EMM441_A2_S4_HookesLaw.ipynb

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              {
                "output_type": "display_data",
                "data": {
                  "text/plain": "<Figure size 1000x500 with 1 Axes>",
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PSXd2/m5X+/btlZmZqZkzZ9q0T506VRaLJc9XorJlZmbmWKK/fPnyCgoKsvn+eXh43PZS/rl97z7//PMcz695eHhIUp4+ssDRx58tP5Z+N+OZLVdXVy1dulRNmjRRp06dtGPHDpvt27dv11tvvaWYmBgNGTJEw4YN08yZMxUfH+/owwNQSHFlC0CxUr9+fUVHR2vu3LlKTk5Wq1attGPHDn3wwQfq3Lmz2rRpI0kqV66cRowYoTFjxqhdu3Z66KGHdPDgQb377rtq0qSJevbsmev4ZcuWVVxcnFq0aKHIyEglJCSoQoUKmjVrllq0aKG6deuqb9++qly5spKSkrR161b99ttvNp+NdL3169drwIAB6tq1q6pXr66rV6/qo48+krOzs7p06XLL483IyMj16pevr6/69eunjh076qOPPpK3t7dq1aqlrVu3au3atbk+85MdpA4ePKiJEyda21u2bKlVq1bJzc3N+plOebFq1SodOHBAV69eVVJSktavX6+4uDiFhIToq6++UsmSJa197T1/t6tTp05q06aNXnnlFR07dkz169fXmjVrtGLFCsXExNgsBpEXFy5cUMWKFfXYY4+pfv368vT01Nq1a7Vz506bq4mNGjXSkiVLFBsbqyZNmsjT01OdOnW66dgdO3bU2LFj1adPHzVv3lw//vijFi1alOO5pSpVqsjHx0dz5sxR6dKl5eHhoaZNm+b6HJijjz/b7S79bg9HP7OVzd3d3XrV+sEHH1R8fLzq1Kmjy5cvKzo6WtWqVdOECRMkSWPGjNF///tf9enTRz/++KM16AIoxgpgBUQAcJhbLSN+/bLs2TIyMowxY8YYYWFhhouLixEcHGyMGDEix7LOhnFtqfcaNWoYLi4uhr+/v/H8888bf/755y3f48iRI0ZgYKBRs2ZN448//jAMwzB++eUXo1evXkZAQIDh4uJiVKhQwejYsaPxxRdf3PD4fv31V+PJJ580qlSpYpQsWdLw9fU12rRpY6xdu/aW5yY6OvqGy19XqVLFMAzD+PPPP40+ffoYZcuWNTw9PY2oqCjjwIEDRkhIiBEdHZ1jzPLlyxuSjKSkJGtbQkKCIcmIiIi4ZU2GkXN5b1dXVyMgIMBo27at8c477xipqam57peX83ej+ZC9LHj29+L6c/T35fEvXLhgDB482AgKCjJcXFyMatWqGW+++abNUvKGcW0589yWdL/+3KWnpxvDhg0z6tevb5QuXdrw8PAw6tevb7z77rs2+1y8eNH497//bfj4+NgsT5699HtuS/9fvnzZGDJkiBEYGGi4u7sb4eHhxtatW41WrVrlWGZ9xYoVRq1atYwSJUrYLAP/96XfHX381/e9naXfb/axAPkht3lx9uxZo1atWkZAQIBx+PBhY/DgwYazs7Oxfft2m37fffedUaJECeP555/Pz5IB3KUshlEAT7ECAAAAQBHHM1sAAAAAYALCFgAAAACYgLAFAAAAACYo0LC1adMmderUSUFBQbJYLFq+fLnNdsMwNGrUKAUGBsrd3V2RkZE6fPiwTZ/z58+rR48e8vLyko+Pj5566qk8faghAAAAAJipQMPWpUuXVL9+fc2aNSvX7ZMnT9b06dM1Z84cbd++XR4eHoqKitLly5etfXr06KGff/5ZcXFx+vrrr7Vp0yY988wz+XUIAAAAAJCru2Y1QovFomXLlqlz586Srl3VCgoK0pAhQzR06FBJUkpKivz9/bVw4UJ1795d+/fvV61atbRz5041btxYkrR69Wq1b99ev/32m4KCggrqcAAAAAAUc3fthxofPXpUiYmJioyMtLZ5e3uradOm2rp1q7p3766tW7fKx8fHGrQkKTIyUk5OTtq+fbseeeSRXMdOT09Xenq69XVWVpbOnz8vPz8/WSwW8w4KAAAAwF3NMAxduHBBQUFBcnK6sxsB79qwlZiYKEny9/e3aff397duS0xMVPny5W22lyhRQr6+vtY+uZk0aZLGjBnj4IoBAAAAFBUnT55UxYoV72iMuzZsmWnEiBGKjY21vk5JSVGlSpV06NAh+fr6FmBlKOoyMjK0YcMGtWnTRi4uLgVdDoow5hryC3MN+YW5hvxy/vx5Va9eXaVLl77jse7asBUQECBJSkpKUmBgoLU9KSlJDRo0sPY5c+aMzX5Xr17V+fPnrfvnxs3NTW5ubjnafX195efn54DqgdxlZGSoVKlS8vPz4x8KmIq5hvzCXEN+Ya4hvzni8aK79nO2wsLCFBAQoHXr1lnbUlNTtX37djVr1kyS1KxZMyUnJ2vXrl3WPuvXr1dWVpaaNm2a7zUDAAAAQLYCvbJ18eJFHTlyxPr66NGj2rNnj3x9fVWpUiXFxMRo/PjxqlatmsLCwjRy5EgFBQVZVyysWbOm2rVrp759+2rOnDnKyMjQgAED1L17d1YiBAAAAFCgCjRsfffdd2rTpo31dfZzVNHR0Vq4cKFefPFFXbp0Sc8884ySk5PVokULrV69WiVLlrTus2jRIg0YMED333+/nJyc1KVLF02fPj3fjwUAAAAArlegYat169a62cd8WSwWjR07VmPHjr1hH19fXy1evNiM8gAAAADAbnftM1sAAAAAUJgRtgAAAADABIQtAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwASELQAAAAAwAWELAAAAAExA2AIAAAAAExC2AAAAAMAEhC0AAAAAMAFhCwAAAABMQNgCAAAAABMQtgAAAADABIQtAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwASELQAAAAAwAWELAAAAAExA2AIAAAAAExC2AAAAAMAEhC0AAAAAMAFhCwAAAABMQNgCAAAAABMQtgAAAADABIQtAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwASELQAAAAAwAWELAAAAAExA2AIAAAAAExC2AAAAAMAEhC0AAAAAMAFhCwAAAABMQNgCAAAAABMQtgAAAADABIQtAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwASELQAAAAAwAWELAAAAAExA2AIAAAAAExC2AAAAAMAEhC0AAAAAMAFhCwAAAABMQNgCAAAAABMQtgAAAADABIQtAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwASELQAAAAAwAWELAAAAAExA2AIAAAAAExC2AAAAAMAEhC0AAAAAMAFhCwAAAABMQNgCAAAAABMQtgAAAADABIQtAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwAR3ddjKzMzUyJEjFRYWJnd3d1WpUkXjxo2TYRjWPoZhaNSoUQoMDJS7u7siIyN1+PDhAqwaAAAAAO7ysPXGG29o9uzZmjlzpvbv36833nhDkydP1owZM6x9Jk+erOnTp2vOnDnavn27PDw8FBUVpcuXLxdg5QAAAACKuxIFXcDNbNmyRQ8//LA6dOggSQoNDdUnn3yiHTt2SLp2VWvatGl69dVX9fDDD0uSPvzwQ/n7+2v58uXq3r17gdUOAAAAoHi7q8NW8+bNNXfuXB06dEjVq1fX3r17lZCQoClTpkiSjh49qsTEREVGRlr38fb2VtOmTbV169Ybhq309HSlp6dbX6empkqSMjIylJGRYeIRobjLnl/MM5iNuYb8wlxDfmGuIb84co7d1WFr+PDhSk1NVY0aNeTs7KzMzExNmDBBPXr0kCQlJiZKkvz9/W328/f3t27LzaRJkzRmzJgc7Rs2bFCpUqUceARA7uLi4gq6BBQTzDXkF+Ya8gtzDWZLS0tz2Fh3ddj67LPPtGjRIi1evFi1a9fWnj17FBMTo6CgIEVHR9s97ogRIxQbG2t9nZqaquDgYLVp00Z+fn6OKB3IVUZGhuLi4tS2bVu5uLgUdDkowphryC/MNeQX5hryy7lz5xw21l0dtoYNG6bhw4dbbwesW7eujh8/rkmTJik6OloBAQGSpKSkJAUGBlr3S0pKUoMGDW44rpubm9zc3HK0u7i48MOLfMFcQ35hriG/MNeQX5hrMJsj59ddvRphWlqanJxsS3R2dlZWVpYkKSwsTAEBAVq3bp11e2pqqrZv365mzZrla60AAAAAcL27+spWp06dNGHCBFWqVEm1a9fW999/rylTpujJJ5+UJFksFsXExGj8+PGqVq2awsLCNHLkSAUFBalz584FWzwAAACAYu2uDlszZszQyJEj1a9fP505c0ZBQUF69tlnNWrUKGufF198UZcuXdIzzzyj5ORktWjRQqtXr1bJkiULsHIAAAAAxd1dHbZKly6tadOmadq0aTfsY7FYNHbsWI0dOzb/CgMAAACAW7irn9kCAAAAgMKKsAUAAAAAJiBsAQAAAIAJCFsAAAAAYALCFgAAAACYgLAFAAAAACYgbAEAAACACQhbAAAAAGACwhYAAAAAmICwBQAAAAAmIGwBAAAAgAkIWwAAAABgAsIWAAAAAJiAsAUAAAAAJiBsAQAAAIAJCFsAAAAAYALCFgAAAACYgLAFAAAAACYgbAEAAACACQhbAAAAAGACwhYAAAAAmICwBQAAAAAmIGwBAAAAgAkIWwAAAABgAsIWAAAAAJiAsAUAAAAAJiBsAQAAAIAJCFsAAAAAYALCFgAAAACYgLAFAAAAACYgbAEAAACACQhbAAAAAGACwhYAAAAAmICwBQAAAAAmIGwBAAAAgAkIWwAAAABgAsIWAAAAAJiAsAUAAAAAJiBsAQAAAIAJHBa2kpOTHTUUAAAAABR6doWtN954Q0uWLLG+7tatm/z8/FShQgXt3bvXYcUBAAAAQGFlV9iaM2eOgoODJUlxcXGKi4vTqlWr9OCDD2rYsGEOLRAAAAAACqMS9uyUmJhoDVtff/21unXrpgceeEChoaFq2rSpQwsEAAAAgMLIritbZcqU0cmTJyVJq1evVmRkpCTJMAxlZmY6rjoAAAAAKKTsurL16KOP6t///reqVaumc+fO6cEHH5Qkff/996patapDCwQAAACAwsiusDV16lSFhobq5MmTmjx5sjw9PSVJp0+fVr9+/RxaIAAAAAAURnaFLRcXFw0dOjRH++DBg++4IAAAAAAoCuwKW5J0+PBhbdiwQWfOnFFWVpbNtlGjRt1xYQAAAABQmNkVtubNm6fnn39eZcuWVUBAgCwWi3WbxWIhbAEAAAAo9uwKW+PHj9eECRP00ksvOboeAAAAACgS7Fr6/c8//1TXrl0dXQsAAAAAFBl2ha2uXbtqzZo1jq4FAAAAAIoMu24jrFq1qkaOHKlt27apbt26cnFxsdk+aNAghxQHAAAAAIWVXWFr7ty58vT0VHx8vOLj4222WSwWwhYAAACAYs+usHX06FFH1wEAAAAARYpdz2xdzzAMGYbhiFoAAAAAoMiwO2x9+OGHqlu3rtzd3eXu7q569erpo48+cmRtAAAAAFBo2XUb4ZQpUzRy5EgNGDBA4eHhkqSEhAQ999xzOnv2rAYPHuzQIgEAAACgsLErbM2YMUOzZ89Wr169rG0PPfSQateurddee42wBQAAAKDYs+s2wtOnT6t58+Y52ps3b67Tp0/fcVEAAAAAUNjZFbaqVq2qzz77LEf7kiVLVK1atTsuCgAAAAAKO7tuIxwzZowef/xxbdq0yfrM1rfffqt169blGsIAAAAAoLix68pWly5dtH37dpUtW1bLly/X8uXLVbZsWe3YsUOPPPKIo2sEAAAAgELHritbktSoUSN9/PHHjqwFAAAAAIoMu65sOTs768yZMznaz507J2dn5zsuCgAAAAAKO7vClmEYubanp6fL1dX1jgoCAAAAgKLgtm4jnD59uiTJYrHoP//5jzw9Pa3bMjMztWnTJtWoUcOhBZ46dUovvfSSVq1apbS0NFWtWlULFixQ48aNJV0LfqNHj9a8efOUnJys8PBwzZ49m1URAQAAABSo2wpbU6dOlXQt4MyZM8fmlkFXV1eFhoZqzpw5Divuzz//VHh4uNq0aaNVq1apXLlyOnz4sMqUKWPtM3nyZE2fPl0ffPCBwsLCNHLkSEVFRWnfvn0qWbKkw2oBAAAAgNtxW2Hr6NGjkqQ2bdpo6dKlNqHHDG+88YaCg4O1YMECa1tYWJj1z4ZhaNq0aXr11Vf18MMPS5I+/PBD+fv7a/ny5erevbup9QEAAADAjdi1GuH8+fNND1qS9NVXXykqKkpdu3ZVfHy8KlSooH79+qlv376SroW/xMRERUZGWvfx9vZW06ZNtXXr1huGrfT0dKWnp1tfp6amSpIyMjKUkZFh4hGhuMueX8wzmI25hvzCXEN+Ya4hvzhyjtkVtqpWraqKFSuqVatWat26tVq1aqWqVas6rKhsv/76q2bPnq3Y2Fi9/PLL2rlzpwYNGiRXV1dFR0crMTFRkuTv72+zn7+/v3VbbiZNmqQxY8bkaN+wYYNKlSrl2IMAchEXF1fQJaCYYK4hvzDXkF+YazBbWlqaw8ayGDdaWvAmTp06pY0bNyo+Pl7x8fE6fPiwgoKC1KpVK7Vp00ZPP/20Q4pzdXVV48aNtWXLFmvboEGDtHPnTm3dulVbtmxReHi4fv/9dwUGBlr7dOvWTRaLRUuWLMl13NyubAUHB+v06dPy8/NzSO1AbjIyMhQXF6e2bdvKxcWloMtBEcZcQ35hriG/MNeQX86dO6fAwEClpKTIy8vrjsay68pWhQoV1KNHD/Xo0UOSdPjwYU2YMEGLFi3Sp59+6rCwFRgYqFq1atm01axZU19++aUkKSAgQJKUlJRkE7aSkpLUoEGDG47r5uYmNze3HO0uLi788CJfMNeQX5hryC/MNeQX5hrM5sj5ZdfnbKWlpWnNmjV6+eWX1bx5c9WrV0979+7VgAEDtHTpUocVFx4eroMHD9q0HTp0SCEhIZKuLZYREBCgdevWWbenpqZq+/btatasmcPqAAAAAIDbZdeVLR8fH5UpU0Y9evTQ8OHDFRERYcqCGYMHD1bz5s01ceJEdevWTTt27NDcuXM1d+5cSdc+7ysmJkbjx49XtWrVrEu/BwUFqXPnzg6vBwAAAADyyq6w1b59eyUkJOjTTz9VYmKiEhMT1bp1a1WvXt2hxTVp0kTLli3TiBEjNHbsWIWFhWnatGnW2xcl6cUXX9SlS5f0zDPPKDk5WS1atNDq1av5jC0AAAAABcqusLV8+XJJ0g8//KD4+HitWbNGI0eOVIkSJdS6dWstWrTIYQV27NhRHTt2vOF2i8WisWPHauzYsQ57TwAAAAC4U3aFrWx169bV1atXdeXKFV2+fFnffPONlixZ4tCwBQAAAACFkV0LZEyZMkUPPfSQ/Pz81LRpU33yySeqXr26vvzyS/3xxx+OrhEAAAAACh27rmx98sknatWqlZ555hlFRETI29vb0XUBAAAAQKFmV9jauXOno+sAAAAAgCLFrtsIJWnz5s3q2bOnmjVrplOnTkmSPvroIyUkJDisOAAAAAAorOwKW19++aWioqLk7u6u77//Xunp6ZKklJQUTZw40aEFAgAAAEBhZFfYGj9+vObMmaN58+bJxcXF2h4eHq7du3c7rDgAAAAAKKzsClsHDx5Uy5Ytc7R7e3srOTn5TmsCAAAAgELPrrAVEBCgI0eO5GhPSEhQ5cqV77goAAAAACjs7Apbffv21QsvvKDt27fLYrHo999/16JFizR06FA9//zzjq4RAAAAAAodu5Z+Hz58uLKysnT//fcrLS1NLVu2lJubm4YOHaqBAwc6ukYAAAAAKHTsClsWi0WvvPKKhg0bpiNHjujixYuqVauWPD09HV0fAAAAABRKdoWtbK6urqpVq5ajagEAAACAIsOusHXp0iW9/vrrWrdunc6cOaOsrCyb7b/++qtDigMAAACAwsqusPX0008rPj5eTzzxhAIDA2WxWBxdFwAAAAAUanaFrVWrVul///ufwsPDHV0PAAAAABQJdi39XqZMGfn6+jq6FgAAAAAoMuwKW+PGjdOoUaOUlpbm6HoAAAAAoEiw6zbCt99+W7/88ov8/f0VGhoqFxcXm+27d+92SHEAAAAAUFjZFbY6d+7s4DIAAAAAoGixK2yNHj3a0XUAAAAAQJFi1zNbAAAAAICbI2wBAAAAgAkIWwAAAABgAsIWAAAAAJiAsAUAAAAAJrBrNcLMzEwtXLhQ69at05kzZ5SVlWWzff369Q4pDgAAAAAKK7vC1gsvvKCFCxeqQ4cOqlOnjiwWi6PrAgAAAIBCza6w9emnn+qzzz5T+/btHV0PAAAAABQJdj2z5erqqqpVqzq6FgAAAAAoMuwKW0OGDNE777wjwzAcXQ8AAAAAFAl23UaYkJCgDRs2aNWqVapdu7ZcXFxsti9dutQhxQEAAABAYWVX2PLx8dEjjzzi6FoAAAAAoMiwK2wtWLDA0XUAAAAAQJHChxoDAAAAgAnyfGWrYcOGWrduncqUKaN77733pp+ttXv3bocUBwAAAACFVZ7D1sMPPyw3NzdJUufOnc2qBwAAAACKhDyHrdGjR+f6ZwAAAABATjyzBQAAAAAmIGwBAAAAgAkIWwAAAABgAsIWAAAAAJiAsAUAAAAAJsjzaoTZsrKyFB8fr82bN+v48eNKS0tTuXLldO+99yoyMlLBwcFm1AkAAAAAhUqer2z99ddfGj9+vIKDg9W+fXutWrVKycnJcnZ21pEjRzR69GiFhYWpffv22rZtm5k1AwAAAMBdL89XtqpXr65mzZpp3rx5atu2rVxcXHL0OX78uBYvXqzu3bvrlVdeUd++fR1aLAAAAAAUFnkOW2vWrFHNmjVv2ickJEQjRozQ0KFDdeLEiTsuDgAAAAAKqzzfRniroHU9FxcXValSxa6CAAAAAKAouK3VCHv16qULFy5YX+/du1cZGRkOLwoAAAAACrvbCluLFi3SX3/9ZX0dERGhkydPOrwoAAAAACjsbitsGYZx09cAAAAAgGv4UGMAAAAAMMFtf6jxvn37lJiYKOnala0DBw7o4sWLNn3q1avnmOoAAAAAoJC67bB1//3329w+2LFjR0mSxWKRYRiyWCzKzMx0XIUAAAAAUAjdVtg6evSoWXUAAAAAQJFyW2ErJCTErDoAAAAAoEjJ8wIZJ06cuK2BT506ddvFAAAAAEBRkeew1aRJEz377LPauXPnDfukpKRo3rx5qlOnjr788kuHFAgAAAAAhVGebyPct2+fJkyYoLZt26pkyZJq1KiRgoKCVLJkSf3555/at2+ffv75ZzVs2FCTJ09W+/btzawbAAAAAO5qeb6y5efnpylTpuj06dOaOXOmqlWrprNnz+rw4cOSpB49emjXrl3aunUrQQsAAABAsXfbS7+7u7vrscce02OPPWZGPQAAAABQJOT5yhYAAAAAIO8IWwAAAABgAsIWAAAAAJiAsAUAAAAAJiBsAQAAAIAJ7A5bH330kcLDwxUUFKTjx49LkqZNm6YVK1Y4rLi/e/3112WxWBQTE2Ntu3z5svr37y8/Pz95enqqS5cuSkpKMq0GAAAAAMgLu8LW7NmzFRsbq/bt2ys5OVmZmZmSJB8fH02bNs2R9Vnt3LlT7733nurVq2fTPnjwYP33v//V559/rvj4eP3+++969NFHTakBAAAAAPLKrrA1Y8YMzZs3T6+88oqcnZ2t7Y0bN9aPP/7osOKyXbx4UT169NC8efNUpkwZa3tKSormz5+vKVOm6L777lOjRo20YMECbdmyRdu2bXN4HQAAAACQV7f9ocaSdPToUd1777052t3c3HTp0qU7Lurv+vfvrw4dOigyMlLjx4+3tu/atUsZGRmKjIy0ttWoUUOVKlXS1q1b9c9//jPX8dLT05Wenm59nZqaKknKyMhQRkaGw+sHsmXPL+YZzMZcQ35hriG/MNeQXxw5x+wKW2FhYdqzZ49CQkJs2levXq2aNWs6pLBsn376qXbv3q2dO3fm2JaYmChXV1f5+PjYtPv7+ysxMfGGY06aNEljxozJ0b5hwwaVKlXqjmsGbiUuLq6gS0AxwVxDfmGuIb8w12C2tLQ0h41lV9iKjY1V//79dfnyZRmGoR07duiTTz7RpEmT9J///MdhxZ08eVIvvPCC4uLiVLJkSYeNO2LECMXGxlpfp6amKjg4WG3atJGfn5/D3gf4u4yMDMXFxalt27ZycXEp6HJQhDHXkF+Ya8gvzDXkl3PnzjlsLLvC1tNPPy13d3e9+uqrSktL07///W8FBQXpnXfeUffu3R1W3K5du3TmzBk1bNjQ2paZmalNmzZp5syZ+uabb3TlyhUlJyfbXN1KSkpSQEDADcd1c3OTm5tbjnYXFxd+eJEvmGvIL8w15BfmGvILcw1mc+T8sitsSVKPHj3Uo0cPpaWl6eLFiypfvrzDisp2//3351hwo0+fPqpRo4ZeeuklBQcHy8XFRevWrVOXLl0kSQcPHtSJEyfUrFkzh9cDAAAAAHll9wIZV69eVbVq1VSqVCnrc06HDx+Wi4uLQkNDHVJc6dKlVadOHZs2Dw8P+fn5WdufeuopxcbGytfXV15eXho4cKCaNWt2w8UxAAAAACA/2LX0e+/evbVly5Yc7du3b1fv3r3vtKbbMnXqVHXs2FFdunRRy5YtFRAQoKVLl+ZrDQAAAADwd3Zd2fr+++8VHh6eo/2f//ynBgwYcMdF3czGjRttXpcsWVKzZs3SrFmzTH1fAAAAALgddl3ZslgsunDhQo72lJQUZWZm3nFRAAAAAFDY2RW2WrZsqUmTJtkEq8zMTE2aNEktWrRwWHEAAAAAUFjZdRvh66+/rlatWumee+5RRESEJGnz5s1KTU3V+vXrHVogAAAAABRGdl3Zql27tn744Qd169ZNZ86c0YULF9SrVy8dOHAgx+qBAAAAAFAc3faVrYyMDLVr105z5szRxIkTzagJAAAAAAq9276y5eLioh9++MGMWgAAAACgyLDrNsKePXtq/vz5jq4FAAAAAIoMuxbIuHr1qt5//32tXbtWjRo1koeHh832KVOmOKQ4AAAAACis7ApbP/30kxo2bChJOnTokM02i8Vy51UBAAAAQCFnV9jasGGDo+sAAAAAgCLFrme2rvfbb7/pt99+c0QtAAAAAFBk2BW2srKyNHbsWHl7eyskJEQhISHy8fHRuHHjlJWV5egaAQAAAKDQses2wldeeUXz58/X66+/rvDwcElSQkKCXnvtNV2+fFkTJkxwaJEAAAAAUNjYFbY++OAD/ec//9FDDz1kbatXr54qVKigfv36EbYAAAAAFHt23UZ4/vx51ahRI0d7jRo1dP78+TsuCgAAAAAKO7vCVv369TVz5swc7TNnzlT9+vXvuCgAAAAAKOzsuo1w8uTJ6tChg9auXatmzZpJkrZu3aqTJ09q5cqVDi0QAAAAAAoju65stWrVSocOHdIjjzyi5ORkJScn69FHH9XBgwcVERHh6BoBAAAAoNC5rStbv/76q8LCwmSxWBQUFMRCGAAAAABwA7d1ZatatWr6448/rK8ff/xxJSUlObwoAAAAACjsbitsGYZh83rlypW6dOmSQwsCAAAAgKLArme2AAAAAAA3d1thy2KxyGKx5GgDAAAAANi6rQUyDMNQ79695ebmJkm6fPmynnvuOXl4eNj0W7p0qeMqBAAAAIBC6LbCVnR0tM3rnj17OrQYAAAAACgqbitsLViwwKw6AAAAAKBIYYEMAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwASELQAAAAAwAWELAAAAAExA2AIAAAAAExC2AAAAAMAEhC0AAAAAMAFhCwAAAABMQNgCAAAAABMQtgAAAADABIQtAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwASELQAAAAAwAWELAAAAAExA2AIAAAAAExC2AAAAAMAEhC0AAAAAMAFhCwAAAABMQNgCAAAAABMQtgAAAADABIQtAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwASELQAAAAAwAWELAAAAAExA2AIAAAAAExC2AAAAAMAEhC0AAAAAMAFhCwAAAABMQNgCAAAAABMQtgAAAADABIQtAAAAADABYQsAAAAATEDYAgAAAAATELYAAAAAwAR3ddiaNGmSmjRpotKlS6t8+fLq3LmzDh48aNPn8uXL6t+/v/z8/OTp6akuXbooKSmpgCoGAAAAgGvu6rAVHx+v/v37a9u2bYqLi1NGRoYeeOABXbp0ydpn8ODB+u9//6vPP/9c8fHx+v333/Xoo48WYNUAAAAAIJUo6AJuZvXq1TavFy5cqPLly2vXrl1q2bKlUlJSNH/+fC1evFj33XefJGnBggWqWbOmtm3bpn/+858FUTYAAAAA3N1h6+9SUlIkSb6+vpKkXbt2KSMjQ5GRkdY+NWrUUKVKlbR169Ybhq309HSlp6dbX6empkqSMjIylJGRYVb5gHV+Mc9gNuYa8gtzDfmFuYb84sg5VmjCVlZWlmJiYhQeHq46depIkhITE+Xq6iofHx+bvv7+/kpMTLzhWJMmTdKYMWNytG/YsEGlSpVyaN1AbuLi4gq6BBQTzDXkF+Ya8gtzDWZLS0tz2FiFJmz1799fP/30kxISEu54rBEjRig2Ntb6OjU1VcHBwWrTpo38/PzueHzgRjIyMhQXF6e2bdvKxcWloMtBEcZcQ35hriG/MNeQX86dO+ewsQpF2BowYIC+/vprbdq0SRUrVrS2BwQE6MqVK0pOTra5upWUlKSAgIAbjufm5iY3N7cc7S4uLvzwIl8w15BfmGvIL8w15BfmGszmyPl1V69GaBiGBgwYoGXLlmn9+vUKCwuz2d6oUSO5uLho3bp11raDBw/qxIkTatasWX6XCwAAAABWd/WVrf79+2vx4sVasWKFSpcubX0Oy9vbW+7u7vL29tZTTz2l2NhY+fr6ysvLSwMHDlSzZs1YiRAAAABAgbqrw9bs2bMlSa1bt7ZpX7BggXr37i1Jmjp1qpycnNSlSxelp6crKipK7777bj5XCgAAAAC27uqwZRjGLfuULFlSs2bN0qxZs/KhIgAAAADIm7v6mS0AAAAAKKwIWwAAAABgAsIWAAAAAJiAsAUAAAAAJiBsAQAAAIAJCFsAAAAAYALCFgAAAACYgLAFAAAAACYgbAEAAACACQhbAAAAAGACwhYAAAAAmICwBQAAAAAmIGwBAAAAgAkIWwAAAABgAsIWAAAAAJigREEXABQamZnS5s3S6dNSYKAUESE5Oxd0VQAAALhLEbaAvFi6VHrhBem33/6/rWJF6Z13pEcfLbi6AAAAcNfiNkLgVpYulR57zDZoSdKpU9faly4tmLoAAABwVyNsATeTmXntipZh5NyW3RYTc60fAAAAcB3CFnAzmzfnvKJ1PcOQTp681g8AAAC4DmELuJnTpx3bDwAAAMUGYQu4mcBAx/YDAABAsUHYAm4mIuLaqoMWS+7bLRYpOPhaPwAAAOA6hC3gZpydry3vLuUMXNmvp03j87YAAACQA2ELuJVHH5W++EKqUMG2vWLFa+18zhYAAABywYcaA3nx6KPSww9fW3Xw9Olrz2hFRHBFCwAAADdE2ALyytlZat26oKsAAABAIcFthAAAAABgAsIWAAAAAJiAsAUAAAAAJiBsAQAAAIAJCFsAAAAAYALCFgAAAACYgLAFAAAAACYgbAEAAACACQhbAAAAAGACwhYAAAAAmICwBQAAAAAmIGwBAAAAgAkIWwAAAABgAsIWAAAAAJiAsAUAAAAAJiBsAQAAAIAJCFsAAAAAYALCFgAAAACYgLAFAAAAACYgbAEAAACACQhbAAAAAGACwhYAAAAAmICwBQAAAAAmIGwBAAAAgAkIWwAAAABgAsIWAAAAAJiAsAUAAAAAJiBsAQAAAIAJCFsAAAAAYALCFgAAAACYgLAFAAAAACYgbAEAAACACQhbAAAAAGACwhYAAAAAmICwBQAAAAAmIGwBAAAAgAkIWwAAAABgAsIWAAAAAJiAsAUAAAAAJigyYWvWrFkKDQ1VyZIl1bRpU+3YsaOgSwIAAABQjBWJsLVkyRLFxsZq9OjR2r17t+rXr6+oqCidOXOmoEsDAAAAUEwVibA1ZcoU9e3bV3369FGtWrU0Z84clSpVSu+//35BlwYAAACgmCpR0AXcqStXrmjXrl0aMWKEtc3JyUmRkZHaunVrrvukp6crPT3d+jolJUWSdP78eXOLRbGXkZGhtLQ0nTt3Ti4uLgVdDoow5hryC3MN+YW5hvySnQkMw7jjsQp92Dp79qwyMzPl7+9v0+7v768DBw7kus+kSZM0ZsyYHO3Vq1c3pUYAAAAAhcu5c+fk7e19R2MU+rBljxEjRig2Ntb6Ojk5WSEhITpx4sQdn1DgZlJTUxUcHKyTJ0/Ky8uroMtBEcZcQ35hriG/MNeQX1JSUlSpUiX5+vre8ViFPmyVLVtWzs7OSkpKsmlPSkpSQEBArvu4ubnJzc0tR7u3tzc/vMgXXl5ezDXkC+Ya8gtzDfmFuYb84uR058tbFPoFMlxdXdWoUSOtW7fO2paVlaV169apWbNmBVgZAAAAgOKs0F/ZkqTY2FhFR0ercePG+sc//qFp06bp0qVL6tOnT0GXBgAAAKCYKhJh6/HHH9cff/yhUaNGKTExUQ0aNNDq1atzLJpxI25ubho9enSutxYCjsRcQ35hriG/MNeQX5hryC+OnGsWwxFrGgIAAAAAbBT6Z7YAAAAA4G5E2AIAAAAAExC2AAAAAMAEhC0AAAAAMEGxD1uzZs1SaGioSpYsqaZNm2rHjh0FXRKKmEmTJqlJkyYqXbq0ypcvr86dO+vgwYMFXRaKgddff10Wi0UxMTEFXQqKoFOnTqlnz57y8/OTu7u76tatq++++66gy0IRk5mZqZEjRyosLEzu7u6qUqWKxo0bJ9Z3w53atGmTOnXqpKCgIFksFi1fvtxmu2EYGjVqlAIDA+Xu7q7IyEgdPnz4tt+nWIetJUuWKDY2VqNHj9bu3btVv359RUVF6cy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        "<a href=\"https://colab.research.google.com/github/walkerjian/Physics/blob/main/EMM441_A2_S4_HookesLaw.