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diff --git a/sample_notebooks/MandalaManoj pruthvi/Chapter_4_Radian_Measure.ipynb b/sample_notebooks/MandalaManoj pruthvi/Chapter_4_Radian_Measure.ipynb new file mode 100755 index 00000000..212743ac --- /dev/null +++ b/sample_notebooks/MandalaManoj pruthvi/Chapter_4_Radian_Measure.ipynb @@ -0,0 +1,658 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Chapter 4 Radian Measure" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.1 page.no:96" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Radian measure is 0.314159 rad\n", + "(or)\n", + "Radian measure is (pi/10)rad\n" + ] + } + ], + "source": [ + "#To convert a degree measure to radians\n", + "from math import pi\n", + "\n", + "deg=18 # degree measure\n", + "radian=deg*(pi/180) # radian measure\n", + "print \"Radian measure is %f rad\\n(or)\"%radian\n", + "print \"Radian measure is (pi/%.0f)rad\"%(1/(radian/pi))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.2 page.no:96" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Degree measure is 20 degree\n" + ] + } + ], + "source": [ + "#To convert a radian meeasure to degree\n", + "from math import pi\n", + "\n", + "radian=pi/9 # radian measure\n", + "deg=radian/(pi/180) # degree measure\n", + "print \"Degree measure is %.0f degree\"%deg" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.3 page.no:99" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Length of arc intercepted =2.4 cm\n" + ] + } + ], + "source": [ + "#To determine length of the intercepted arc\n", + "r=2. #radius of circle\n", + "theta=1.2 # central angle in radian\n", + "s=r*theta # length of arc\n", + "print \"Length of arc intercepted =%.1f cm\"%s" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.4 page.no:99" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Length of arc intercepted = 7.16 ft \n" + ] + } + ], + "source": [ + "#To determine length of the arc intercepted\n", + "from math import pi\n", + "\n", + "r=10 #radius of circle\n", + "theta=41*(pi/180) # central angle in radian\n", + "s=r*theta # length of arc\n", + "print \"Length of arc intercepted = %.2f ft \"%s" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.5 page.no:100" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Measure of central angle = 0.40 rad\n", + " \n", + "Measure of central angle =22.92 degree\n" + ] + } + ], + "source": [ + "#To determine angle in radians and degrees\n", + "from math import pi\n", + "\n", + "r=5. #radius of circle\n", + "s=2. #length of arc\n", + "theta = s/r #central angle in radian\n", + "print \"Measure of central angle = %.2f rad\\n \"%theta\n", + "print \"Measure of central angle =%.2f degree\"%(theta*(180/pi))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.6 page.no:100" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Length of the rope =13.4 ft\n" + ] + } + ], + "source": [ + "#To determine the length of the rope\n", + "from math import sqrt,pi,atan,acos\n", + "\n", + "d=8. #distance between places in feet\n", + "r=2. #radius of cylinder in feet\n", + "#from the figure\n", + "DA=d/2\n", + "BE=r\n", + "DE=3 #distance from centre of container to wall\n", + "AE=sqrt(DE**2 + DA**2) # pythagoras theorem\n", + "AB=sqrt(AE**2 - BE**2) # pythagoras theorem\n", + "#all angles below are in radians\n", + "angle_AED = atan((d/2)/DE)\n", + "angle_AEB = acos(BE/AE)\n", + "angle_BEC = pi - (angle_AED + angle_AEB)\n", + "arc_BC = BE*angle_BEC #length of arc BC\n", + "L = 2*(AB + arc_BC) #length of rope\n", + "print \"Length of the rope =%.1f ft\"%L" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.7 page.no:101" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Length of belt around pulley = 71.4 cm\n" + ] + } + ], + "source": [ + "#To determine the length of the belt around the pulleys\n", + "from math import pi,sqrt,asin\n", + "\n", + "AE= 