path: root/sample_notebooks/NirenNegandhi/ch2.ipynb

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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter 2: Special Theory of Relativity"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.2, Page 34"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "ly = 9.46e+015;    # Distance travelled by light in an year, m\n",
      "c = 3e+008;    # Speed of light, m/s\n",
      "L = 4.30*ly;    # Distance of Alpha Centauri from earth, m\n",
      "T0 = 16*365.25*24*60*60;    # Proper time in system K_prime, s\n",
      "\n",
      "#Calculations\n",
      "# As time measured on earth, T = 2*L/v = T0_prime/sqrt(1-(v/c)^2), solving for v\n",
      "v = sqrt(4*L**2/(T0**2+4*L**2/c**2));    # Speed of the aircraft, m/s\n",
      "gama = 1/sqrt(1-(v/c)**2);    # Relativistic factor\n",
      "T = gama*T0/(365.25*24*60*60);    # Time interval as measured on Earth, y\n",
      "\n",
      "#Results\n",
      "print \"The speed of the aircraft = %4.2e m/s\" %v\n",
      "print \"The time interval as measured on earth = %4.1f y\"%T\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The speed of the aircraft = 1.42e+08 m/s\n",
        "The time interval as measured on earth = 18.2 y\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.3, Page 38"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "L0 = 4.30;    # Distance of Alpha Centauri from earth, ly\n",
      "c = 3e+008;    # Speed of light, m/s\n",
      "T = 8;    # Proper time in system K_prime, y\n",
      "\n",
      "#Calculations\n",
      "# As v/c = L0*sqrt(1-(v/c)^2)/(c*T) or bita = L0*sqrt(1-bita^2)/(c*T), solving for bita\n",
      "bita = sqrt(L0**2/(T**2 + L0**2));    # Boost parameter\n",
      "v = L0*sqrt(1-bita**2)/T;    # Speed of the aircraft, c units\n",
      "\n",
      "#Results\n",
      "print \"The boost parameter = %5.3f\"%bita\n",
      "print \"The speed of the aircraft = %5.3fc units\"%v"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The boost parameter = 0.473\n",
        "The speed of the aircraft = 0.473c units\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.4, Page 40"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 1;    # For simplicity assume speed of light to be unity, m/s\n",
      "bita = 0.600;    # Boost parameter\n",
      "gama = 1/sqrt(1-bita**2);    # Relativistic factor\n",
      "u_x_prime = 0;    # Speed of the protons in spaceship frame along x-axis, m/s\n",
      "u_y_prime = 0.99*c;    # Speed of the protons in spaceship frame along y-axis, m/s\n",
      "u_z_prime = 0;    # Speed of the protons in spaceship frame along z-axis, m/s\n",
      "v = 0.60*c;    # Speed of the spaceship w.r.t. space station, m/s\n",
      "\n",
      "#Calculations\n",
      "u_x = (u_x_prime + v)/(1 + v/c**2*u_x_prime);    # Speed of protons in space station frame along x-axis, m/s\n",
      "u_y = u_y_prime/(gama*(1 + v/c**2*u_x_prime));    # Speed of protons in space station frame along y-axis, m/s\n",
      "u_z = u_z_prime/(gama*(1 + v/c**2*u_x_prime));    # Speed of protons in space station frame along y-axis, m/s\n",
      "u = sqrt(u_x**2 + u_y**2 + u_z**2);    # The speed of the protons measured by an observer in the space station, m/s\n",
      "\n",
      "#Result\n",
      "print \"The speed of the protons measured by an observer in the space station = %5.3fc units\"%u"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The speed of the protons measured by an observer in the space station = 0.994c units\n"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.5, Page 45"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 2.998e+008;    # Speed of light in free space, m/s\n",
      "v = 7712;    # Speed of the space shuttle, m/s\n",
      "bita = v/c;    # Boost parameter\n",
      "T_loss = 295.02;    # Total measured loss in time, ps/sec\n",
      "Add_T_loss = 35.0;    # Additional time loss for moving clock from general relativity prediction, ps/s\n",
      "\n",
      "#Calculations\n",
      "# From time dilation relation, T0_prime = T*sqrt(1 - bita^2), solving for (T - T0_prime)/T\n",
      "Calc_T_loss = bita**2/2*1e+012;    # Expected time lost per sec by the moving clock, ps/sec\n",
      "Measured_T_loss = T_loss + Add_T_loss;    # Total measured loss in time per sec as per special relativity, ps/s\n",
      "percent_T_loss = (Calc_T_loss - Measured_T_loss)/Calc_T_loss*100;    # Percentage deviation of measured and the calculated time loss per sec\n",
