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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Chapter 8 Friction"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Ex 8.1 Page No 305"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "F = 69.3 N\n",
+ "NC = 236.2 N\n",
+ "x = -9.08 mm\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Example 8.1\n",
+ "import math\n",
+ "\n",
+ "# Variable Declaration\n",
+ "P = 80 #[Newton]\n",
+ "\n",
+ "# Calculation\n",
+ "# Using +ΣF_x(right) = 0+ΣMO(counterclockwise) = 0\n",
+ "F = round(80*math.cos(math.pi*30/180),1) #[Newton]\n",
+ "\n",
+ "# Using +ΣF_y(upward) = 0\n",
+ "NC = round(80*math.sin(math.pi*30/180)+196.2,1) #[Newton]\n",
+ "\n",
+ "# Using +ΣMO(counterclockwise) = 0\n",
+ "x = (80*math.cos(math.pi*30/180)*0.2-80*math.sin(math.pi*30/180)*0.4)/NC #[meter]\n",
+ "\n",
+ "# Result\n",
+ "print\"F = \",(F),\"N\"\n",
+ "print\"NC = \",(NC),\"N\"\n",
+ "print\"x = \",(round(x*1000,2)),\"mm\"\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Ex 8.2 Page No 306"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "us = 0.466\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Ex 8.2\n",
+ "import math\n",
+ "\n",
+ "# Calculation\n",
+ "# W*sin25 = us(W*cos25)\n",
+ "us = round(math.tan(math.pi*25/180),3)\n",
+ "\n",
+ "# Result\n",
+ "print\"us = \",(us)\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Ex 8.3 Page No 307"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "us = 0.228\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Ex 8.3\n",
+ "import numpy as np\n",
+ "\n",
+ "# Calculation\n",
+ "coeff = [1, -4.619, 1]\n",
+ "us = np.roots(coeff)\n",
+ "\n",
+ "# Result\n",
+ "# Finding the smallest root\n",
+ "print\"us = \",(round(min(us),3))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Ex 8.4 Page No 308"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "us_min = 0.268\n",
+ "ug_min = 0.0893\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Ex 8.4\n",
+ "import math\n",
+ "\n",
+ "# Calculation\n",
+ "# Using +ΣF_x(right) = 0,FA = F and NA = N for bottom pipe\n",
+ "# us_min = F/N\n",
+ "us_min = round(math.sin(math.pi*30/180)/(1+math.cos(math.pi*30/180)),3)\n",
+ "# Let smallest required coefficient of static friction be ug_min\n",
+ "# ug_min = F/NC\n",
+ "ug_min = round(0.2679*0.5/1.5,4)\n",
+ "\n",
+ "# Result\n",
+ "print\"us_min = \",(us_min)\n",
+ "print\"ug_min = \",(ug_min)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Ex 8.5 Page No 309"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "P = 266.7 N\n",
+ "NC = 400 N\n",
+ "FC = 200.0 N\n",
+ "FB = 66.7 N\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Ex 8.5\n",
+ "import numpy as np\n",
+ "\n",
+ "# Variable Declaration\n",
+ "uB = 0.2\n",
+ "uC = 0.5\n",
+ "\n",
+ "# Calculation\n",
+ "# Post slips only at B\n",
+ "# FB = uB*NB\n",
+ "FB = uB*400 #[Newton]\n",
+ "# Using +ΣMC(counterclockwise) = 0\n",
+ "P = FB/0.25 #[Newton]\n",
+ "# Using +ΣF_y(upward) = 0\n",
+ "NC = 400 #[Newton]\n",
+ "# Using +ΣF_x(right) = 0\n",
+ "FC = P-FB #[Newton]\n",
+ "\n",
+ "# Post slips only at C\n",
+ "FC = uC*NC #[Newton]\n",
+ "# Using +ΣF_x(right) = 0 and # Using +ΣMC(counterclockwise) = 0\n",
+ "a = np.array([[1,-1],[-0.25,1]])\n",
+ "b = np.array([200,0])\n",
+ "x = np.linalg.solve(a, b)\n",
