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+{

+ "metadata": {

+  "name": "chapter 15.ipynb"

+ },

+ "nbformat": 3,

+ "nbformat_minor": 0,

+ "worksheets": [

+  {

+   "cells": [

+    {

+     "cell_type": "heading",

+     "level": 1,

+     "metadata": {},

+     "source": [

+      "CHAPTER 15: MOMENTS OF INERTIA"

+     ]

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-11, Page no 305"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "y1=1 #in\n",

+      "y2=4 #in\n",

+      "d1=2.2-1 #in\n",

+      "d2=4-2.2 #in\n",

+      "A1=12 #in**2\n",

+      "A2=8 #in**2\n",

+      "b1=6 #in\n",

+      "b2=2 #in\n",

+      "h1=2 #in\n",

+      "h2=4 #in\n",

+      "\n",

+      "#Calculations\n",

+      "y_bar=(A1*y1+A2*y2)/(A1+A2) #in\n",

+      "I1=(12**-1)*(b1)*(h1**3) #in**4\n",

+      "I2=(12**-1)*(b2)*(h2**3) #in**4\n",

+      "#Using Parallel Axes Theorem\n",

+      "I=(I1+(A1*d1**2))+(I2+(A2*d2**2)) #in**4\n",

+      "\n",

+      "#Result\n",

+      "print'The moment of inertia is',round(I,1),\"in**4\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The moment of inertia is 57.9 in**4\n"

+       ]

+      }

+     ],

+     "prompt_number": 2

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-12, Page no 306"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "d=60 #mm diameter of the hole\n",

+      "#Areas\n",

+      "At=100*100 #mm**2\n",

+      "Ab=200*100 #mm**2\n",

+      "Ac=((pi/4)*d**2) #mm**2\n",

+      "bt=100 #mm\n",

+      "ht=100 #mm\n",

+      "bb=200 #mm\n",

+      "hb=100 #mm\n",

+      "#Distance of centroids of each area\n",

+      "yt=150 #mm\n",

+      "yb=50 #mm\n",

+      "yc=150 #mm\n",

+      "\n",

+      "#Calculations\n",

+      "y_bar=((At*yt)+(Ab*yb)-(Ac*yc))/(At+Ab-Ac) #mm\n",

+      "#Distances\n",

+      "dt=yt-y_bar #mm\n",

+      "db=y_bar-yb #mm\n",

+      "dc=yc-y_bar #mm\n",

+      "#Values of Inertia\n",

+      "It=(12**-1)*(bt)*(ht**3) #mm**4\n",

+      "Ib=(12**-1)*(bb)*(hb**3) #mm**4\n",

+      "Ic=(4**-1)*(pi)*((d/2)**4) #mm**4\n",

+      "#Moment of inertia\n",

+      "I=((It+At*dt**2)+(Ib+Ab*db**2)-(Ic+Ac*dc**2)) #mm**4\n",

+      "\n",

+      "#Result\n",

+      "print'The moment of inertia is',round(I,1),\"mm**4\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The moment of inertia is 77156533.6 mm**4\n"

+       ]

+      }

+     ],

+     "prompt_number": 7

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-14, Page no 308"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "b1=2 #in\n",

+      "b2=4 #in\n",

+      "h1=8 #in\n",

+      "h2=2 #in\n",

+      "bo=8 #in\n",

+      "ho=8 #in\n",

+      "bi=4 #in\n",

+      "hi=4 #in\n",

+      "\n",

+      "#Calculations\n",

+      "I1=(12**-1)*(b1)*(h1**3) #in**4\n",

+      "I2=(12**-1)*(b2)*(h2**3) #in**4\n",

+      "I=2*(I1+I2) #in**4\n",

+      "Io=(12**-1)*(bo)*(ho**3) #in**4\n",

+      "Ii=(12**-1)*(bi)*(hi**3) #in**4\n",

+      "I_bar=Io-Ii #in**4\n",

+      "\n",

+      "#Result\n",

+      "print'The moment of inertia is',round(I_bar),\"in**4\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The moment of inertia is 320.0 in**4\n"

+       ]

+      }

+     ],

+     "prompt_number": 8

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-15, Page no 308"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "b1=75 #mm\n",

