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diff --git a/Engineering_Mechanics,_Schaum_Series_by_McLean/chapter10_5.ipynb b/Engineering_Mechanics,_Schaum_Series_by_McLean/chapter10_5.ipynb new file mode 100755 index 00000000..3ce0f857 --- /dev/null +++ b/Engineering_Mechanics,_Schaum_Series_by_McLean/chapter10_5.ipynb @@ -0,0 +1,426 @@ +{
+ "metadata": {
+ "name": "chapter10.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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