{ "cells": [ { "cell_type": "markdown", "id": "857d9e7f-b782-4ea5-93a2-b38383677a0f", "metadata": {}, "source": [ "# Phys-A0110 Yliopistofysiikan perusteet (TFM)\n", "## Luentoesimerkki\n", "## Putoava kartio neliöllisellä vastusvoimalla" ] }, { "cell_type": "code", "execution_count": 4, "id": "f138022b-c4a1-4e32-ac27-09154b6e3e4f", "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "from scipy.integrate import odeint\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from matplotlib import animation, rc\n", "from IPython.display import HTML\n", "\n", "plt.rcParams['text.usetex'] = True\n" ] }, { "cell_type": "code", "execution_count": 7, "id": "70763e21-8a6c-4269-86b3-89a12db196a1", "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#parametrien arvot\n", "\n", "halkaisija = 0.1 # kartionn halkaisija [m]\n", "pintaala = np.pi*halkaisija*halkaisija/4 # kartion poikkipinta-ala [m^2]\n", "muotovakio = 0.5 # kartiolle, dimensioton\n", "ilmantiheys = 1.225 # NTP arvo, [kg/m^3]\n", "massa = 0.003 # kartion massa [kg]\n", "g = 9.81 # putoamiskiihtyvyys [m/s^2]\n", "\n", "#alkuehdot\n", "\n", "y0 = 2 # pudotuskorkeus [m]\n", "v0 = 0 # alkunopeus [m/s]\n", "\n", "x0 = [y0,v0] # muodostetaan alkuehdoista vektori \n", "\n", "# muuttujan alustus\n", "\n", "t = np.linspace(0,1,1000) # määritellään aikahaarukka [0:1] [s] ja tarkkuus (1000 pistettä) jolla yhtälöparia ratkotaan\n", "\n", "#määritellään differentiaaliyhtälö\n", "\n", "def liikeyhtalo(x,t): \n", " y = x[0] ## korkeus\n", " v = x[1] ## nopeus\n", " \n", "#liikeyhtälö dx/dt = v, dv/dt = -mg - 0.5*C*A*rho*v^2\n", " dxdt = v # korkeuden aikaderivaatta = nopeus\n", " if v > 0:\n", " dvdt = (-massa*g - 0.5*muotovakio*pintaala*ilmantiheys*v*v)/massa\n", " else:\n", " dvdt = (-massa*g + 0.5*muotovakio*pintaala*ilmantiheys*v*v)/massa\n", " \n", " return [dxdt,dvdt] \n", "\n", "#ratkaistaan differentiaaliyhtälöpari. Tällä rivillä siis ratkaistaan numeerisesti koko ongelma.\n", "\n", "x = odeint(liikeyhtalo,x0,t)\n", "\n", "ys = x[:,0] # poimitaan ratkaisusta korkeudet y(t)\n", "vs = x[:,1] # poimitaan ratkaisusta nopeudet v(t)\n", "\n", "# piirretään ratkaisut\n", "\n", "plt.plot(t,ys,label=\"korkeus\")\n", "plt.title(\"Putoava kappale\",fontsize=25)\n", "plt.xlabel(r\"Aika t\",fontsize=25)\n", "plt.ylabel(r\"Korkeus\",fontsize=25)\n", "plt.legend(loc=\"upper right\")\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 8, "id": "f969116f-0306-4a1f-9ab7-4faad6814731", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(t,-vs,label=\"nopeus\")\n", "plt.title(\"Putoava kappale\",fontsize=25)\n", "plt.xlabel(r\"Aika t\",fontsize=25)\n", "plt.ylabel(r\"Vauhti\",fontsize=25)\n", "\n", "terminaalinopeus = np.ones(t.size)*np.sqrt(2*massa*g/(muotovakio*pintaala*ilmantiheys))\n", "plt.plot(t,terminaalinopeus, label =\"terminaalinopeus\")\n", "\n", "plt.legend(loc=\"lower right\")\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "id": "7ca0ef92-f953-46b0-a647-409c0c79e7c3", "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": "3.9.5" } }, "nbformat": 4, "nbformat_minor": 5 }