+      "b2=12 #mm\n",

+      "h1=12 #mm\n",

+      "h2=162 #mm\n",

+      "d1=75 #mm\n",

+      "\n",

+      "#Calculations\n",

+      "A=(h2*b2)+(2*b1*h1) #mm**2\n",

+      "I1=(12**-1)*(b1)*(h1**3)+(b1*h1*d1**2) #mm**4\n",

+      "I2=(12**-1)*(b2)*(h2**3) #mm**4\n",

+      "I_bar=2*I1+I2 #mm**4\n",

+      "k=sqrt(I_bar/A) #mm\n",

+      "\n",

+      "#Result\n",

+      "print'The axial moment of inertia is',round(I_bar,1),\"mm**4\"\n",

+      "print'The radius of gyration is',round(k,1),\"mm\"\n",

+      "\n",

+      "# Here value of k is off by 0.1 mm"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The axial moment of inertia is 14398128.0 mm**4\n",

+        "The radius of gyration is 62.0 mm\n"

+       ]

+      }

+     ],

+     "prompt_number": 4

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-20, Page no 311"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "r=50 #mm\n",

+      "\n",

+      "#Calculations\n",

+      "Ixy=(8**-1)*(50**4) #mm**4\n",

+      "\n",

+      "#Result\n",

+      "print'The moment of inertia is',round(Ixy,2),\"mm**4\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The moment of inertia is 781250.0 mm**4\n"

+       ]

+      }

+     ],

+     "prompt_number": 11

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-24, Page no 314"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#The notation has been changed for ease\n",

+      "\n",

+      "#Calculations\n",

+      "x=(5*1*3.5+8*1*0.5)/(5*1+8*1) #in\n",

+      "y=(5*1*0.5+8*1*4)/13 #in\n",

+      "#Moment of inertia \n",

+      "Ix=(12**-1)*(5)*(1**3)+(5*2.15*2.15)+(12**-1)*(1*8**3)+(8*1.35**2) #in**4\n",

+      "Iy=(12**-1)*(1)*(5**3)+(5*1.85*1.85)+(12**-1)*(8)*(1**3)+(8*1.15**2) #in**4\n",

+      "Ixy=(8*1*(-1.15)*1.35)+(5*1*1.85*(-2.15)) #in**4\n",

+      "#Mohr circle calculations\n",

+      "d=0.5*(Ix+Iy) #distance to center of the cirlce \n",

+      "r=sqrt((21**2)+(32.3**2)) \n",

+      "maxI=d+r #in**4\n",

+      "theta=arctan(32.3/21)*(180/pi) #degrees maxI occurs at this angle\n",

+      "minI=d-r #in**4\n",

+      "\n",

+      "#Result\n",

+      "print'The moment of inertias are as follows:'\n",

+      "print'Ix=',round(Ix,1),\"in**4\"\n",

+      "print'Iy=',round(Iy,1),\"in**4\"\n",

+      "print'Ixy=',round(Ixy,1),\"in**4\"\n",

+      "print'maxI=',round(maxI,1),\"in**4\"\n",

+      "print'minI=',round(minI,1),\"in**4\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The moment of inertias are as follows:\n",

+        "Ix= 80.8 in**4\n",

+        "Iy= 38.8 in**4\n",

+        "Ixy= -32.3 in**4\n",

+        "maxI= 98.3 in**4\n",

+        "minI= 21.2 in**4\n"

+       ]

+      }

+     ],

+     "prompt_number": 16

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-25, Page no 315"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Notations have been changed\n",

+      "\n",

+      "#Calculations\n",

+      "x=-(25*125*0.5*125+25*100*0.5*25)/(25*125+25*100) #mm\n",

+      "y=(25*125*0.5*25+25*100*75)/5625 #mm \n",

+      "Iy=(12**-1)*25*125**3+25*125*(62.5-40.3)**2+(12**-1)*100*25**3+100*25*(40.3-12.5)**2 #mm**4\n",

+      "Ix=Iy #mm**4 for L-section\n",

+      "#The second computation checks the first\n",

+      "Ixy=(125*25*22.2*27.8)+(100*25*(-27.8)*(-34.7)) #mm**4\n",

+      "#Mohr Circle analysis\n",

+      "Imax=Ix+Ixy #mm**4\n",

+      "Imin=Ix-Ixy #mm**4\n",

+      "\n",

+      "#Result\n",

+      "print'The moment of inertias are as follows:'\n",

+      "print'Ix=',round(Ix,2),\"mm**4\"\n",

+      "print'Iy=',round(Iy,2),\"mm**4\"\n",

+      "print'Ixy=',round(Ixy,2),\"mm**4\"\n",

+      "print'Imax=',round(Imax),\"mm**4\"\n",

+      "print'Imin=',round(Imin,2),\"mm**4\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The moment of inertias are as follows:\n",

