xFygcyuBDgAAbKRmx/X2+vCnc56GftL681bzenAtgQ4AAGyqw9QbwiXJyaRHqd3B7oKfn44T6AAAwMZpNoVrj2g/v+6+c7R1+13YZAIdAADYKEVRbOXymvCz5mg0WCmBDgAAbJqDXB7NXtTouRFzpiLQAQCAjXHF6Hly+Ti0ebIpHFMR6AAAwCbZy+WR7eOqqs4X9FpG0JmKQAcAADbJ+Oj56QJfa3zXduvcuZFABwAANkIzvX182vlCorkoiu0k22M3z/OMdXpIoAMAAJvivTXhVVUtagR9fPT8rKoqgc6NBDoAALApljnlfPy1FrURHT0i0AEAgE21kBHtZir93tjNAp1bCXQAAGBTjE9xX9Tu7eNxfrLAqfT0iEAHAACYr4Oxj5+v5CroHIEOAAAwJ0VR7Oby7u0nVVU5Xo2JCHQAAGBTLGOa+bOxj58u4TXpCYEOAABsikWtOU+SFEWxn8vr3J9be840BDoAALApxqeab83riZud29uj5ydVVRk9ZyoCHQAA2AjNaHb7aLXxs8rv4kUugv80yeM5PjcbQqADAACbpL2j+nZRFNvX3nNCRVEc5iL2T5M8qqpqodPp6SeBDgAAbIyqqo5yeRR9/Ei0qRRF8SzJfvOhOOdOiqqqVn0NAAAAS9MchfaqddODWTZza0bOR3F+kuSxOOcujKADAAAbpTmX/EnrpsNmk7eJFEWxXRTFq1zE+fOqqoycc2dG0AEAgI1UFMVeksPUm7udpR4Bv3YkvYn4g9RhvpV61PxJVVVn1z0GpiHQAQCAjdVsEjeK7iQ5Th3ep1VVnTbT4bdTn2++lzrMj5IcNyPxMDcCHQAA2HhNqO+l3o19u/k1cpJ6hP00dZibys5CCHQAAABYAzaJAwAAgDUg0AEAAGANCHQAAABYAwIdAAAA1sD/D9BkoQF34J1lAAAAAElFTkSuQmCC",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 6,
     "metadata": {
      "image/png": {
       "width": 500
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='decrochage.png', width=500)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2ae0ecd1-ddba-4073-b055-23d02b5dbe11",
   "metadata": {},
   "source": [
    "On a un repère $(X,Y,Z)$ au point $O$. L'échelle tourne autour de $O$ d'un angle $\\theta(t)$, dépendent du temps.\n",
    "\n",
    "Le repère mobile $(x,y,z)$ attaché à l'échelle, au centre de masse $G$ (sur lequel est appliqué le poids $mg$), est relié au repère fixe par un rotation:\n",
    "$$\\begin{Bmatrix}\\hat{\\bm x} \\\\ \\hat{\\bm y} \\\\ \\hat{\\bm z} \\end{Bmatrix} = R_z(\\theta)\\begin{Bmatrix}\\hat{\\bm X} \\\\ \\hat{\\bm Y} \\\\ \\hat{\\bm Z} \\end{Bmatrix}$$\n",
    "\n",
    "Les coordonées du centre de masse sont\n",
    "$$\\bm G = \\frac{l}{2}\\big(\\cos\\theta \\hat{\\bm X} + \\sin\\theta \\hat{\\bm Y}\\big) = \\frac{l}{2} \\hat{\\bm x}$$\n",
    "ce qui donne la vitesse\n",
    "$$\\dot{\\bm G} = \\frac{l \\dot\\theta}{2}\\big(-\\sin\\theta \\hat{\\bm X} + \\cos\\theta \\hat{\\bm Y}\\big) = \\frac{l \\dot\\theta}{2} \\hat{\\bm y}$$\n",
    "et l'accelération\n",
    "\\begin{align*}\n",
    "\\ddot{\\bm G} &= -\\frac{l}{2}\\big(\\ddot\\theta \\sin\\theta + \\dot\\theta^2 \\cos\\theta \\big) \\hat{\\bm X} + \\frac{l}{2}\\big(\\ddot\\theta \\cos\\theta - \\dot\\theta^2 \\sin\\theta \\big)\\hat{\\bm Y} \\\\\n",
    "&= \\frac{l \\ddot\\theta}{2} \\hat{\\bm y}-\\frac{l \\dot\\theta^2}{2} \\hat{\\bm x}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fdb5b725-1c78-4004-9a14-cc155087a1ce",
   "metadata": {},
   "source": [
    "### Moment cinétique\n",
    "\n",
    "La vitesse angulaire du centre de masse, dans le repère mobile de centre $O$ est\n",
    "$$\\bm \\omega = \\dot{\\theta} \\hat{\\bm z} = \\dot{\\theta} \\hat{\\bm Z}$$\n",
    "(on note que la rotation ne change pas l'axe $Z$).\n",
    "\n",
    "Le moment cinétique (par rapport à $O$) et le moment sont:\n",
    "$$\\bm L = I \\bm \\omega = I\\dot{\\theta} \\hat{\\bm Z}, \\quad \\frac{\\D \\bm L}{\\D t} = I\\ddot{\\theta} \\hat{\\bm Z}$$\n",
    "toujours au centre de masse. \n",
    "\n",
    "La variation du moment cinétique $\\dot{\\bm L}$, satisfait:\n",
    "$$\\dot{\\bm L} = I\\dot{\\bm \\omega} + \\bm \\omega \\times I \\bm \\omega = \\bm M_O$$\n",
    "où $\\bm M_O$ est le moment des forces (en $O$)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff24c7f5-fd3d-49f9-a349-a3a1df7e184f",
   "metadata": {},
   "source": [
    "### Équations de mouvement\n",
    "\n",
    "Le forces sont:\n",
    "\n",