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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        "#Hookes Law\n",
        "\n",
        "## Engage\n",
        "\n",
        "Start your lesson by setting up a simple demonstration. Hang a spring from a fixed point and attach various weights to it. With each weight, measure how much the spring stretches. Display these results in front of the class. Ask students to observe the changes in the length of the spring as more weight is added. Here are some questions to encourage discussion:\n",
        "\n",
        "- What do you notice about the spring as the weight increases?\n",
        "- Do you think there is a pattern in how the spring stretches with more weight?\n",
        "- What might happen if we keep adding more weight?\n",
        "\n",
        "This hands-on activity will help students visually grasp the relationship between weight (force) and spring extension.\n",
        "\n",
        "## Pose Problem\n",
        "\n",
        "Introduce Hooke's Law: It is a principle of physics that states the force $ F $ needed to extend or compress a spring by some distance $ x $ scales linearly with respect to that distance. That is, $ F = kx $, where:\n",
        "- $ F $ is the force applied to the spring,\n",
        "- $ x $ is the extension or compression of the spring (how much it stretches or shrinks),\n",
        "- $ k $ is the spring constant that depends on the material and make of the spring.\n",
        "\n",
        "Explain that this law helps in understanding how much force is needed to stretch or compress various materials, which is useful in numerous engineering and scientific applications.\n",
        "\n",
        "## Present the Function\n",
        "\n",
        "Write the formula $ F = kx $ on the board and explain each component:\n",
        "- **F (Force)**: This is the total force applied to the spring. In our case, it's due to the weights hung from the spring. This can be measured in newtons (N).\n",
        "- **x (Extension)**: This is how much the spring has stretched from its original length, measured in meters (m).\n",
        "- **k (Spring Constant)**: This is a constant for each spring, indicating how stiff the spring is. The higher the value of $ k $, the stiffer the spring, meaning it’s harder to stretch. This is measured in newtons per meter (N/m).\n",
        "\n",
        "Now, let's move to a Python code that will allow students to interactively explore Hooke's Law by changing the spring constant $ k $ and the extension $ x $ to see how it affects the force $ F $.\n",
        "\n",
        "## Python Code for Interactive Plot\n",
        "\n",