5. #radius of first pulley in cm\n", + "BF= 8. #radius of second pulley in cm\n", + "AB=15. #distance between centre of pulleys in cm\n", + "#from the figure\n", + "CF=AE #parallel side of rectangle ACFE\n", + "BC= BF- CF\n", + "AC = sqrt(AB**2 - BC**2) #by pythagoras theorem\n", + "EF=AC# parallel side of rectangle ACFE 14\n", + "angle_EAC = pi/2\n", + "angle_BAC = asin(BC/AB)\n", + "angle_DAE = pi - angle_EAC - angle_BAC\n", + "angle_ABC = angle_DAE #AE and BF are parallel\n", + "angle_GBF= pi - angle_ABC\n", + "arc_DE=AE*angle_ABC # length of arc DE\n", + "arc_FG=BF*angle_GBF # length of arc FG\n", + "L=2*(arc_DE + EF + arc_FG) #length of belt\n", + "print \"Length of belt around pulley = %.1f cm\"%L" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.8 page.no:103" + ] + }, + { + "cell_type": "code", + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Area of sector = 1.6∗pi cmˆ2\n", + "(or)\n", + "Area of sector = 5.026548 cmˆ2\n" + ] + } + ], + "source": [ + "#To find the area of sector of circle\n", + "from math import pi\n", + "\n", + "theta= pi/5 # angle in radian\n", + "r=4. #radius in cm\n", + "A=r*r*theta/2 #Area of sector\n", + "print \"Area of sector = %.1f∗pi cmˆ2\\n(or)\"%(A/pi)\n", + "print \"Area of sector = %f cmˆ2\"%A" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.9 page.no:103" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Area of sector =12.51 mˆ2\n" + ] + } + ], + "source": [ + "#To determine area of sector of a circle\n", + "from math import pi\n", + "\n", + "theta= 117*(pi/180) # angle in radian\n", + "r=3.5 #radius in m\n", + "A=r*r*theta/2 #Area of sector\n", + "print \"Area of sector =%.2f mˆ2\"%A" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.10 page.no:104" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Area of sector =27 cmˆ2\n", + "\n", + "Note: Angle subtended by arc = 0.666667 rad\n" + ] + } + ], + "source": [ + "#To determine area of sector of circle\n", + "\n", + "s=6. #arc length in cm\n", + "r=9. #radius in cm\n", + "A=r*s/2 #Area of sector\n", + "print \"Area of sector =%.0f cmˆ2\\n\"%A\n", + "print \"Note: Angle subtended by arc = %f rad\"%(s/r)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.11 page.no:104" + ] + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Area enclosed by belt pulley system = 338.71 cmˆ2 \n" + ] + } + ], + "source": [ + "#To determine area insude belt pulley system\n", + "from math import pi,sqrt,asin\n", + "\n", + "AE= 5. #radius of first pulley\n", + "BF= 8. #radius of second pulley\n", + "AB=15. #distance between centre of pulleys\n", + "#from the figure\n", + "CF=AE\n", + "BC= BF- CF\n", + "AC = sqrt(AB**2 - BC**2)\n", + "#from the figure\n", + "angle_EAC = pi/2\n", + "angle_BAC = asin(BC/AB)\n", + "angle_DAE = pi - angle_EAC - angle_BAC\n", + "angle_ABC = angle_DAE #AE and BF are parallel\n", + "angle_GBF= pi - angle_ABC\n", + "area_DAE = AE**2*angle_DAE/2 #area of sector DAE\n", + "area_GBF = BF**2*angle_GBF/2 #area of sector GBF\n", + "area_AEFC = AE*AC #area of rectangle AEFC\n", + "area_ABC = AC*BC/2 #area of triangle ABC\n", + "area_K =2*( area_DAE + area_AEFC + area_ABC +area_GBF)\n", + "print \"Area enclosed by belt pulley system = %.2f cmˆ2 \"%area_K\n", + "#Note: answer differs from book due to approximations by them " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.12 page.no:105" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Required area of segment = 1.408 square units\n" + ] + } + ], + "source": [ + "#To determine area of segment formed by a chord in circle\n", + "from math import acos,sin\n", + "\n", + "radius = 2.\n", + "chord = 3.