ex-14-091.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "6e963640",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "from oetils import init_plotting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "af0eba87",
   "metadata": {},
   "outputs": [],
   "source": [
    "W = init_plotting(latex=True, sans=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "e6b73ffd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>L</th>\n",
       "      <th>T</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>12.0</td>\n",
       "      <td>6.96</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>10.0</td>\n",
       "      <td>6.36</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>8.0</td>\n",
       "      <td>5.70</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>6.0</td>\n",
       "      <td>4.95</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>5.0</td>\n",
       "      <td>4.54</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>4.0</td>\n",
       "      <td>4.08</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>3.0</td>\n",
       "      <td>3.60</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>2.5</td>\n",
       "      <td>3.35</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>2.3</td>\n",
       "      <td>3.27</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      L     T\n",
       "0  12.0  6.96\n",
       "1  10.0  6.36\n",
       "2   8.0  5.70\n",
       "3   6.0  4.95\n",
       "4   5.0  4.54\n",
       "5   4.0  4.08\n",
       "6   3.0  3.60\n",
       "7   2.5  3.35\n",
       "8   2.3  3.27"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = 2.00  # Horizontal amplitude [m]\n",
    "g = 9.81  # Acceleration of gravity [m/s^2]\n",
    "data = pd.DataFrame({\n",
    "    'L': [12.00, 10.00, 8.00, 6.00, 5.00, 4.00, 3.00, 2.50, 2.30],  # Pendulum length [m]\n",
    "    'T': [6.96, 6.36, 5.70, 4.95, 4.54, 4.08, 3.60, 3.35, 3.27]     # Period [s]\n",
    "})\n",
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "53cc05be",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 3000x2086.34 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Least squares fit for five largest values\n",
    "x = np.stack([np.ones(5), data.L[:5]]).T\n",
    "y = data['T'][:5] ** 2\n",
    "w = np.linalg.lstsq(x, y)[0]\n",
    "\n",
    "# Plot\n",
    "fig, ax = plt.subplots()\n",
    "ax.set_xlabel(r\"$L\\quad/\\quad\\mathrm{m}$\")\n",
    "ax.set_ylabel(r\"$T^2\\quad/\\quad\\mathrm{s^2}$\")\n",
    "ax.plot(data.L[:5], data['T'][:5] ** 2, 'r.', label=\"Data\")\n",
    "ax.plot(data.L[5:], data['T'][5:] ** 2, 'g.', label=\"Additional data\")\n",
    "x = np.linspace(data.L.min(), data.L.max())\n",
    "ax.plot(x, sum(w[i] * x**i for i in range(2)), label=\"Fit\")\n",
    "ax.plot(x, 4 * np.pi**2 * x / g, 'k--', label=\"Theory (small-angle)\")\n",
    "ax.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "d6d53e73",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 3000x2086.34 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# A closer look at the deviation from the theory\n",
    "fig, ax = plt.subplots()\n",
    "ax.set_xlabel(r\"$L\\quad/\\quad\\mathrm{m}$\")\n",
    "ax.set_ylabel(r\"$T\\quad/\\quad 2 \\pi \\sqrt{L / g}$\")\n",
    "ax.plot(data.L[:5], data['T'][:5] / (2 * np.pi * np.sqrt(data.L[:5] / g)), 'r.', label=\"Data\")\n",
    "ax.plot(data.L[5:], data['T'][5:] / (2 * np.pi * np.sqrt(data.L[5:] / g)), 'g.-', label=\"Additional data\")\n",
    "x = np.linspace(data.L.min(), data.L.max())\n",
    "ax.plot(x, np.sqrt(sum(w[i] * x**i for i in range(2))) / (2 * np.pi * np.sqrt(x / g)), label=\"Fit\")\n",
    "ax.axvline(2.65, c='k', ls=':', label='$5\\%$ error')\n",
    "ax.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "b2772796",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "A 5% error is attaned when theta ≈ 49°\n"
     ]
    }
   ],
   "source": [
    "print(f\"A 5% error is attaned when theta ≈ {np.arcsin(D / 2.65) * 180 / np.pi:.0f}°\")"
   ]
  }
 ],
 "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": "3.10.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}