        "The following Python code uses `ipywidgets` and `matplotlib` to create an interactive plot where students can adjust $ k $ and $ x $ using sliders and observe the resulting force $ F $:\n"
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            "text/plain": [
              "interactive(children=(FloatSlider(value=5.0, description='k', max=10.0, min=0.1), FloatSlider(value=5.0, descr…"
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              "<function __main__.plot_hooke_law(k, x)>"
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              "<div style=\"max-width:800px; border: 1px solid var(--colab-border-color);\"><style>\n",
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              "         var(--colab-secondary-surface-color); padding: 8px 12px;\n",
              "         border-bottom: 1px solid var(--colab-border-color);\"><b>plot_hooke_law</b><br/>def plot_hooke_law(k, x)</pre><pre class=\"function-repr-contents function-repr-contents-collapsed\" style=\"\"><a class=\"filepath\" style=\"display:none\" href=\"#\">/content/&lt;ipython-input-1-34e254b050d1&gt;</a>&lt;no docstring&gt;</pre></div>"
            ]
          },
          "metadata": {},
          "execution_count": 1
        }
      ],
      "source": [
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "from ipywidgets import interact\n",
        "\n",
        "def plot_hooke_law(k, x):\n",
        "    F = k * x  # Calculate the force based on Hooke's Law\n",
        "    plt.figure(figsize=(10, 5))\n",
        "    plt.plot(x, F, 'ro')  # Plot the force vs extension as a red dot\n",
        "    plt.xlim(0, 10)\n",
        "    plt.ylim(0, 100)\n",
        "    plt.xlabel('Extension (x) in meters')\n",
        "    plt.ylabel('Force (F) in newtons')\n",
        "    plt.title('Hooke\\'s Law Demonstration: F = kx')\n",
        "    plt.grid(True)\n",
        "    plt.show()\n",
        "\n",
        "# Create interactive sliders for spring constant and extension\n",
        "interact(plot_hooke_law, k=(0.1, 10.0, 0.1), x=(0, 10, 0.1))"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "from ipywidgets import interact, FloatSlider, Button, VBox, HBox, Textarea\n",
        "\n",
        "# Default data points for extension (x values)\n",
        "x_data = np.array([0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0])\n",
        "\n",
        "# Function to plot the graph based on the spring constant and extensions\n",
        "def plot_hooke_law(k):\n",
        "    # Calculate force using Hooke's Law\n",
        "    F = k * x_data\n",
        "    plt.figure(figsize=(10, 5))\n",
        "    plt.plot(x_data, F, 'bo-', label=f'Spring Constant k = {k:.1f} N/m')\n",
        "    plt.xlim(0, max(x_data) + 1)\n",
        "    plt.ylim(0, max(F) + 10)\n",
        "    plt.xlabel('Extension (x) in meters')\n",
        "    plt.ylabel('Force (F) in newtons')\n",
        "    plt.title('Hooke\\'s Law: F = kx')\n",
        "    plt.legend()\n",
        "    plt.grid(True)\n",
        "    plt.show()\n",
