Lab 4.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Measuring Cp/Cv\n",
    "<bk>\n",
    "Shreya Parjan (+Alex)\n",
    "<bk>\n",
    "1 Oct 2018"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The purpose of this lab is to measure gamma, the ratio of the specific heat capacity under constant pressure to the specific heat capacity under constant volume, for air and argon. A plunger apparatus was used, in which we observed the plunger's oscillation frequency at varying volumes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Imports"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats\n",
    "import scipy.optimize\n",
    "import math as m\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Measured Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "initPressure = 102900 #Pa\n",
    "mass = .07 #kg\n",
    "g = 9.8 #m/s^2\n",
    "diameter = .03048 #m\n",
    "\n",
    "airVolume = np.array([40,30,20,46,36])/(100**3) #m^3\n",
    "airPeriod = np.array([(142/3),(124/3),(70/2),(200/4),(220/5)])/1000 #s\n",
    "\n",
    "argonVolume = np.array([40,30,46,36,20])/(100**3) #m^3\n",
    "argonPeriod = np.array([(220/5),(190/5),(188/4),(206/5),(196/6)])/1000 #s\n",
    "\n",
    "area = np.pi*((diameter/2)**2) #m^2\n",
    "syringePressure = ((initPressure*area)+(mass*g))/area #Pa"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Graphs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0.00224044 0.00170844 0.001225   0.0025     0.001936  ]\n",
      "y-intercept =  0.00022574673882571602\n",
      "slope =  49.309041831782764\n",
      "uncertainties:  [8.13801576e-05 2.29044588e+00]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "airPeriodSquared = airPeriod**2\n",
    "print(airPeriodSquared)\n",
    "def straight_line_model(x, A, B):\n",
    "    return A + B * x\n",
    "\n",
    "popt_air, pcov = scipy.optimize.curve_fit(straight_line_model, airVolume, airPeriodSquared)\n",
    "yIntercept = popt_air[0]\n",
    "slope = popt_air[1]\n",
    "print('y-intercept = ',yIntercept)\n",
    "print('slope = ',slope)\n",
    "\n",
    "airVolume_for_plot = np.linspace(airVolume.min(), airVolume.max(), 50)\n",
    "best_fit_line = straight_line_model(airVolume_for_plot, popt_air[0], popt_air[1])\n",
    "\n",
    "#calculates uncertainties for both variables\n",
    "popt_air_uncertainties = np.sqrt(np.diag(pcov)) # get the diagonal elements then take sqrt\n",
    "print('uncertainties: ', popt_air_uncertainties)\n",
    "\n",
    "plt.plot(airVolume, airPeriodSquared, 'bo') #without '-' after bo, plots points alone\n",
    "plt.plot(airVolume_for_plot, best_fit_line, 'b') # adds best fit line\n",
    "\n",
    "plt.xlabel('Air Volume (m^3)') # label the x-axis\n",
    "plt.ylabel('Period Squared (s^2)', size=12) # label the y-axis\n",
    "\n",
    "plt.title('Period Squared vs. Air Volume') # title\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "y-intercept =  0.00015074691081365849\n",
      "slope =  44.184979967896965\n",
      "uncertainties:  [7.31440098e-05 2.05863936e+00]\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "argonPeriodSquared = argonPeriod**2\n",
    "\n",
    "def straight_line_model(x, A, B):\n",
    "    return A + B * x\n",
    "\n",
    "popt_argon, pcov = scipy.optimize.curve_fit(straight_line_model, argonVolume, argonPeriodSquared)\n",
    "yIntercept = popt_argon[0]\n",
    "slope = popt_argon[1]\n",
    "print('y-intercept = ',yIntercept)\n",
    "print('slope = ',slope)\n",
    "\n",
    "argonVolume_for_plot = np.linspace(argonVolume.min(), argonVolume.max(), 50)\n",
    "best_fit_line = straight_line_model(argonVolume_for_plot, popt_argon[0], popt_argon[1])\n",
    "\n",
    "#calculates uncertainties for both variables\n",
    "popt_argon_uncertainties = np.sqrt(np.diag(pcov)) # get the diagonal elements then take sqrt\n",
    "print('uncertainties: ', popt_argon_uncertainties)\n",
    "\n",
    "plt.plot(argonVolume, argonPeriodSquared, 'bo') #without '-' after bo, plots points alone\n",
