{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Zeroth and First Moments of Mass Example\n",
    "\n",
    "Find the distance $\\mu$ such that the mass center is closest to the line $\\hat{\\mathbf{n}}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sm\n",
    "import sympy.physics.mechanics as me\n",
    "me.init_vprinting()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "m, u = sm.symbols('m, mu')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = me.ReferenceFrame('A')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Locations of each particle:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "p1 = A.y + A.z\n",
    "p2 = A.x + A.z\n",
    "p3 = A.x + A.y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Mass center vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 m + \\mu}{3 m + \\mu}\\mathbf{\\hat{a}_x} + \\frac{m + \\mu}{3 m + \\mu}\\mathbf{\\hat{a}_y} + \\frac{3 m}{3 m + \\mu}\\mathbf{\\hat{a}_z}$"
      ],
      "text/plain": [
       "2⋅m + μ        m + μ          3⋅m\n",
       "─────── a_x + ─────── a_y + ─────── a_z\n",
       "3⋅m + μ       3⋅m + μ       3⋅m + μ"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_star = (m*p1 + 2*m*p2 + u*p3)/(m + 2*m + u)\n",
    "p_star"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the line:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{3}}{3}\\mathbf{\\hat{a}_x} + \\frac{\\sqrt{3}}{3}\\mathbf{\\hat{a}_y} + \\frac{\\sqrt{3}}{3}\\mathbf{\\hat{a}_z}$"
      ],
      "text/plain": [
       "√3       √3       √3\n",
       "── a_x + ── a_y + ── a_z\n",
       "3        3        3"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = (A.x + A.y + A.z)/sm.sqrt(3)\n",
    "n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Form a vector that has a magnitude equal to the distance of interest:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\mu}{3 \\left(3 m + \\mu\\right)}\\mathbf{\\hat{a}_x} + \\frac{- m + \\frac{\\mu}{3}}{3 m + \\mu}\\mathbf{\\hat{a}_y} + \\frac{m - \\frac{2 \\mu}{3}}{3 m + \\mu}\\mathbf{\\hat{a}_z}$"
      ],
      "text/plain": [
       "                        μ           2⋅μ\n",
       "                   -m + ─       m - ───\n",
       "     μ                  3            3\n",
       "─────────── a_x + ─────── a_y + ─────── a_z\n",
       "3⋅(3⋅m + μ)       3⋅m + μ       3⋅m + μ"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = p_star - (p_star.dot(n)*n)\n",
    "D.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{\\frac{\\mu^{2} + \\left(- 3 m + \\mu\\right)^{2} + \\left(3 m - 2 \\mu\\right)^{2}}{\\left(3 m + \\mu\\right)^{2}}}}{3}$"
      ],
      "text/plain": [
       "      _________________________________\n",
       "     ╱  2             2              2 \n",
       "    ╱  μ  + (-3⋅m + μ)  + (3⋅m - 2⋅μ)  \n",
       "   ╱   ─────────────────────────────── \n",
       "  ╱                        2           \n",
       "╲╱                (3⋅m + μ)            \n",
       "───────────────────────────────────────\n",
       "                   3                   "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D_mag = D.magnitude().simplify()\n",
    "D_mag"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Find $\\mu$ such that "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{\\frac{\\mu^{2} + \\left(- 3 m + \\mu\\right)^{2} + \\left(3 m - 2 \\mu\\right)^{2}}{\\left(3 m + \\mu\\right)^{2}}} \\left(3 m + \\mu\\right)^{2} \\left(\\frac{- 18 m + 12 \\mu}{2 \\left(3 m + \\mu\\right)^{2}} - \\frac{\\mu^{2} + \\left(- 3 m + \\mu\\right)^{2} + \\left(3 m - 2 \\mu\\right)^{2}}{\\left(3 m + \\mu\\right)^{3}}\\right)}{3 \\left(\\mu^{2} + \\left(- 3 m + \\mu\\right)^{2} + \\left(3 m - 2 \\mu\\right)^{2}\\right)}$"
      ],
      "text/plain": [
       "      _________________________________                                       \n",
       "     ╱  2             2              2             ⎛                2         \n",
       "    ╱  μ  + (-3⋅m + μ)  + (3⋅m - 2⋅μ)            2 ⎜-18⋅m + 12⋅μ   μ  + (-3⋅m \n",
       "   ╱   ─────────────────────────────── ⋅(3⋅m + μ) ⋅⎜──────────── - ───────────\n",
       "  ╱                        2                       ⎜           2              \n",
       "╲╱                (3⋅m + μ)                        ⎝2⋅(3⋅m + μ)               \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                  ⎛ 2             2              2⎞           \n",
       "                                3⋅⎝μ  + (-3⋅m + μ)  + (3⋅m - 2⋅μ) ⎠           \n",
       "\n",
       "                     \n",
       "    2              2⎞\n",
       "+ μ)  + (3⋅m - 2⋅μ) ⎟\n",
       "────────────────────⎟\n",
       "         3          ⎟\n",
       "(3⋅m + μ)           ⎠\n",
       "─────────────────────\n",
       "                     \n",
       "                     "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D_mag.diff(u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{5 m}{3}$"
      ],
      "text/plain": [
       "5⋅m\n",
       "───\n",
       " 3 "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.solve(D_mag.diff(u), u)[0]"
   ]
  }
 ],
 "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": "3.6.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}