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+{

+ "metadata": {

+  "name": "chapter 10.ipynb"

+ },

+ "nbformat": 3,

+ "nbformat_minor": 0,

+ "worksheets": [

+  {

+   "cells": [

+    {

+     "cell_type": "heading",

+     "level": 1,

+     "metadata": {},

+     "source": [

+      "CHAPTER 10: FIRST MOMENTS AND CENTROIDS"

+     ]

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 10.10-5, Page no 160"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "import numpy as np\n",

+      "\n",

+      "#Initilization of variables\n",

+      "r=50 #mm\n",

+      "L1=75 #mm\n",

+      "L2=pi*r #mm\n",

+      "L3=61.2 #mm\n",

+      "# as theta1=45 degrees & theta2=60 degrees\n",

+      "sintheta1=sqrt(2)**-1\n",

+      "costheta1=sqrt(2)**-1\n",

+      "sintheta2=sqrt(3)*2**-1\n",

+      "costheta2=2**-1\n",

+      "\n",

+      "#Calculations\n",

+      "x_bar=np.array([(L1/2)*costheta1,L1*costheta1+r,L1*costheta1+100+(L3/2)*costheta2]) #mm\n",

+      "y_bar=np.array([(L1/2)*sintheta1,L1*sintheta1+(2*r)/pi,(L3/2)*sintheta2]) #mm\n",

+      "#Centroid Calculations\n",

+      "x=(L1*x_bar[0]+L2*x_bar[1]+L3*x_bar[2])/(L1+L2+L3) #mm\n",

+      "y=(L1*y_bar[0]+L2*y_bar[1]+L3*y_bar[2])/(L1+L2+L3) #mm\n",

+      "\n",

+      "#Result\n",

+      "print'The centroid is as follows:'\n",

+      "print'x=',round(x,1),\"mm\"\n",

+      "print'y=',round(y,1),\"mm\"\n",

+      "\n",

+      "# The answer may wary due to decimal point descrepancy"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The centroid is as follows:\n",

+        "x= 97.0 mm\n",

+        "y= 57.7 mm\n"

+       ]

+      }

+     ],

+     "prompt_number": 4

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 10.10-6, Page no 160"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "theta=75 #degrees\n",

+      "theta1=30 #degrees\n",

+      "sintheta=0.96\n",

+      "costheta=0.25\n",

+      "sintheta1=2**-1\n",

+      "costheta1=sqrt(3)*2**-1\n",

+      "alpha=(150*pi)/180 #rad\n",

+      "r=1\n",

+      "lhor=14 #in\n",

+      "\n",

+      "#calculations\n",

+      "a=((2*r)/alpha)*sintheta #in\n",

+      "p=90-theta\n",

+      "sinp=0.259\n",

+      "y=-a*sinp #in\n",

+      "#Length of arc\n",

+      "l=r*alpha #in\n",

+      "#Slope length calculations\n",

+      "DF=7 #in\n",

+      "AB=DF #in\n",

+      "BC=1 #in\n",

+      "BF=BC*costheta1 #in\n",

+      "FC=BC*sintheta1 #in\n",

+      "DC=DF+FC #in\n",

+      "EC=DC/costheta1 #in\n",

+      "#Centroid of EC is at G\n",

+      "yslope=0.5*EC*sintheta1+BF #in\n",

+      "#Y of composite figure\n",

+      "Y=((2*l*y)+14*-1+(2*EC*yslope))/(2*l+lhor+2*EC) #in\n",

+      "\n",

+      "#Result\n",

+      "print'The centroid is at Y=',round(Y,2),\"in\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The centroid is at Y= 1.03 in\n"

+       ]

+      }

+     ],

+     "prompt_number": 13

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 10.10-11, Page no 163"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "a=100 #mm\n",

+      "b=150 #mm\n",

+      "A1=2*10**4 #mm**2\n",

+      "A2=5*10**3 #mm**2\n",

+      "A3=(pi*(a/2)**2)/2 #mm**2\n",

+      "\n",

+      "#Calculations\n",

+      "x=(A1*a+A2*(133.3)-A3*b)/(A1+A2-A3) #mm\n",

+      "y=(A1*a*0.5+A2*(116.66)-A3*((4*a*0.5)/(3*pi)))/(A1+A2-A3) #mm\n",

+      "\n",

+      "#Result\n",

+      "print'The centroidal distances are'\n",

+      "print'x=',round(x,1),\"mm\"\n",

+      "print'y=',round(y,1),\"mm\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The centroidal distances are\n",

+        "x= 98.6 mm\n",

+        "y= 71.2 mm\n"

+       ]

+      }

+     ],

+     "prompt_number": 14

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 10.10-16, Page no 166"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "V=np.array([1728*10**3,432*10**3,7.54*10**3])\n",

+      "x_bar=np.array([60,140,60]) #mm\n",

+      "y_bar=np.array([30,20,30]) #mm\n",

+      "\n",

+      "#Calculations\n",

+      "x=(V[0]*x_bar[0]+V[1]*x_bar[1]+V[2]*x_bar[2])/(V[0]+V[1]+V[2]) #mm\n",

+      "y=(V[0]*y_bar[0]+V[1]*y_bar[1]+V[2]*y_bar[2])/(V[0]+V[1]+V[2]) #mm\n",

+      "z=120 #mm from symmetry\n",

+      "\n",

+      "#Result\n",

+      "print'The centroid is at'\n",

+      "print'x=',round(x,1),\"mm\"\n",

+      "print'y=',round(y,1),\"mm\"\n",

+      "print'z=',round(z,1),\"mm\"\n",

+      "\n",

+      "#Decimal accuracy causes discrepancy in answers\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The centroid is at\n",

