path: root/Applied_Physics_for_Engineers/chapter_1.ipynb

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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Chapter 1: Relativistic Mechanics"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.2, Page 26"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 3e+008;    # Speed of light in vacuum, m/s\n",
      "delta_x = 2.45e+03;    # Space difference, m\n",
      "delta_t = 5.35e-06;    # Time difference, s\n",
      "\n",
      "#Calculations\n",
      "v = 0.855*c;    # Speed of frame S_prime, m/s\n",
      "delta_x_prime = 1/sqrt(1-v**2/c**2)*(delta_x - v*(delta_t))*1e-03;    # Distance between two flashes as measured in S_prime frame, km\n",
      "delta_t_prime = 1/sqrt(1-v**2/c**2)*(delta_t - v/c**2*delta_x)*1e+006;    # Time between two flashes as measured in S_prime\n",
      "\n",
      "#Results\n",
      "print \"The distance between two flashes as measured in S_prime frame = %4.2f km\"%delta_x_prime\n",
      "print \"The time between two flashes as measured in S_prime frame = %4.2f micro-second\"%delta_t_prime\n",
      "\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The distance between two flashes as measured in S_prime frame = 2.08 km\n",
        "The time between two flashes as measured in S_prime frame = -3.15 micro-second\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.4, Page 27"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "c = Symbol('c')\n",
      "\n",
      "#Variable declaration\n",
      "c = 1;            # Speed of light in vacuum, m/s\n",
      "u_x_prime = c;    # Velocity of photon as measured in S_prime frame, m/s\n",
      "v = c;            # Velocity of frame S_prime relative to S frame, m/s\n",
      "\n",
      "#Calculations\n",
      "u_x = (u_x_prime + v)/(1+v*u_x_prime/c**2);\n",
      "if u_x == 1: \n",
      "    ux = 'c';\n",
      "else: \n",
      "    ux = string(u_x)+'c';    \n",
      "\n",
      "\n",
      "#Result\n",
      "print \"The speed of one photon as observed by the other is %c\"%ux\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The speed of one photon as observed by the other is c\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.6, Page 28"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "\n",
      "#Variable declaration\n",
      "a = 1;               # For simplicity assume length of semi minor axis to be unity, m\n",
      "c = 3e+08;           # Speed of light, m/s\n",
      "\n",
      "\n",
      "#Calculations\n",
      "#From equation 1-v^2/c^2=1/4, we derive the following expression\n",
      "v = math.sqrt(3*c**2/4)    # Velocity at which surface area of lamina reduces to half in S-frame, m/s\n",
      "\n",
      "\n",
      "print \"The velocity at which surface area of lamina reduces to half in S-frame = %4.2e m/s\"%v\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The velocity at which surface area of lamina reduces to half in S-frame = 2.60e+08 m/s\n"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.7, Page 29"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Variable declaration\n",
      "m0 = 1;    # For simplicity assume the rest mass of stick to be unity, kg\n",
      "m = 1.5*m0;    # Mass of the moving stick, kg\n",
      "L0 = 1;    # Assume resting length of the stick to be unity, m\n",
      "\n",
      "#Calculations\n",
      "# As m = m0/sqrt(1-v^2/c^2) = m0*gama, solving for gama\n",
      "gama = m/m0;    # Relativistic factor\n",
      "L = L0/gama;    # Contracted length of the metre stick, m\n",
      "\n",
      "#Result\n",
      "print \"The contracted length of the metre stick = %4.2f m\"%L\n",
      " \n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The contracted length of the metre stick = 0.67 m\n"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.8, Page 29"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 3e+008;    # Speed of light in vacuum, m/s\n",
      "tau0 = 2e-008;    # Mean lifetime of meson at rest, m/s\n",
