lecture 02.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "556b31ce",
   "metadata": {},
   "source": [
    "## Общая физика. Механика. Лекция 2. Кинематика материальной точки."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "18944c05",
   "metadata": {},
   "source": [
    "*(c) 2021 Alexey Shcherbakov. These lectures are licensed under a Creative <a href=\"https://creativecommons.org/licenses/by/4.0/\">Commons Attribution License CC-BY 4.0</a>. All code contained herein is licensed under an <a href=\"https://opensource.org/licenses/MIT\">MIT license</a>. The lectures were generated from a <a href=\"https://justinbois.github.io/bootcamp/2017/lessons/l37_jupyter.ipynb\">Jupyter notebook</a>.*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e815697",
   "metadata": {},
   "source": [
    "### План лекции:\n",
    "* основные понятия\n",
    "* ускорение материальной точки\n",
    "* радиус кривизны траектории\n",
    "* баллистическое движение\n",
    "* движение по окружности\n",
    "* циклоида"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb5b3cfd",
   "metadata": {},
   "source": [
    "### Основные понятия\n",
    "В начале курса мы будем иметь дело с простыми случаями и рассматривать движение материальных точек - таких физических объектов, размерами которых в условиях задачи можно пренебречь. Говоря о кинематике, мы подразумеваем, что изучаются закономерности движения без выяснения причин, по которым это движение происходит. Напомним основные понятия, известные из школьного курса общей физики, относящиеся к кинематике материальной точки.\n",
    "\n",
    "Для задания положения материальной точки в пространстве нужно ввести систему отсчета - физческое тело и привязанную к нему систему координат, например, декартову. В общем случае введение обобщенных координат подразумевает использование минимального числа независимых параметров, которые однозначно определяют положение тела в пространстве (в двумерном пространстве - две координаты, в трехмерном - три и т.п.). Например, полярные координаты $r,\\;\\varphi$ на плоскости или сферические координаты $r,\\;\\theta,\\;\\varphi$ в пространстве.\n",
    "\n",
    "Положение материальной точки в заданной системе отсчета можно по-разному. Можно ввести зависимый от времени радиус-вектор ${\\bf r}(t)$, или зафиксировать траекторию и отсчитывать путь, пройденный по этой траектории от некоторой начальной точки $s=s(t)$. Смещением будет вектор\n",
    "$$\\Delta{\\bf r}(t) = \\Delta{\\bf r}(t) - \\Delta{\\bf r}(0)$$\n",
    "причем, очевидно, что $|\\Delta{\\bf r}(t)| \\leq s(t)$. Вектор скорости ${\\bf v}(t)$ всегда направлен по касательной к траектории.\n",
    "\n",
    "<figure>\n",
    "    <center>\n",
    "    <img src=\"./fig/lect02_fig01.png\" width=\"300\" align=\"center\"/>\n",
    "        <figcaption>Траектория, перемещение, путь и вектор скорости материальной точки</figcaption>\n",
    "    </center>\n",
    "</figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5bb71022",
   "metadata": {},
   "source": [
    "### Ускорение материальной точки\n",
    "Рассмотрим декартову систему координат и зададим положение материальной точки радиус-вектором\n",
    "$${\\bf r}\\left(t\\right) = x\\hat{{\\bf e}}_{x}+y\\hat{{\\bf e}}_{y}+z\\hat{{\\bf e}}_{z} = \\left(x\\left(t\\right),y\\left(t\\right),z\\left(t\\right)\\right)$$\n",
    "Скорость есть производная радиус-вектора по времени:\n",
    "$${\\bf v} = \\dfrac{d{\\bf r}}{dt} = \\dfrac{dx}{dt}\\hat{{\\bf e}}_{x}+\\dfrac{dy}{dt}\\hat{{\\bf e}}_{y}+\\dfrac{dz}{dt}\\hat{{\\bf e}}_{z} = v_{x}\\hat{{\\bf e}}_{x}+v_{y}\\hat{{\\bf e}}_{y}+v_{z}\\hat{{\\bf e}}_{z}$$\n",