      "T = 6.05e+05;    # Total time of the seven-day mission, s\n",
      "delta_T = Calc_T_loss*1e-012*T;    # The total time difference between the moving and stationary clocks during the entire shuttle flight, s\n",
      "\n",
      "#Results\n",
      "print \"The expected time lost per second for the moving clock = %6.2f ps\"%Calc_T_loss\n",
      "print \"The percentage deviation of measured and the calculated time loss per sec for moving clock = %3.1f percent\"%percent_T_loss   #answer differs due to rounding errors\n",
      "print \"The total time difference between the moving and stationary clocks during the entire shuttle flight = %3.1f ms\"%(delta_T/1e-003)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The expected time lost per second for the moving clock = 330.86 ps\n",
        "The percentage deviation of measured and the calculated time loss per sec for moving clock = 0.3 percent\n",
        "The total time difference between the moving and stationary clocks during the entire shuttle flight = 0.2 ms\n"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.8, Page 57"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "f0 = 1;    # For simplicity assume frequency of the signals sent by Frank, Hz\n",
      "# Outbound trip\n",
      "bita = -0.8;    # Boost parameter for receding frames\n",
      "\n",
      "#Calculations&Results\n",
      "f = sqrt(1+bita)/sqrt(1-bita)*f0;    # The frequency of the signals received by Mary in outbound trip, Hz\n",
      "print \"The frequency of the signals received by Mary in outbound trip = f0/%d\", ceil(f*9)\n",
      "# Return trip\n",
      "bita = +0.8;    # Boost parameter for approaching frames\n",
      "f = sqrt(1+bita)/sqrt(1-bita)*f0;    # The frequency of the signals received by Mary in return trip, Hz\n",
      "print \"The frequency of the signals received by Mary in return trip = %df0\"%f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The frequency of the signals received by Mary in outbound trip = f0/%d 3.0\n",
        "The frequency of the signals received by Mary in return trip = 3f0\n"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.11, Page 64"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "q = 1.6e-019;    # Charge on an electron, C\n",
      "V = 25e+003;    # Accelerating potential, volt\n",
      "K = q*V;    # Kinetic energy of electrons, J\n",
      "m = 9.11e-031;    # Rest mass of an electron, kg\n",
      "c = 3.00e+08;    # Speed of light, m/s\n",
      "\n",
      "#Calculations\n",
      "# From relativistic kinetic energy formula, K = (gama - 1)*m*C^2, solving for gama\n",
      "gama = 1 + K/(m*c**2);    # Relativistic factor\n",
      "bita = sqrt((gama**2-1)/gama**2);    # Boost parameter\n",
      "u = bita*c;    # Speed of the electrons, m/s\n",
      "# From non-relativistic expression, K = 1/2*m*u^2, solving for u\n",
      "u_classical = sqrt(2*K/m);    # Non-relativistic speed of the electrons, m/s\n",
      "u_error = (u_classical - u)/u_classical*100;    # Percentage error in speed of electrons, m/s\n",
      "\n",
      "#Results\n",
      "print \"The relativistic speed of the accelerated electrons = %4.2e m/s\"%u\n",
      "print \"The classical speed is about %d percent greater than the relativistic speed\"%ceil(u_error)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The relativistic speed of the accelerated electrons = 9.04e+07 m/s\n",
        "The classical speed is about 4 percent greater than the relativistic speed\n"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.13, Page 69"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 1;    # For simplicity assume peed of light to be unity, m/s\n",
      "K = 2.00;    # Kinetic energy of protons, GeV\n",
      "E0 = 0.938;    # Rest mass of a proton, GeV\n",
      "E = K + E0;    # Total energy of the proton, GeV\n",
      "\n",
      "#Calculations\n",
      "# From relativistic mass energy relation, E^2 = (p*c)^2+E0^2, solving for p\n",
      "p = sqrt(E**2 - E0**2)/c;    # Momentum of the protons, GeV/c\n",
      "# As E = gama*E0, solving for gama\n",
      "gama = E/E0;    # Relativistic factor\n",
      "bita = sqrt((gama**2-1)/gama**2);    # Boost parameter\n",
      "v = bita*3.00e+08;    # Speed of 2 GeV proton, m/s\n",
      "\n",
      "#Results\n",
      "print \"The energy of each initial proton = %5.3f GeV\"%E\n",
      "print \"The momentum of each initial proton = %4.2f GeV/c\"%p\n",