    "* le poids $-mg \\hat{\\bm X}$\n",
    "* la force normale en $O$: $\\bm N = N \\hat{\\bm X}$\n",
    "* et la force de frottement statique: $\\bm f = f \\hat{\\bm Y}$\n",
    "\n",
    "L'équation de Newton est:\n",
    "$$ m \\ddot G = (N-mg) \\hat{\\bm X} + f \\hat{\\bm Y}$$\n",
    "ou, en remplaçant l'accelération du centre de masse par son expression, on obtient\n",
    "$$N = mg - \\frac{ml}{2}\\big(\\ddot\\theta \\sin\\theta + \\dot\\theta^2 \\cos\\theta \\big)$$\n",
    "et\n",
    "$$f =  \\frac{ml}{2}\\big(\\ddot\\theta \\cos\\theta - \\dot\\theta^2 \\sin\\theta \\big)$$\n",
    "\n",
    "Le terme de rotation de l'équation d'Euler s'annule $\\bm \\omega\\times I\\bm\\omega=0$, d'où:\n",
    "$$I_G \\dot{\\bm \\omega} = \\bm G \\times \\big(-mg \\hat{\\bm X}\\big)$$\n",
    "pusique seule le poids exerce un moment par rapport à $O$:\n",
    "$$\\ddot{\\theta} = \\frac{3g}{2l} \\sin\\theta$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1bcdf024-0c71-4568-a1c2-963ce4fa0e71",
   "metadata": {},
   "source": [
    "## Frottement\n",
    "\n",
    "Au point de contact de deux solides on a une force de réaction:\n",
    "$$\\bm R = \\bm T + \\bm N$$\n",
    "qu'on décompose en partie tangentielle et normale à la surface de contact.\n",
    "\n",
    "Frottement dynamique: la force s'oppose au mouvement tangentiel,\n",
    "$$\\bm T = - \\mu_d N \\hat{\\bm v}_t$$\n",
    "où $\\hat{\\bm v}_t$ est la direction du mouvement au point de contact, et $N$ est la grandeur de la force normale en ce point.\n",
    "\n",
    "Frottement statique ($\\bm V_t = 0$):\n",
    "$$T = \\mu_s N$$\n",
    "\n",
    "Le frottement dynamique est plus petit que le frottement statique: $\\mu_d < \\mu_s$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "953f984e-931c-4e87-ac5d-535aac5beb32",
   "metadata": {},
   "source": [
    "En utilisant les formules:\n",
    "\n",
    "* $\\ddot\\theta = (3g/2l)\\sin\\theta$\n",
    "* et $\\dot\\theta^2 = (3g/l)(1 - \\cos\\theta)$ (on suppose qu'on part du sommet à vitesse nulle)\n",
    "\n",
    "pour exprimer les forces en fonction de l'angle, on obtient:\n",
    "$$N = \\frac{mg}{4}(3\\cos\\theta - 1)^2$$\n",
    "et\n",
    "$$f = \\frac{3mg}{4} \\sin\\theta(3\\cos\\theta - 2)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c575d807-2da8-42b1-bfc5-3a7146908544",
   "metadata": {},
   "source": [
    "### Condition de glissement\n",
    "\n",
    "Le décrochage se produit au moment ou les forces $N$ et $f$ atteignent la limite du frottement statique $f = \\mu N$:\n",
    "$$\\frac{3mg}{4} \\sin\\theta(3\\cos\\theta - 2) = \\mu \\frac{mg}{4}(3\\cos\\theta - 1)^2$$\n",
    "ou\n",
    "$$\\sin\\theta(3\\cos\\theta - 2)=\\frac{\\mu}{3}(3\\cos\\theta - 1)^2$$\n",
    "\n",
    "On cherche d'abord une solution graphique. On choisit une valeur du coefficient de frottement $\\mu \\sim 0.1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "d6cdcaf1-07f7-47fa-9ca0-5c1233f92112",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Graphe (on écrit theta = t)\n",
    "mu = 0.1\n",
    "t = np.linspace(0, np.pi/2, 129)\n",
    "f = np.sin(t)*(3*np.cos(t) - 2)\n",
    "N = (mu/3)*(3*np.cos(t) - 1)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "776b45a5-9a2a-4b11-a746-30623fb6d980",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x324 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(180*t/np.pi, f, color = '#800', label = r\"$f$\")\n",
    "ax.plot(180*t/np.pi, N, color = '#008', label = r\"$N$\")\n",
    "ax.set_xlabel('angle en degrés')\n",
    "ax.set_ylim(-0.1,0.4)\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "56b44948-6eff-44de-9dea-37db838e2825",