    "Ускорение – это скорость изменения скорости:\n",
    "$${\\bf a}=\\dfrac{d{\\bf v}}{dt}=\\dfrac{dv_{x}}{dt}\\hat{{\\bf e}}_{x}+\\dfrac{dv_{y}}{dt}\\hat{{\\bf e}}_{y}+\\dfrac{dv_{z}}{dt}\\hat{{\\bf e}}_{z}=a_{x}\\hat{{\\bf e}}_{x}+a_{y}\\hat{{\\bf e}}_{y}+a_{z}\\hat{{\\bf e}}_{z}=\\dfrac{d^{2}{\\bf r}}{dt^{2}}$$\n",
    "\n",
    "Обозначим путь вдоль траектории как $s$. Его еще называют естественным параметром движения, поскольку он задает положение материальной точки на траектории, отсчитываемое от некоторой начальной точки. Введем единичный вектор касательной к траектории $\\hat{\\tau} = d{\\bf r}/ds$. Тогда скорость можно выразить как\n",
    "$${\\bf v} = \\frac{d{\\bf r}}{dt} = \\frac{d{\\bf r}}{ds}\\frac{ds}{dt} = v\\left(t\\right)\\hat{\\tau}\\left(t\\right)$$\n",
    "а, зная скорость, путь будет вычисляться через интеграл\n",
    "$$v(t) = \\frac{ds}{dt} \\; \\Rightarrow\\;s=\\intop_{t_{1}}^{t_{2}}v\\left(t\\right)dt$$\n",
    "\n",
    "В общем случае движения материальной точки по криволинейной траектории в каждой точке траектории (при определенных математических ограничениях) можно ввести вектор касательной $\\hat{\\tau}$ и вектор нормали $\\hat{\\bf n}$. Оказывается, что ускорение материальной точки раскладывается по этим двум векторам:\n",
    "$${\\bf a}=\\dfrac{d{\\bf v}}{dt}=\\dfrac{d}{dt}\\left(v\\hat{\\tau}\\right)=\\dfrac{dv}{dt}\\hat{\\tau}+v\\dfrac{d\\hat{\\tau}}{dt}=$$\n",
    "$$=a_{\\tau}\\hat{\\tau}+v\\dfrac{d{\\bf \\tau}}{ds}\\dfrac{ds}{dt}=a_{\\tau}\\hat{\\tau}+v^{2}\\dfrac{d\\hat{\\tau}}{ds}=a_{\\tau}\\hat{\\tau}+\\dfrac{v^{2}}{R}\\hat{\\bf n}$$\n",
    "\n",
    "Чтобы получить значение производной $d\\hat{\\tau}/ds$ можно рассмотреть конечное приращение вектора $\\hat{\\tau}$ и ввести окружность, касательную в данной точке траектории (движение материальной точки в малой области представляется как движение по окружности):\n",
    "$$|\\Delta\\hat{\\bf \\tau}| = 2|\\hat{\\bf \\tau}|\\sin(\\alpha/2) = 2|\\hat{\\bf \\tau}|\\frac{|\\Delta{\\bf r}|/2}{R}$$\n",
    "$$\\Rightarrow \\frac{d\\hat{\\bf \\tau}}{ds} = \\hat{\\bf n} \\lim_{\\Delta s\\rightarrow 0} \\frac{|\\Delta\\hat{\\bf \\tau}|}{\\Delta s} = \\hat{\\bf n} \\lim_{\\Delta s\\rightarrow 0} \\frac{|\\Delta\\hat{\\bf \\tau}|}{|\\Delta {\\bf r}|} = \\frac{1}{R}\\hat{\\bf n}$$\n",
    "\n",
    "<figure>\n",
    "    <center>\n",
    "    <img src=\"./fig/lect02_fig02.png\" width=\"200\" align=\"center\"/>\n",
    "        <figcaption>К выводу нормальной компоненты вектора ускорения</figcaption>\n",
    "    </center>\n",
    "</figure>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "56a0bd8d",
   "metadata": {},
   "source": [
    "### Частные случаи\n",
    "\n",
    "Рассмотрим некоторые частные случаи, хорошо известные из школько курса физики.\n",
    "\n",
    "1. При одномерном движении начальные условия определяют неизвестные константы интегрирования: $$v_{x}\\left(t\\right)=\\dfrac{dx}{dt}=\\int adt=at+C_{1}\\Rightarrow x\\left(t\\right)=\\int\\left(at+C_{1}\\right)dt=\\dfrac{1}{2}at^{2}+C_{1}t+C_{2}$$ $$x\\left(0\\right)=x_{0},v\\left(0\\right)=v_{0}\\Rightarrow C_{1}=v_{0},C_{2}=x_{0}$$\n",
    "2. При движении тел, брошенных под углом к горизонту, $$\\begin{array}{c} a_{x}=0\\cos\\varphi\\\\ a_{y}=-g \\end{array} \\Rightarrow \\begin{array}{c} v_{x}=v_{0}\\cos\\varphi\\\\ v_{y}=v_{0}\\sin\\varphi-gt \\end{array} \\Rightarrow \\begin{array}{c}\n",
    "x=x_{0}+v_{0}t\\cos\\varphi\\\\\n",
    "y=y_{0}+v_{0}t\\sin\\varphi-\\dfrac{1}{2}gt^{2}\n",
    "\\end{array}$$ Условие касания земли дает полное время полета: $$y=0\\Rightarrow v_{0}t\\sin\\varphi-\\dfrac{1}{2}gt^{2}=0\\Rightarrow T=2\\dfrac{v_{0}}{g}\\sin\\varphi$$ Дальность полета и высота: $$\\begin{array}{c}\n",