+        "Ix= 7671443.75 mm**4\n",

+        "Iy= 7671443.75 mm**4\n",

+        "Ixy= 4340275.0 mm**4\n",

+        "Imax= 12011719.0 mm**4\n",

+        "Imin= 3331168.75 mm**4\n"

+       ]

+      }

+     ],

+     "prompt_number": 18

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-30, Page no 320"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "rho=490 #lb/ft**3\n",

+      "t=0.02 #in\n",

+      "d=4 #in\n",

+      "r=d/2 #in\n",

+      "g=32.2 #ft/s**2\n",

+      "\n",

+      "#Calculations\n",

+      "W=(pi*r**2*t*rho)*1728**-1 #lb\n",

+      "#Mass\n",

+      "m=W*g**-1 #slugs\n",

+      "#Momemt of inertia\n",

+      "I=(4**-1)*m*(r*12**-1)**2 #slug-ft**2\n",

+      "\n",

+      "#Result\n",

+      "print'The moment of inertia is',round(I,6),\"slug-ft**2\""

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The moment of inertia is 1.5e-05 slug-ft**2\n"

+       ]

+      }

+     ],

+     "prompt_number": 18

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-36, Page no 322"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "#The integration involves variables hence the direct formula is being used in this coding\n",

+      "m=500 #kg\n",

+      "R=0.25 #m\n",

+      "h=0.5 #m\n",

+      "\n",

+      "#Calculations\n",

+      "Ix=(3*10**-1)*m*R**2 #kg.m**2\n",

+      "Iy=(3*5**-1)*m*((4**-1)*R**2+h**2) #kg.m**2\n",

+      "\n",

+      "#Result\n",

+      "print'Hence proved that Ix=',round(Ix,2),\"kg.m**2\" \n",

+      "print'and Iy=',round(Iy,1),\"kg.m**2\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "Hence proved that Ix= 9.38 kg.m**2\n",

+        "and Iy= 79.7 kg.m**2\n"

+       ]

+      }

+     ],

+     "prompt_number": 27

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 15.15-37, Page no 323"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "delta=450 #lb/ft**3\n",

+      "h1=9*12**-1 #ft\n",

+      "h2=10*12**-1 #ft\n",

+      "ro1=4*12**-1 #ft\n",

+      "ri1=2*12**-1 #ft\n",

+      "ro2=18*12**-1 #ft\n",

+      "ri2=16*12**-1 #ft\n",

+      "a=2.5*24**-1 #ft\n",

+      "b=3.5*24**-1 #ft\n",

+      "l=1 #ft\n",

+      "g=32.2 #ft/s**2\n",

+      "\n",

+      "#Calculations\n",

+      "Whub=(pi*ro1**2-pi*ri1**2)*h1*delta #lb\n",

+      "Wrim=(pi*ro2**2-pi*ri2**2)*h2*delta #lb\n",

+      "#For one spoke\n",

+      "Wspoke=(pi*a*b*l*delta) #lb\n",

+      "#Moment of inertia calculations\n",

+      "Ihub=0.5*(Whub*g**-1)*(ro1**2+ri1**2) #lb-s**2-ft\n",

+      "Irim=0.5*(Wrim*g**-1)*(ro2**2+ri2**2) #lb-s**2-ft\n",

+      "Ispoke=6*((12**-1)*(Wspoke*g**-1)*l**2+(Wspoke*g**-1)*h2**2) #lb-s**2-ft\n",

+      "Iwheel=Ihub+Irim+Ispoke #lb-s**2-ft\n",

+      "\n",

+      "#Result\n",

+      "print'The moment of inertia of the wheel is',round(Iwheel,1),\"lb-s**s-ft\""

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The moment of inertia of the wheel is 38.1 lb-s**s-ft\n"

+       ]

+      }

+     ],

+     "prompt_number": 20

+    }

+   ],

+   "metadata": {}

+  }

+ ]

+}
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