      "\n",
      "#Calculations\n",
      "v = 0.8*c;    # Velocity of moving meason, m/s\n",
      "tau = tau0/sqrt(1-v**2/c**2);    # Mean lifetime of meson in motion, m/s\n",
      "\n",
      "#Result\n",
      "print \"The mean lifetime of meson in motion = %4.2e s\"%tau\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The mean lifetime of meson in motion = 3.33e-08 s\n"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.9, Page 30"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 3e+008;    # Speed of light in vacuum, m/s\n",
      "delta_t0 = 59;    # Reading of the moving clock for each hour, min\n",
      "delta_t = 60;    # Reading of the stationary clock for each hour, min\n",
      "\n",
      "#Calculations\n",
      "# As from Time Dilation, delta_t = delta_t0/sqrt(1-v^2/c^2), solving for v\n",
      "v = sqrt(((delta_t**2-delta_t0**2)*c**2)/delta_t**2)\n",
      "\n",
      "#Result\n",
      "print \"The speed at which the moving clock ticks slow = %4.2e m/s\"%v\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The speed at which the moving clock ticks slow = 5.45e+07 m/s\n"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.10, Page 30"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 3e+008;    # Speed of light in vacuum, m/s\n",
      "tau0 = 2.5e-008;    # Mean lifetime of meson at rest, m/s\n",
      "\n",
      "#Calculations\n",
      "v = 0.8*c;    # Velocity of moving meason, m/s\n",
      "tau = tau0/sqrt(1-v**2/c**2);    # Mean lifetime of meson in motion, m/s\n",
      "N0 = 1;    # Assume initial flux of meson beam to be unity, watt/Sq.m\n",
      "N = N0*exp(-2);    # Meson flux after time t, watt/Sq.m\n",
      "# As N = N0*e^(-t/tau), which on comparing gives\n",
      "t = 2*tau;    # Time during which the meson beam flux reduces, s\n",
      "d = 0.8*c*t;    # The distance that the meson beam can travel before reduction in its flux, m\n",
      "\n",
      "#Result\n",
      "print \"The distance that the meson beam can travel before reduction in its flux = %2d m\"%d\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The distance that the meson beam can travel before reduction in its flux = 20 m\n"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.11, Page 31"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 3e+008;    # Speed of light in vacuum, m/s\n",
      "E0 = 1;    # Rest energy of particle, unit\n",
      "\n",
      "#Calculations\n",
      "E = 3*E0;    # Energy of relativistically moving particle, unit\n",
      "# E = m*c^2 and E0 = m0*c^2\n",
      "# With m = m0/sqrt(1-v^2/c^2), we have\n",
      "v = c*sqrt(1-(E0/E)**2);    # Velocity of the moving particle, m/s\n",
      "\n",
      "#Result\n",
      "print \"The velocity of the moving particle = %4.2e m/s\"%v\n",
      "#answer differs due to rounding-off errors"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The velocity of the moving particle = 3.00e+08 m/s\n"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.12, Page 32"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 3e+008;    # Speed of light in vacuum, m/s\n",
      "m0 = 9.1e-031;    # Rest mass of electron, kg\n",
      "\n",
      "#Calculations\n",
      "m = 11*m0;    # Mass of relativistically moving electron, kg\n",
      "E_k = (m-m0)*c**2/(1.6e-019*1e+06);    # Kinetic energy of moving electron, MeV\n",
      "# As m = m0/sqrt(1-v^2/c^2), solving for v\n",
      "v = c*sqrt(1-(m0/m)**2);    # The velocity of the moving electron, m/s\n",
      "p = m*v;    # Momentum of moving electron, kg-m/s\n",
      "\n",
      "#Results\n",
      "print \"The kinetic energy of moving electron = %4.2f MeV\"%E_k\n",
      "print \"The momentum of moving electron = %4.2e kg-m/s\"%p\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The kinetic energy of moving electron = 5.12 MeV\n",
        "The momentum of moving electron = 2.99e-21 kg-m/s\n"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.13, Page 32"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 3e+008;    # Speed of light in vacuum, m/s\n",