      "print \"The speed of each initial proton = %3.1e m/s\"%v\n",
      "print \"The relativistic factor, gama = %4.2f\"%gama\n",
      "print \"The boost parameter, beta = %5.3f\"%bita"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The energy of each initial proton = 2.938 GeV\n",
        "The momentum of each initial proton = 2.78 GeV/c\n",
        "The speed of each initial proton = 2.8e+08 m/s\n",
        "The relativistic factor, gama = 3.13\n",
        "The boost parameter, beta = 0.948\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.15, Page 71"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Variable declaration\n",
      "E_d = 1875.6;    # Rest mass energy of the deuterium, MeV\n",
      "E_pi = 139.6;    # Rest mass energy of the pion, MeV\n",
      "E_p = 938.3;    # Rest mass energy of the proton, MeV\n",
      "\n",
      "#Calculation\n",
      "K = 1./2*(E_d + E_pi - 2*E_p);    # Minimum kinetic energy of the protons, MeV \n",
      "\n",
      "#Result\n",
      "print \"The minimum kinetic energy of the protons = %2d MeV\"%K"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The minimum kinetic energy of the protons = 69 MeV\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.16, Page 72"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Variable declaration\n",
      "u = 931.5;    # Energy equivalent of 1 amu, MeV\n",
      "c = 1;    # Speed of light in vacuum, m/s\n",
      "\n",
      "#Calculations\n",
      "m_e = 0.000549*u;    # Rest mass of an electron, MeV/c^2\n",
      "m_p = 1.007276*u;    # Rest mass of a proton, MeV/c^2\n",
      "m_n = 1.008665*u;    # Rest mass of a neutron, MeV/c^2\n",
      "m_He = 4.002603*u;    # Rest mass of a helium, MeV/c^2\n",
      "M_He = m_He - 2*m_e;    # Mass of He nucleus, MeV/c^2\n",
      "E_B_He = (2*m_p + 2*m_n - M_He)*c**2;    # Binding energy of the He nucleus, MeV\n",
      "\n",
      "#Result\n",
      "print \"The binding energy of the He nucleus = %4.1f MeV\"%E_B_He"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The binding energy of the He nucleus = 28.3 MeV\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.17, Page 72"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Variable declaration\n",
      "u = 931.5;    # Energy equivalent of 1 amu, MeV/u\n",
      "c = 1;    # For simplicity assume speed of light in vacuum to be unity, m/s\n",
      "E_B = 4.24;    # The dissociationenergy of the NaCl molecule, MeV\n",
      "\n",
      "#Calculations\n",
      "M = 58.44*u;    # Energy corresponding to molecular mass of NaCl, MeV\n",
      "f_r = E_B/M;    # The fractional mass increase of the Na and Cl atoms\n",
      "\n",
      "#Result\n",
      "print \"The fractional mass increase of the Na and Cl atoms when they are not bound together in NaCl = %4.1e\"%(f_r/1e+006)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The fractional mass increase of the Na and Cl atoms when they are not bound together in NaCl = 7.8e-11\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 2.18, Page 72"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 1;    # For simplicity assume speed of light in vacuum to be unity, m/s\n",
      "E0_n = 940;    # Rest energy of a neutron, MeV\n",
      "E0_pi = 140;    # Rest energy of a pion, MeV\n",
      "p_n = 4702;    # Momentum of the neutron, MeV/c\n",
      "p_pi = 169;    # Momentum of the pion, MeV/c\n",
      "\n",
      "#Calculations\n",
      "E_n = sqrt((p_n*c)**2+E0_n**2);    # Total energy of the neutron, MeV\n",
      "E_pi = sqrt((p_pi*c)**2+E0_pi**2);    # Total energy of the pion, MeV\n",
      "E = E_n + E_pi;    # Total energy of the reaction, MeV\n",
      "p_sigma = p_n + p_pi;    # Momentum of the sigma particle, MeV/c\n",
      "E0_sigma = sqrt(E**2 - (p_sigma*c)**2);    # Rest mass energy of the sigma particle, MeV\n",
      "m_sigma = E0_sigma/c**2;    # Rest mass of sigma particle, MeV/c^2;\n",
      "K = E - E0_sigma;    # Kinetic energy of sigma particle, MeV\n",
      "\n",
      "#Results\n",
      "print \"The rest mass of sigma particle = %4d MeV/c^2\"%ceil(m_sigma)\n",
      "print \"The kinetic energy of sigma particle = %4d MeV\"%ceil(K)\n",
      "\n",
      "#Answers differ due to rounding errors"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The rest mass of sigma particle = 1192 MeV/c^2\n",
        "The kinetic energy of sigma particle = 3824 MeV\n"
       ]
      }
     ],
     "prompt_number": 12
    }
   ],
   "metadata": {}
  }
 ]
}