        "\n",
        "# Slider for adjusting the spring constant\n",
        "k_slider = FloatSlider(value=1.0, min=0.1, max=10.0, step=0.1, description='Spring Constant k')\n",
        "\n",
        "# Text area for entering custom extension data\n",
        "data_input = Textarea(\n",
        "    value='0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0',\n",
        "    placeholder='Enter comma-separated extension values',\n",
        "    description='Extensions:',\n",
        "    disabled=False\n",
        ")\n",
        "\n",
        "# Button to update data\n",
        "update_button = Button(description='Update Data')\n",
        "\n",
        "# Function to update data based on input\n",
        "def update_data(b):\n",
        "    global x_data\n",
        "    try:\n",
        "        # Convert input string to numpy array of floats\n",
        "        x_data = np.array(list(map(float, data_input.value.split(','))))\n",
        "        plot_hooke_law(k_slider.value)  # Re-plot graph with new data\n",
        "    except ValueError:\n",
        "        print(\"Please enter valid comma-separated numbers.\")\n",
        "\n",
        "# Set the button's on_click function to update the data\n",
        "update_button.on_click(update_data)\n",
        "\n",
        "# Display widgets and initial plot\n",
        "display(VBox([HBox([data_input, update_button]), k_slider]))\n",
        "interact(plot_hooke_law, k=k_slider)\n"
      ],
      "metadata": {
        "colab": {
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        "outputId": "34319280-cf87-48d7-e0df-e92999456a92"
      },
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "VBox(children=(HBox(children=(Textarea(value='0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0', description='Extensions:', p…"
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          "output_type": "display_data",
          "data": {
            "text/plain": [
              "interactive(children=(FloatSlider(value=1.0, description='Spring Constant k', max=10.0, min=0.1), Output()), _…"
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              "model_id": "33285fd758e44af7a2b868415556871a"
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            "text/plain": [
              "<function __main__.plot_hooke_law(k)>"
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              "<div style=\"max-width:800px; border: 1px solid var(--colab-border-color);\"><style>\n",
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              "         var(--colab-secondary-surface-color); padding: 8px 12px;\n",
              "         border-bottom: 1px solid var(--colab-border-color);\"><b>plot_hooke_law</b><br/>def plot_hooke_law(k)</pre><pre class=\"function-repr-contents function-repr-contents-collapsed\" style=\"\"><a class=\"filepath\" style=\"display:none\" href=\"#\">/content/&lt;ipython-input-2-4bb92b07452c&gt;</a>&lt;no docstring&gt;</pre></div>"
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}