\n", + "#Use law of cosines\n", + "cos_theta = (radius**2+radius**2-chord**2)/(2*radius*radius)\n", + "theta=acos(cos_theta) #subtended central angle in radians\n", + "area_K=radius**2*(theta-sin(theta))/2\n", + "print \"Required area of segment = %.3f square units\"%area_K" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.13 page.no:106" + ] + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Area of intersection of 2 circles =7.66 cm ˆ2 \n" + ] + } + ], + "source": [ + "#To determine area of intersection of 2 circles\n", + "from math import acos\n", + "\n", + "d=7. #distance between centres in cm\n", + "r1= 5. #radius of first circle in cm\n", + "r2= 4. #radius of second circle in cm\n", + "#use law of cosines\n", + "cos_alpha=(d**2+ r1**2 - r2**2 ) /(2*d*r1)\n", + "cos_beeta=(d**2+ r2**2 - r1**2 ) /(2*d*r2)\n", + "#from the geometry of the figure\n", + "#all the angles below are in radians\n", + "alpha= acos(cos_alpha)\n", + "beeta= acos(cos_beeta)\n", + "angle_BAC = alpha\n", + "angle_ABC = beeta\n", + "angle_CAD =2* angle_BAC\n", + "angle_CBD =2* angle_ABC\n", + "#required area = area at segment CD in circle at A and at B\n", + "area_K = r1**2*(angle_CAD-sin(angle_CAD))/2 + r2 **2*(angle_CBD-sin(angle_CBD))/2\n", + "print \"Area of intersection of 2 circles =%.2f cm ˆ2 \"%area_K" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.14 page.no:109" + ] + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Angular speed= 2.094395 radian/sec\n", + "\n", + "Linear speed=6.283185m/sec\n", + "\n", + "(or)\n", + "\n", + "Angular speed= 0.666667∗pi radian/sec\n", + " \n", + "Linear speed = 2.000000∗pi m/sec \n" + ] + } + ], + "source": [ + "#To find linear and angular speed of a moving object\n", + "from math import pi\n", + "t=0.5 #time in second\n", + "r= 3 #radius in m of the circle\n", + "theta = pi/3 # central angle in radian\n", + "w = theta/t #angular speed in rad /sec\n", + "v=w*r#linear speed in m/sec\n", + "print \"Angular speed= %f radian/sec\\n\"%w\n", + "print \"Linear speed=%fm/sec\"%v\n", + "print \"\\n(or)\\n\\nAngular speed= %f∗pi radian/sec\\n \"%(w/pi)\n", + "print \"Linear speed = %f∗pi m/sec \"%(v/pi)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.15 page.no:109" + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Linear speed = 12.96 ft/sec\n", + "\n", + "Angular speed= 6.48 radian/sec\n" + ] + } + ], + "source": [ + "#To find linear and angular speed of a moving object\n", + "\n", + "t=2.7 #time in second\n", + "r= 2. #radius in ft of the circle\n", + "s=35. #distance in feet\n", + "v=s/t #linear speed in ft/sec\n", + "w=v/r #angular speed in rad /sec\n", + "print \"Linear speed = %.2f ft/sec\\n\"%v\n", + "print \"Angular speed= %.2f radian/sec\"%w" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.16 page.no:109" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "central angle swept = 7.75 radian \n" + ] + } + ], + "source": [ + "#To find the central angle swept by a moving object\n", + "t=3.1 #time in second\n", + "v= 10 #linear speed in m/sec\n", + "r= 4 #radius in m of the circle\n", + "s=v*t # distance in m\n", + "theta = s/r #central angle swept\n", + "print \"central angle swept = %.2f radian \"%theta" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Exa 4.17 page.no:110" + ] + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Angular speed of larger gear=20 rpm \n" + ] + } + ], + "source": [ + "#To find the angular speed of larger gear interlocked with smaller gear\n", + "r1=5 #radius of larger gear\n", + "r2=4 #radius smaller gear\n", + "w2=25 #angular speed of smaller gear\n", + "# Imagine a particle on outer radii of each gear\n", + "#At any time , for every rotation , circular displacement of each particle is same\n", + "# (or) s1=s2 implies v1∗t=v2∗t\n", + "#v1= v2 implies w1∗r1=w2∗r2\n", + "w1=(w2*r2)/r1 #angular speed of larger gear\n", + "print \"Angular speed of larger gear=%.0f rpm \"%w1" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 2", + "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": "2.7.10" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} |