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+{

+ "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

+}