   "metadata": {},
   "source": [
    "On calcule maintenant la valeur de $\\theta$ numériquement"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "aa04366b-8d66-431f-98a2-a31c3c79d544",
   "metadata": {},
   "outputs": [],
   "source": [
    "import scipy\n",
    "from scipy.optimize import root_scalar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "27e993e7-6e2b-4036-b82c-be51da98462a",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fN(t):\n",
    "    mu = 0.1\n",
    "    f = np.sin(t)*(3*np.cos(t) - 2)\n",
    "    N = (mu/3)*(3*np.cos(t) - 1)**2\n",
    "    return f-N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "86baaa4f-13b5-4711-a3ec-d4bec0f9cb35",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 46.8853927590844$"
      ],
      "text/plain": [
       "46.88539275908442"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = root_scalar(fN, x0 = 0.5, x1 = 1.3)\n",
    "180*sol.root/np.pi"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e15d6545-a3b2-428f-bf2b-40477b9e63c2",
   "metadata": {},
   "source": [
    "On trouve finalement que pour $\\mu = 0.1$ l'angle de décrochage est de $\\theta \\approx 47^\\circ$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "10aa814a-8a6a-4f0f-8aa7-1452bfbd401d",
   "metadata": {},
   "source": [
    "# Énergie cinétique\n",
    "\n",
    "L'énergie cinétique de rotation est \n",
    "$$E_\\omega = \\frac{1}{2} \\bm \\omega I \\bm \\omega$$\n",
    "avec $I$ calculé par rapport à l'axe de rotation $O$ du corps. Selon le problème il faut ajouter l'énergie cinétique de mouvement de centre de masse; ce n'est pas le cas présent.\n",
    "\n",
    "En utilisant les calculs précédents de $\\bm \\omega$ et $I$ on obtient\n",
    "$$E_\\omega = \\frac{ml^2}{6}\\dot\\theta^2$$\n",
    "\n",
    "L'équation de conservation de l'énergie\n",
    "$$\\dot E_\\omega = \\bm F \\cdot \\bm v$$\n",
    "où $\\bm F$ sont les forces **conservatives** (ici la gravité), nous conduit à l'équation de mouvement de la barre:\n",
    "$$\\frac{ml^2}{3}\\dot\\theta \\ddot\\theta = \\frac{mgl}{2}\\sin\\theta \\dot \\theta $$\n",
    "pusique $\\bm v = \\dot{\\bm G}$ (la composante $\\hat{\\bm Y}$ de $\\bm v$ s'annule avec le produit scalaire, la force étant dans la direction $\\hat{\\bm X}$). On en déduit\n",
    "$$\\ddot\\theta = \\frac{3g}{2l} \\sin \\theta  $$\n",
    "équation identique à celle d'Euler trouvée précédemment."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d32844ce-b1f5-4053-b898-47469eb81d64",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d031ab69-8bb6-41e9-abc3-a5ae56aaef87",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "eb6a2000-5268-4fac-85a8-85368cb1e408",
   "metadata": {},
   "source": [
    "# Brouillon"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8231b8cb-c1c4-4d7e-8eff-d606f707111f",
   "metadata": {},
   "outputs": [],
   "source": [
    "# font\n",
    "# matplotlib.rcParams['font.serif'] = \"STIX Two Text\"\n",
    "# matplotlib.rcParams['mathtext.fontset'] = 'stix'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "c50aa76e-346d-4c08-b6cc-92d66ce9f37f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x324 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(t, f, color = '#800', label = r\"$f$\")\n",
    "ax.plot(t, N, color = '#008', label = r\"$N$\")\n",
    "ax.set_ylim(-0.1,0.4)\n",
    "ax.legend();"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c8c1fd1e-f67a-42a9-ad2b-0e7f5b78559a",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.11.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}