Cinematica_Exemplo01.ipynb

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   "source": [
    "# Cinemática - Exemplo 1a\n",
    "## Professor Paulo Victor"
   ]
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  {
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   "source": [
    "### Obtenha as velocidades lineares e angulares e as acelerações lineares e angulares da formiga P.  Considere a formiga parada no ventilador.\n",
    "![Formiga1](formiga.png)"
   ]
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   "source": [
    "from IPython.display import Video\n",
    "\n",
    "Video(\"formiga.mp4\",width=700)"
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       "    <path style=\"fill:#0000ff;fill-opacity:1;stroke-width:0.264583\" id=\"path252\" d=\"m 75.786994,136.8825 c -0.40051,0.50973 -0.648981,1.09762 -0.750657,1.73396 -0.07119,0.64938 0.179141,1.18788 0.569145,1.68602 0.385432,0.44205 0.892567,0.66899 1.452753,0.79443 0.584332,0.10402 1.168977,0.0659 1.740469,-0.0808 0.665832,-0.22962 1.030158,-0.52079 1.368859,-1.13717 0.278561,-0.64363 0.200107,-1.21016 -0.04598,-1.84392 -0.246483,-0.57581 -0.638998,-1.05925 -1.070584,-1.50518 -0.389551,-0.38993 -0.876855,-0.6175 -1.397939,-0.7713 -0.544915,-0.13949 -0.969579,-0.0262 -1.433759,0.2679 -0.436324,0.32663 -0.685051,0.80207 -0.88642,1.29518 -0.07549,0.20034 -0.135789,0.40673 -0.161017,0.61967 -0.01302,0.0844 0.0069,0.15418 0.02742,0.23726 0.184621,0.50351 0.896683,0.24241 0.712062,-0.26109 v 0 c 7.94e-4,0.005 0.01598,0.10244 0.01426,0.10806 0.01355,-0.14446 0.06375,-0.2828 0.110199,-0.41909 0.140351,-0.34977 0.302602,-0.69949 0.598959,-0.94548 0.250253,-0.17313 0.502068,-0.25609 0.807127,-0.17399 0.395989,0.11207 0.767466,0.27836 1.065131,0.57134 0.363865,0.37293 0.698145,0.77563 0.909849,1.25702 0.163549,0.39802 0.249891,0.79659 0.07927,1.21094 -0.185066,0.38172 -0.486061,0.62174 -0.890278,0.75636 -0.454242,0.12315 -0.921415,0.14872 -1.387751,0.073 -0.389117,-0.0794 -0.744651,-0.22589 -1.018995,-0.52509 -0.264858,-0.32601 -0.452482,-0.6658 -0.412771,-1.10125 0.06904,-0.50436 0.280924,-0.9663 0.591019,-1.37058 0.336661,-0.41744 -0.253694,-0.89356 -0.590357,-0.47611 z\"/>\n",
       "    <path style=\"fill:#0000ff;fill-opacity:1;stroke-width:0.264583\" id=\"path254\" d=\"m 76.061528,138.92085 c 0.799563,-0.0648 1.602018,-0.0753 2.403769,-0.0766 0.178591,0.0224 0.368625,-0.0244 0.54606,0.0111 0.531905,0.0684 0.628653,-0.68382 0.09675,-0.75223 v 0 c -0.213969,-0.0232 -0.429025,-0.0147 -0.644055,-0.0173 -0.824026,10e-4 -1.648766,0.0126 -2.470507,0.0796 -0.534125,0.0481 -0.466143,0.80344 0.06798,0.75537 z\"/>\n",
       "  </g>\n",
       "</svg>"
      ],
      "text/plain": [
       "<IPython.core.display.SVG object>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.core.display import SVG\n",
    "SVG(filename='ex01.svg')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Primeiramente, importaremos a biblioteca de computação simbólica e a sub-biblioteca de mecânica."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sy\n",
    "import sympy.physics.mechanics as me"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Criaremos o sistema referencial inercial e o sistema  referencial móvel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = me.ReferenceFrame('N') # Sistema inercial\n",
    "A = me.ReferenceFrame('A') # Sistema móvel"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A seguir, definiremos as varíaveis dinâmicas do nosso sistema. No caso desse problema, temos que o ângulo $\\theta$ está variando no tempo, porque o ventilador está girando. A princípio, consideraremos que a formiga está parada no vetilador."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "theta = me.dynamicsymbols('theta')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Como a formiga está parada no ventilador, teremos que $l$ é uma constante."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "l = sy.symbols('l')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definiremos a orientação do referencial móvel em relação ao inercial:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "A.orient(N,'Axis',[theta,N.z])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A velocidade angular de $A$ em relação a $N$ pode ser obtido por:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\dot{\\theta}\\mathbf{\\hat{n}_z}$"
      ],
      "text/plain": [
       "theta'*N.z"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.ang_vel_in(N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A aceleração angular de $A$ em relação a $N$ pode ser obtido por:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\ddot{\\theta}\\mathbf{\\hat{n}_z}$"
      ],
      "text/plain": [
       "theta''*N.z"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.ang_acc_in(N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Criaremos o ponto $O$, que é  a origem do sitema inercial."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "O = me.Point('O')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Já que este é um ponto fixo, precisamos  definir a velocidade dele como sendo zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "O.set_vel(N,0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Criaremos agora o ponto $P$ que representa a formiga."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = me.Point('P')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos definir a posição do ponto $P$ em relação ao ponto $O$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "P.set_pos(O, l*A.x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos imprimir a posição deste ponto com o comando:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle l\\mathbf{\\hat{a}_x}$"
      ],
      "text/plain": [
       "l*A.x"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.pos_from(O)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Obtemos a velocidade absoluta do ponto P com o seguinte comando:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle l \\dot{\\theta}\\mathbf{\\hat{a}_y}$"
      ],
      "text/plain": [
       "l*theta'*A.y"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.pos_from(O).dt(N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vamos definir essa velocidade como sendo a velocidade do ponto $P$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "P.set_vel(N,P.pos_from(O).dt(N))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Após defirmos a velocidade da partícula, podemos imprimi-la com o seguinte comando:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle l \\dot{\\theta}\\mathbf{\\hat{a}_y}$"
      ],
      "text/plain": [
       "l*theta'*A.y"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.vel(N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A aceleração do ponto $P$ em relação ao referencial $N$ pode ser facilmente obtida pelo comando:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -  l \\dot{\\theta}^{2}\\mathbf{\\hat{a}_x} + l \\ddot{\\theta}\\mathbf{\\hat{a}_y}$"
      ],
      "text/plain": [
       "- l*theta'**2*A.x + l*theta''*A.y"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P.acc(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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