    "plt.plot(argonVolume_for_plot, best_fit_line, 'b') # adds best fit line\n",
    "\n",
    "plt.xlabel('Argon Volume (m^3)') # label the x-axis\n",
    "plt.ylabel('Period Squared (s^2)', size=12) # label the y-axis\n",
    "\n",
    "plt.title('Period Squared vs. Argon Volume') # title\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Gamma for air =  1.0137391715867612\n",
      "The uncertainty in gammaAir is:  0.04708902501771737\n"
     ]
    }
   ],
   "source": [
    "slopeUncertaintyAir = (popt_air_uncertainties[1])\n",
    "slopeAir = popt_air[1] #Slope is equivalent to this expression, found from (20) in the lab handout: (Pressure_e*Area^2*gamma)/(4pi*m)\n",
    "gammaAir = (4*np.pi**2*mass)/(slopeAir*syringePressure*area**2)\n",
    "print(\"Gamma for air = \",gammaAir)\n",
    "\n",
    "#to find the uncertainty in gammaAir, dG, we use the equation (dG/gammaAir) = (dSlope/slopeAir)\n",
    "#dG = (dSlope/slopeAir)*gammaAir\n",
    "gammaAirUncertainty = (slopeUncertaintyAir/slopeAir)*gammaAir\n",
    "print(\"The uncertainty in gammaAir is: \",gammaAirUncertainty)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "Gamma for argon =  1.1313008912668197\n",
      "The uncertainty in gammaArgon is:  0.05270887401128482\n"
     ]
    }
   ],
   "source": [
    "slopeUncertaintyArgon = (popt_argon_uncertainties[1])\n",
    "slopeArgon = popt_argon[1] #Slope is equivalent to this expression, found from (20) in the lab handout: (Pressure_e*Area^2*gamma)/(4pi*m)\n",
    "gammaArgon = (4*np.pi**2*mass)/(slopeArgon*syringePressure*area**2)\n",
    "print(\"Gamma for argon = \",gammaArgon)\n",
    "\n",
    "gammaArgonUncertainty = (slopeUncertaintyArgon/slopeArgon)*gammaArgon\n",
    "print(\"The uncertainty in gammaArgon is: \",gammaArgonUncertainty)\n",
    "\n",
    "#to find the uncertainty in gammaArgon, dG, we use the equation (dG/gammaArgon) = (dSlope/slopeArgon)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Discussion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "gammaArgon = 1.13 +/- 0.05\n",
    "gammaAir = 1.01 +/- 0.05\n",
    "\n",
    "Using equation 13 in the handout, the true value for the gammas can be found using: gamma = 1 + 2/f\n",
    "So the true gammaArgon is 1.67 and gammaAir is 1.40\n",
    "\n",
    "We know that gamma equals Cp/Cv. Because the value for gammaAir is about 28% greater than that of gammaArgon, we consider the difference between the gammas for the two gases to be significant. It is reasonable that gammaArgon is less than gammaAir because argon is a monatomic ideal gas while air is a compound with larger molecules.\n",
    "\n",
    "We concur with the student because a non-zero thermal conductivity (friction) would decrease the period, resulting in a lower period squared per unit volume and, thus a higher slope. Because gamma and the ratio of the period squared to the volume are directly proportional by (20) in the lab handout, we would expect a lower gamma than expected."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Stop to Thinks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. The voltage measurements and extrema are sensitive to the velocity of the plunger because generating a voltage relies on a change in magnetic field which can only be done through motion. It does not matter whether we observe position or velocity for calculating frequency.\n",
    "\n",
    "2. We assumed that the apparatus was frictionless and there was no air resistance. Therefore, we neglected how these assumptions would affect the period.\n",
    "\n",
    "3. Because the period and omega are indirectly proportional, if the process was instead adiabatic, the period would decrease (now accounting for friction) so omega would increase.\n",
    "\n",
    "4. It's better to measure over several oscillations because a multiple measurements reduce uncertainty in a calculation."
   ]
  }
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