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author | hardythe1 | 2015-01-28 14:31:21 +0530 |
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committer | hardythe1 | 2015-01-28 14:31:21 +0530 |
commit | 53f72e6790ff23b43c8f6a0b69d6386940671429 (patch) | |
tree | 7745af6dbf2f5b2972b23f9f5a7a19c695a27321 /Applied_Physics_for_Engineers/chapter_1.ipynb | |
parent | 7b78be04fe05bf240417e22f74b3fc22e7a77d19 (diff) | |
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diff --git a/Applied_Physics_for_Engineers/chapter_1.ipynb b/Applied_Physics_for_Engineers/chapter_1.ipynb new file mode 100755 index 00000000..659af082 --- /dev/null +++ b/Applied_Physics_for_Engineers/chapter_1.ipynb @@ -0,0 +1,558 @@ +{ + "metadata": { + "name": "" + }, + "nbformat": 3, + "nbformat_minor": 0, + "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": {} + } + ] +}
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