+        "x= 75.9 mm\n",

+        "y= 28.0 mm\n",

+        "z= 120.0 mm\n"

+       ]

+      }

+     ],

+     "prompt_number": 17

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 10.10-17, Page no 166"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "# Here tx=30 degrees,ty=45 degrees& tz=60 degrees,, thus\n",

+      "sintx=2**-1\n",

+      "costx=sqrt(3)*2**-1\n",

+      "sinty=sqrt(2)**-1\n",

+      "costy=sqrt(2)**-1\n",

+      "sintz=sqrt(3)*2**-1\n",

+      "costz=2**-1\n",

+      "\n",

+      "#Calculations\n",

+      "V=np.array([10,15,25]) #in**3\n",

+      "x_bar=np.array([4,12,24]) #in\n",

+      "y_bar=np.array([4*costx,-6*costy,-4*costz])\n",

+      "z_bar=np.array([-4*sintx,6*sinty,-4*sintz])\n",

+      "#Centroid calculations\n",

+      "x=(V[0]*x_bar[0]+V[1]*x_bar[1]+V[2]*x_bar[2])/(V[0]+V[1]+V[2]) #in\n",

+      "y=(V[0]*y_bar[0]+V[1]*y_bar[1]+V[2]*y_bar[2])/(V[0]+V[1]+V[2]) #in\n",

+      "z=(V[0]*z_bar[0]+V[1]*z_bar[1]+V[2]*z_bar[2])/(V[0]+V[1]+V[2]) #in\n",

+      "\n",

+      "#Result\n",

+      "print'The centroid of three volumes is at'\n",

+      "print'x=',round(x,1),\"in\"\n",

+      "print'y=',round(y,2),\"in\"\n",

+      "print'z=',round(z,2),\"in\"\n",

+      "\n",

+      "# The ans for x is off by 0.4 in"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The centroid of three volumes is at\n",

+        "x= 16.0 in\n",

+        "y= -1.58 in\n",

+        "z= -0.86 in\n"

+       ]

+      }

+     ],

+     "prompt_number": 22

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 10.10-26, Page no 171"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "# Part a\n",

+      "# Pefer textbook for part a.\n",

+      "\n",

+      "# Part b\n",

+      "\n",

+      "# Initilization of variables\n",

+      "w=150 #lb/ft**2\n",

+      "h=2 #ft height of the load\n",

+      "s=8 #ft span\n",

+      "b=2 #ft\n",

+      "\n",

+      "import math\n",

+      "from scipy.integrate import quad\n",

+      "def integrand(x, a, b):\n",

+      "    return x*(150*(x/4)*2)\n",

+      "a=1\n",

+      "b=1\n",

+      "M=quad(integrand, 0, s, args=(a,b))\n",

+      "Rr=M[0]/(2*s) #lb\n",

+      "\n",

+      "# Results\n",

+      "print'The value of M is',round(M[0]),\"lb-ft\"\n",

+      "print'The value of Rr is',round(Rr),\"lb\"\n",

+      "\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The value of M is 12800.0 lb-ft\n",

+        "The value of Rr is 800.0 lb\n"

+       ]

+      }

+     ],

+     "prompt_number": 2

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 10.10-27, Page no 172"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "# Initilization of variables\n",

+      "\n",

+      "rho_m=1000 # kg/m**3\n",

+      "h=0.3 # m height of hole\n",

+      "b=0.6 # m width of hole\n",

+      "\n",

+      "import math\n",

+      "from scipy.integrate import quad\n",

+      "def integrand(y, a, b):\n",

+      "    return y*9.8*rho_m*(1.2-y)*(0.6)\n",

+      "a=1\n",

+      "b=1\n",

+      "I=quad(integrand, 0, h, args=(a,b))\n",

+      "B=I[0]/(2*(0.3))\n",

+      "\n",

+      "# Results\n",

+      "print'The value of B is',round(B),\"N\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The value of B is 441.0 N\n"

+       ]

+      }

+     ],

+     "prompt_number": 5

+    },

+    {

+     "cell_type": "heading",

+     "level": 2,

+     "metadata": {},

+     "source": [

+      "Example 10.10-28, Page no 172"

+     ]

+    },

+    {

+     "cell_type": "code",

+     "collapsed": false,

+     "input": [

+      "import math\n",

+      "\n",

+      "#Initilization of variables\n",

+      "l=62.4 #lb/ft**3\n",

+      "h=12 #ft\n",

+      "f=105 #lb/ft**3\n",

+      "\n",

+      "#Calculations\n",

+      "p1=l*h #lb/ft**2\n",

+      "#Total force on left side\n",

+      "#Simplfying the equation we get a three degree equation in d\n",

+      "#solving for d\n",

+      "p=np.array([3**-1,0,-144,467])\n",

+      "r=roots(p)\n",

+      "d=r[2] #ft\n",

+      "\n",

+      "#Result\n",

+      "print'The value of d is',round(d,2),\"feet\"\n"

+     ],

+     "language": "python",

+     "metadata": {},

+     "outputs": [

+      {

+       "output_type": "stream",

+       "stream": "stdout",

+       "text": [

+        "The value of d is 3.33 feet\n"

+       ]

+      }

+     ],

+     "prompt_number": 24

+    }

+   ],

+   "metadata": {}

+  }

+ ]

+}
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