    "L=x\\left(T\\right)=\\dfrac{v_{0}^{2}}{g}\\sin2\\varphi\\\\\n",
    "H=y\\left(T/2\\right)=\\dfrac{1}{2}\\dfrac{v_{0}^{2}}{g}\\sin^{2}\\varphi\n",
    "\\end{array}$$ Максимальную дальность и высоту можно найти из условия экстремальности: $$\\begin{array}{c}\n",
    "\\dfrac{dL}{d\\varphi}=2\\dfrac{v_{0}^{2}}{g}\\cos2\\varphi=0\\Rightarrow\\varphi=\\pi/4\\\\\n",
    "\\dfrac{dH}{d\\varphi}=\\dfrac{v_{0}^{2}}{2g}\\sin2\\varphi=0\\Rightarrow\\varphi=\\pi/2\n",
    "\\end{array}$$\n",
    "3. При равномерном движении по окружности можно ввести угловую скорость $$\\omega = \\dfrac{d\\varphi}{dt}$$ так что $$ds=Rd\\varphi\\Rightarrow\\dfrac{ds}{dt}=R\\dfrac{d\\varphi}{dt}\\Rightarrow v=R\\omega$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e9c77a0",
   "metadata": {},
   "source": [
    "### Циклоида\n",
    "Часто, если материальная точки совершает сложное движение, его можно представить как суперпозицию более простых. Наглядным примером является движение точек на ободе колеса. Если колесо радиуса $R$ катится без проскальзывания, то скорость нижней точки касания с землей равна нулю, а скорость центра колеса связана с угловой скоростью вращения как $v=\\omega R$. Положение произвольной точки обода есть сумма радиус-вектора центра колеса и вектора, задающего положение относительно центра, а скорость, соответственно, есть сумма скоростей центра и скорости относительно центра:\n",
    "$${\\bf r}={\\bf R}_{c}+{\\bf r}_1$$\n",
    "$${\\bf v}={\\bf v}_{c}+\\omega R\\hat{\\bf \\tau}$$\n",
    "При равномерном качении ${\\bf R}_c = {\\bf V}t$. Если отсчитывать угол поворота колеса относительно вертикали $\\varphi = \\omega t$, то в проекции на оси декартовой системы координат получается параметрическое уравнение траектории:\n",
    "$$x = R \\left( \\omega t + \\sin\\left(\\omega t\\right)\\right)$$\n",
    "$$y = R \\left(1 + \\cos\\left(\\omega t\\right)\\right)$$\n",
    "Эта кривая называется циклоидой. Одно из ее свойств: если ее перевернуть и сделать скат такой же формы, то время соскальзывания (без трения) малых тел с такого ската не будет зависеть от начального положения тела. Такая кривая еще называется изохронной или таутохронной. Другое свойства этой кривой – тела по ней будут съезжать быстрее всего (брахистрохрона). Гюйгенс предложил маятник, движущийся по огибаюдей циклоиды, период которого не зависит от амплитуды колебанийй:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "aae3bc83",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/jpeg": "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\n",
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"800\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/1BdO8J0iynY\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x7f039c6c76a0>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# брахистрона:\n",
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo('1BdO8J0iynY', width=800, height=300)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "5fb36d28",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/jpeg": "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\n",
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"800\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/EI5zwTvfkqA\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x7f039c692400>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# маятник Гюйгенса:\n",
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo('EI5zwTvfkqA', width=800, height=300)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "17878080",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 936x180 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7fe269bdc6a0>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# построим циклоиду:\n",