      "E0 = 0.5;    # Rest energy of the electron, MeV\n",
      "\n",
      "#Calculations\n",
      "v1 = 0.6*c;    # Initial velocity of the electron, m/s\n",
      "v2 = 0.8*c;    # Final velocity of the electron, m/s\n",
      "W = (1/sqrt(1-v2**2/c**2)-1/sqrt(1-v1**2/c**2))*E0;    # The amount of work to be done to increase the speed of the electron, MeV\n",
      "\n",
      "#Result\n",
      "print \"The amount of work to be done to increase the speed of an electron = %4.2e J\"%(W*1e+06*1.6e-019)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The amount of work to be done to increase the speed of an electron = 3.33e-14 J\n"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.14, Page 33"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 1;    # Assume speed of light in vacuum to be unity, unit\n",
      "m0 = 1;    # For simplicity assume rest mass of the particle to be unity, unit\n",
      "\n",
      "#Calculations\n",
      "v = c/sqrt(2);    # Given speed of the particle, m/s\n",
      "gama = 1/sqrt(1-v**2/c**2);    # Relativistic factor\n",
      "m = gama*m0;   # The relativistic mass of the particle, unit\n",
      "p = m*v;    # The relativistic momentum of the particle, unit\n",
      "E = m*c**2;    # The relativistic total eneryg of the particle, unit\n",
      "E_k = (m-m0)*c**2;    # The relativistic kinetic energy of the particle, unit\n",
      "\n",
      "#Results\n",
      "print \"The relativistic mass of the particle = %5.3fm\"%m\n",
      "print \"The relativistic momentum of the particle = %1.0gm0c\"%p\n",
      "print \"The relativistic total energy of the particle = %5.3fm0c^2\"%E\n",
      "print \"The relativistic kinetic energy of the particle = %5.3fm0c^2\"%E_k\n",
      " "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The relativistic mass of the particle = 1.414m\n",
        "The relativistic momentum of the particle = 1m0c\n",
        "The relativistic total energy of the particle = 1.414m0c^2\n",
        "The relativistic kinetic energy of the particle = 0.414m0c^2\n"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.15, Page 34"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 3e+008;    # Speed of light in vacuum, unit\n",
      "m0 = 9.1e-031;    # Rest mass of the electron, kg\n",
      "m = 1.67e-027;    # Rest mass of the proton, kg\n",
      "\n",
      "#Calculations\n",
      "# As m = m0/sqrt(1-v^2/c^2), solving for v\n",
      "v = c*sqrt(1-(m0/m)**2);    # Velocity of the electron, m/s\n",
      "\n",
      "#Result\n",
      "print \"The velocity of the electron to have its mass equal to mass of the proton = %5.3e m/s\"%v\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The velocity of the electron to have its mass equal to mass of the proton = 3.000e+08 m/s\n"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example 1.17, Page 35"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from math import *\n",
      "\n",
      "#Variable declaration\n",
      "c = 3e+008;    # Speed of light in vacuum, unit\n",
      "m0 = 9.1e-031;    # Rest mass of the electron, kg\n",
      "E_k = 0.1*1e+006*1.6e-019;    # Kinetic energy of the electron, J\n",
      "\n",
      "#Calculations&Results\n",
      "v = sqrt(2*E_k/m0);    # Classical speed of the electron, m/s\n",
      "print \"The classical speed of the electron = %5.3e m/s\"%v\n",
      "# As E_k = (m-m0)*c^2 = (1/sqrt(1-v^2/c^2)-1)*m0*c^2, solving for v\n",
      "v = c*sqrt(1-(m0*c**2/(E_k+m0*c**2))**2);    # Relativistic speed of the electron, m/s\n",
      "print \"The relativistic speed of the electron = %5.3e m/s\"%v\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The classical speed of the electron = 1.875e+08 m/s\n",
        "The relativistic speed of the electron = 1.644e+08 m/s\n"
       ]
      }
     ],
     "prompt_number": 24
    }
   ],
   "metadata": {}
  }
 ]
}