From 4a1f703f1c1808d390ebf80e80659fe161f69fab Mon Sep 17 00:00:00 2001 From: Thomas Stephen Lee Date: Fri, 28 Aug 2015 16:53:23 +0530 Subject: add books --- sample_notebooks/AdityaAnand/Chapter_8.ipynb | 390 +++++++++++++++++++++++++++ 1 file changed, 390 insertions(+) create mode 100644 sample_notebooks/AdityaAnand/Chapter_8.ipynb (limited to 'sample_notebooks/AdityaAnand/Chapter_8.ipynb') diff --git a/sample_notebooks/AdityaAnand/Chapter_8.ipynb b/sample_notebooks/AdityaAnand/Chapter_8.ipynb new file mode 100644 index 00000000..cbd1971a --- /dev/null +++ b/sample_notebooks/AdityaAnand/Chapter_8.ipynb @@ -0,0 +1,390 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "###Example 8.2" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(a) The force acting = [0.0, 2.5849394142282115e-26, 0.0] ≈ 0\n", + "(b) The force acting = 2 Gm²\n" + ] + } + ], + "source": [ + "# Importing module\n", + "\n", + "import math\n", + "\n", + "# Variable declaration\n", + "\n", + "G=6.67*pow(10,-11) # Gravitational constant\n", + "m=1 # For convenience,mass is assumed to be unity \n", + "x=30 # The angle between GC and the positive x-axis is 30° and so is the angle between GB and the negative x-axis\n", + "y=math.radians(x) # The angle in radians\n", + "a=math.cos(y)\n", + "b=math.sin(y)\n", + "v1=(0,1,0)\n", + "v2=(-a,-b,0)\n", + "v3=(a,-b,0)\n", + "c=(2*G*pow(m,2))/1 # 2Gm²/1\n", + "\n", + "# Calculation\n", + "\n", + "#(a)\n", + "F1=[y * c for y in v1] # F(GA)\n", + "F2=[y * c for y in v2] # F(GB)\n", + "F3=[y * c for y in v3] # F(GC)\n", + "# From the principle of superposition and the law of vector addition, the resultant gravitational force FR on (2m) is given by\n", + "Fa=[sum(x) for x in zip(F1,F2,F3)]\n", + "\n", + "#(b)\n", + "# By symmetry the x-component of the force cancels out and the y-component survives\n", + "Fb=4-2 # 4Gm² j - 2Gm² j\n", + "\n", + "# Result\n", + "\n", + "print(\"(a) The force acting =\",Fa,\"≈ 0\")\n", + "print(\"(b) The force acting =\",Fb,\"Gm²\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "###Example 8.3 " + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Potential energy of a system of four particles = -5.414213562373095 Gm²/l\n", + "The gravitational potential at the centre of the square = -5.65685424949238 Gm²/l\n" + ] + } + ], + "source": [ + "# Importing module\n", + "\n", + "import math\n", + "\n", + "# Variable declaration\n", + "\n", + "G=6.67*pow(10,-11) # Gravitational constant\n", + "m=1 # For convenience,mass is assumed to be unity \n", + "l=1 # For convenience,side of the square is assumed to be unity \n", + "c=(G*pow(m,2))/l\n", + "n=4 # Number of particles\n", + "\n", + "# Calculation\n", + "\n", + "d=math.sqrt(2)\n", + "# If the side of a square is l then the diagonal distance is √2l\n", + "# We have four mass pairs at distance l and two diagonal pairs at distance √2l \n", + "# Since the Potential Energy of a system of four particles is -4Gm²/l) - 2Gm²/dl\n", + "w=(-n-(2/d)) \n", + "# If the side of a square is l then the diagonal distance from the centre to corner is \n", + "# Since the Gravitational Potential at the centre of the square\n", + "u=-n*(2/d)\n", + "\n", + "# Result\n", + "\n", + "print (\"Potential energy of a system of four particles =\",w,\"Gm²/l\")\n", + "print(\"The gravitational potential at the centre of the square =\",u,\"Gm²/l\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "###Example 8.4" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Minimum speed of the projectile to reach the surface of the second sphere = ( 0.6 GM/R ) ^(1/2)\n" + ] + } + ], + "source": [ + "# Importing module\n", + "\n", + "import math\n", + "\n", + "# Variable declaration\n", + "\n", + "R=1 # For convenience,radii of both the spheres is assumed to be unity \n", + "M=1 # For convenience,mass is assumed to be unity \n", + "m1=M # Mass of the first sphere\n", + "m2=6*M # Mass of the second sphere\n", + "m=1 # Since the mass of the projectile is unknown,take it as unity\n", + "d=6*R # Distance between the centres of both the spheres\n", + "r=1 # The distance from the centre of first sphere to the neutral point N\n", + "\n", + "G=6.67*pow(10,-11) # Gravitational constant\n", + "\n", + "# Calculation\n", + "\n", + "# Since N is the neutral point; GMm/r² = 4GMm/(6R-r)² and we get\n", + "r=2*R\n", + "# The mechanical energy at the surface of M is; Et = m(v^2)/2 - GMm/R - 4GMm/5R\n", + "# The mechanical energy at N is; En = -GMm/2R - 4GMm/4R\n", + "# From the principle of conservation of mechanical energy; Et = En and we get\n", + "v_sqr=2*((4/5)-(1/2))\n", + "\n", + "# Result\n", + "\n", + "print(\"Minimum speed of the projectile to reach the surface of the second sphere =\",\"(\",round(v_sqr,5),\"GM/R\",\")\",\"^(1/2)\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "###Example 8.5 " + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "(i) Mass of Mars = 6.475139697520706e+23 kg\n", + "(ii) Period of revolution of Mars = 684.0033777694376 days\n" + ] + } + ], + "source": [ + "# Importing module\n", + "\n", + "import math\n", + "\n", + "# Variable declaration\n", + "\n", + "π=3.14 # Constant pi\n", + "G=6.67*pow(10,-11) # Gravitational constant\n", + "R=9.4*pow(10,3) # Orbital radius of Mars in km\n", + "T=459*60\n", + "Te=365 # Period of revolution of Earth\n", + "r=1.52 # Ratio of Rms/Res, where Rms is the mars-sun distance and Res is the earth-sun distance. \n", + "\n", + "# Calculation\n", + "\n", + "# (i) \n", + "R=R*pow(10,3)\n", + "# Using Kepler's 3rd law:T²=4π²(R^3)/GMm\n", + "Mm=(4*pow(π,2)*pow(R,3))/(G*pow(T,2))\n", + "\n", + "# (ii)\n", + "# Using Kepler's 3rd law: Tm²/Te² = (Rms^3/Res^3)\n", + "Tm=pow(r,(3/2))*365\n", + "\n", + "\n", + "# Result\n", + "\n", + "print(\"(i) Mass of Mars =\",Mm,\"kg\")\n", + "print(\"(ii) Period of revolution of Mars =\",Tm,\"days\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "###Example 8.6" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Mass of the Earth = 5.967906881559221e+24 kg\n", + "Mass of the Earth = 6.017752855396305e+24 kg\n" + ] + } + ], + "source": [ + "# Importing module\n", + "\n", + "import math\n", + "\n", + "# Variable declaration\n", + "\n", + "g=9.81 # Acceleration due to gravity\n", + "G=6.67*pow(10,-11) # Gravitational constant\n", + "Re=6.37*pow(10,6) # Radius of Earth in m\n", + "R=3.84*pow(10,8) # Distance of Moon from Earth in m\n", + "T=27.3 # Period of revolution of Moon in days\n", + "π=3.14 # Constant pi\n", + "\n", + "# Calculation\n", + "\n", + "# I Method\n", + "# Using Newton's 2nd law of motion:g = F/m = GMe/Re²\n", + "Me1=(g*pow(Re,2))/G\n", + "\n", + "# II Method\n", + "# Using Kepler's 3rd law: T²= 4π²(R^3)/GMe\n", + "T1=T*24*60*60\n", + "Me2=(4*pow(π,2)*pow(R,3))/(G*pow(T1,2))\n", + "\n", + "#Result\n", + "\n", + "print(\"Mass of the Earth =\",Me1,\"kg\")\n", + "print(\"Mass of the Earth =\",Me2,\"kg\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "###Example 8.7 " + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Period of revolution of Moon = 27.5 days\n" + ] + } + ], + "source": [ + "# Importing module\n", + "\n", + "import math\n", + "\n", + "# Variable declaration\n", + "\n", + "k=pow(10,-13) # A constant = 4π² / GME\n", + "Re=3.84*pow(10,5) # Distance of the Moon from the Earth in m\n", + "\n", + "# Calculation\n", + "\n", + "k=pow(10,-13)*(pow(1/(24*60*60),2))*(1/pow((1/1000),3))\n", + "T2=k*pow(Re,3)\n", + "T=math.sqrt(T2) # Period of revolution of Moon in days\n", + "\n", + "# Result\n", + "\n", + "print(\"Period of revolution of Moon =\",round(T,1),\"days\")" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "###Example 8.8 " + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Change in Kinetic Energy = 3124485000.0 J\n", + "Change in Potential Energy = 6248970000.0 J\n" + ] + } + ], + "source": [ + "# Importing module\n", + "\n", + "import math\n", + "\n", + "# Variable declaration\n", + "\n", + "m=400 # Mass of satellite in kg\n", + "Re=6.37*pow(10,6) # Radius of Earth in m\n", + "g=9.81 # Acceleration due to gravity\n", + "\n", + "# Calculation\n", + "\n", + "# Change in energy is E=Ef-Ei\n", + "ΔE=(g*m*Re)/8 # Change in Total energy\n", + "# Since Potential Energy is twice as the change in Total Energy (V = Vf - Vi)\n", + "ΔV=2*ΔE # Change in Potential Energy in J\n", + "\n", + "# Result\n", + "\n", + "print(\"Change in Kinetic Energy =\",round(ΔE,4),\"J\")\n", + "print(\"Change in Potential Energy =\",round(ΔV,4),\"J\")" + ] + } + ], + "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": "3.4.3" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} -- cgit