    "from sympy import *    \n",
    "from sympy.plotting import plot_parametric\n",
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "t = symbols('t')\n",
    "x = t-sin(t)\n",
    "y = 1-cos(t)\n",
    "plt.rcParams['figure.figsize'] = 13, 2.5\n",
    "plot_parametric(x, y, (t, 0, 4*pi), xlim=[0,13], ylim=[0,2.5])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3ffaf5d",
   "metadata": {},
   "source": [
    "### Радиус кривизны траектории\n",
    "\n",
    "При выводе общей формулы разложения ускорения мы воспользовались возможностью приблизить траекторию на достаточно малом участке окружностью некоторого радиуса. Рассмотрим частный случай плоского движения и найдем этот радиус, если известна траектория движения как функция $y = y(x)$. Выразим малое приращение пути с одной стороны через малое приращение угла при движении по касательной окружности, а с другой - через малые приращения координат:\n",
    "$$ds = Rd\\alpha$$\n",
    "$$ds = \\sqrt{\\left(dx\\right)^{2} + \\left(dy\\right)^{2}} = dx \\sqrt{1 + \\left(\\dfrac{dy}{dx}\\right)^{2}} = dx\\sqrt{1+\\left(y'\\right)^{2}}\\Rightarrow\\dfrac{d\\alpha}{dx} = \\dfrac{1}{R}\\sqrt{1+\\left(y'\\right)^{2}}$$\n",
    "Вторая производная функции, задающей траекторию,\n",
    "$$y'\\equiv\\dfrac{dy}{dx}=\\tan\\alpha\\Rightarrow y''=\\dfrac{d}{dx}\\left(\\tan\\alpha\\right)=\\dfrac{1}{\\cos^{2}\\alpha}\\dfrac{d\\alpha}{dx}=\\left(1+\\tan^{2}\\alpha\\right)\\dfrac{d\\alpha}{dx}=\\left(1+\\left(y'\\right)^{2}\\right)\\dfrac{d\\alpha}{dx}$$\n",
    "Тогда подставляя производную $d\\alpha/dx$ из второго выражения в первое, получаем,\n",
    "$$\\dfrac{d\\alpha}{dx}=\\dfrac{1}{R}\\sqrt{1+\\left(y'\\right)^{2}}=\\dfrac{y''}{1+\\left(y'\\right)^{2}}\\Rightarrow R=\\dfrac{\\left(1+\\left(y'\\right)^{2}\\right)^{3/2}}{y''}$$\n",
    "\n",
    "<figure>\n",
    "    <center>\n",
    "    <img src=\"./fig/lect02_fig03.png\" width=\"300\" align=\"center\"/>\n",
    "        <figcaption>К выводу формулы радиуса кривизны траектории</figcaption>\n",
    "    </center>\n",
    "</figure>\n",
    "\n",
    "Найдем радиус кривизны трактории при баллистическом движении материальной точки. С одной стороны, можно использовать выведенное выражение, подставив уравнение траектории - параболу. С другой, можно воспользоваться фактом о том, что полное ускорение тела в любой точке траектории равно $\\bf g$, а скорость направлена по касательной. Зная проекции скорости в некоторый момент времени\n",
    "$$\\begin{array}{c}\n",
    "v_{x}\\left(t_{0}\\right)=v_{0}\\cos\\varphi\\\\\n",
    "v_{y}\\left(t_{0}\\right)=v_{0}\\sin\\varphi-gt_{0}\n",
    "\\end{array}$$\n",
    "можно выразить угол между вектором скорости и вертикалью:\n",
    "$$\\alpha(t_0) = \\arccos\\frac{v_{x}(t_0)}{v(t_0)}$$\n",
    "Тогда проекция полного ускорения на нормаль к тректории\n",
    "$$a_{n} = g\\cos\\alpha = g\\dfrac{v_{x}}{v}$$\n",
    "где $v\\left(t_{0}\\right)=\\sqrt{v_{x}^{2}+v_{y}^{2}}=\\sqrt{v_{0}^{2}+g^{2}t_{0}^{2}-2v_{0}gt_{0}\\sin\\varphi}$ - полная скорость. С другой стороны, в общем случае $a_n = v^2/R$, откуда\n",
    "$$R=\\dfrac{v^{2}}{a_{n}}=\\dfrac{v^{3}}{gv_{x}}.$$\n",
    "\n",
    "<figure>\n",
    "    <center>\n",
    "    <img src=\"./fig/lect02_fig04.png\" width=\"300\" align=\"center\"/>\n",
    "        <figcaption>К выводу формулы радиуса кривизны баллистической траектории</figcaption>\n",
    "    </center>\n",
    "</figure>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "28c53d03",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}