+ "P = round(x[0],1) #[Newton]\n",
+ "FB = round(x[1],1) #[Newton]\n",
+ "\n",
+ "# Result\n",
+ "print\"P = \",(P),\"N\"\n",
+ "print\"NC = \",(NC),\"N\"\n",
+ "print\"FC = \",(FC),\"N\"\n",
+ "print\"FB = \",(FB),\"N\""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "collapsed": true
+ },
+ "source": [
+ "## Ex 8.6 Page No 310"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "NA = 1084.5 N\n",
+ "FA = 162.7 N\n",
+ "FB = 162.7 N\n",
+ "P = 498.2 N\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Ex 8.6\n",
+ "import math\n",
+ "import numpy as np\n",
+ "\n",
+ "# Variable Declaration\n",
+ "usA = 0.15\n",
+ "usB = 0.4\n",
+ "\n",
+ "# Calculation\n",
+ "\n",
+ "# Pipe rolls up incline\n",
+ "# Using +ΣF_x(right) = 0, +ΣMO(counterclockwise) = 0 and FB = 0.4*P\n",
+ "a = np.array([[-1,1],[-400,0.4*400]])\n",
+ "b = np.array([981*math.sin(math.pi*20/180),0])\n",
+ "x = np.linalg.solve(a, b)\n",
+ "FA = round(x[0],1) #[Newton]\n",
+ "P = round(x[1],1) #[Newton]\n",
+ "FB = FA #[Newton]\n",
+ "NA = round(FB+981*math.cos(math.pi*20/180),1) #[Newton]\n",
+ "P = round(981*math.sin(math.pi*20/180)+FA,1) #[Newton]\n",
+ "\n",
+ "# Pipe slides up incline\n",
+ "# Using +ΣMO(counterclockwise) = 0 and FA = 0.15*NA\n",
+ "a = np.array([[-0.15*400,400],[1,-1]])\n",
+ "b = np.array([0,981*math.cos(math.pi*20/180)])\n",
+ "x = np.linalg.solve(a, b)\n",
+ "NA = round(x[0],1) #[Newton]\n",
+ "FB = round(x[1],1) #[Newton]\n",
+ "FA = FB #[Newton]\n",
+ "P = round(FA+981*math.sin(math.pi*20/180),1) #[Newton]\n",
+ "\n",
+ "# Result\n",
+ "print\"NA = \",(NA),\"N\"\n",
+ "print\"FA = \",(FA),\"N\"\n",
+ "print\"FB = \",(FB),\"N\"\n",
+ "print\"P = \",(P),\"N\"\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Ex 8.8 Page No 319"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "M = 6.38 N.m\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Ex 8.8\n",
+ "import math\n",
+ "from __future__ import division\n",
+ "\n",
+ "# Variable Declaration\n",
+ "us = 0.25\n",
+ "W = 2000 #[Newton]\n",
+ "r = 5 #[millimeter]\n",
+ "\n",
+ "# Calculation\n",
+ "phi_s = round(math.degrees(math.atan(us)),2)\n",
+ "theta = round(math.degrees(math.atan(2/(2*math.pi*5))),2)\n",
+ "M = 2*W*r*math.tan(math.pi*(phi_s+theta)/180) #[Newton millimeter]\n",
+ "\n",
+ "# Result\n",
+ "print\"M = \",round(M/1000,2),\"N.m\"\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "collapsed": true
+ },
+ "source": [
+ "## Ex 8.9 Page No 324"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "m = 15.7 kg\n"
+ ]
+ }
+ ],
+ "source": [
+ "# Ex 8.9\n",
+ "import math\n",
+ "from __future__ import division\n",
+ "\n",
+ "# Variable Declaration\n",
+ "us = 0.25\n",
+ "\n",
+ "# Calculation\n",
+ "T1 = round(500/(math.exp(us*(3/4)*math.pi)),1)\n",
+ "W = round(T1/(math.exp(us*(3/4)*math.pi)),1)\n",
+ "m = round(W/9.81,1)\n",
+ "\n",
+ "# Result\n",
+ "print\"m = \",(m),\"kg\""
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "anaconda-cloud": {},
+ "kernelspec": {
+ "display_name": "Python [default]",
+ "language": "python",
+ "name": "python2"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 2
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython2",
+ "version": "2.7.12"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 1
+}
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