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* from \"-!../../../../node_modules/mini-css-extract-plugin/dist/loader.js??ref--8-oneOf-1-0!../../../../node_modules/css-loader/index.js??ref--8-oneOf-1-1!../../../../node_modules/vue-loader/lib/loaders/stylePostLoader.js!../../../../node_modules/postcss-loader/src/index.js??ref--8-oneOf-1-2!../../../../node_modules/sass-loader/dist/cjs.js??ref--8-oneOf-1-3!../../../../node_modules/cache-loader/dist/cjs.js??ref--0-0!../../../../node_modules/vue-loader/lib/index.js??vue-loader-options!./Lesson.vue?vue&type=style&index=0&id=7e08f0f1&prod&lang=scss&\"","var render = function render(){var _vm=this,_c=_vm._self._c;return _c('div',[_c('h3',[_vm._v(\"Rolling Motion\")]),_c('p',[_vm._v(\"\\n    Rolling motion is a combination of translational motion and rotational motion. We see a number of examples\\n    of rolling motion in our daily life, for example- a ball rolling, wheels of a vehicle rolling on the road\\n    etc. Rolling motion can occur with or without slipping (i.e., with or without relative motion at the\\n    point of contact of object with the surface).\\n  \")]),_c('h3',[_vm._v(\"Pure Rolling Motion\")]),_c('p',[_vm._v(\"\\n    In pure rolling motion, there is no relative motion at the point of contact of the object in motion\\n    with the surface. Therefore, no friction or opposing force acts at the surface in contact with the\\n    object.\\n    Center of mass of the object travels with a velocity equal to the velocity of translation and the object\\n    rotates\\n    about its center of mass. An example of pure rolling motion of a disc of radius R is shown in the\\n    figure given below. Let us analyze its motion in detail.\\n  \")]),_vm._m(0),_c('p',[_vm._v(\"\\n    The disc translates with a velocity \\\\(v\\\\) in forward direction and rotates with an angular velocity\\n    \\\\({\\\\omega}\\\\) about\\n    its center. Since translation velocity of the disc is \\\\(v\\\\), velocity of its center of mass is also\\n    \\\\(v\\\\).\\n    Velocity of any point of the disc has two components, one is the translation velocity \\\\(v\\\\) in the same\\n    direction as the velocity of center of mass and other is the velocity on account of rotation of the disc\\n    in a direction perpendicular to the radius vector of the given point with respect to the center and\\n    magnitude equal to \\\\(R{\\\\omega}\\\\). Net velocity of any point of the disc is the vector sum of these two\\n    velocities.\\n  \")]),_c('p',[_vm._v(\"\\n    As the given example is of pure rolling motion, the bottom of the disc must be at rest with respect to the\\n    surface, therefore\\n    $$v-R \\\\omega =0\\n    \\\\\\\\ v=R \\\\omega$$\\n    The direction of velocity of some other points of the disc are as shown in the figure.\\n  \")]),_c('h3',[_vm._v(\"Kinetic Energy of Rolling Motion\")]),_vm._m(1),_c('h3',[_vm._v(\"Instantaneous Axis of Rotation\")]),_c('p',[_vm._v(\"\\n    Pure rolling motion can be simplified and assumed as pure rotational motion (with same \\\\({\\\\omega}\\\\)) about\\n    an axis\\n    passing through the point of contact of the object and the surface and this axis is called instantaneous\\n    axis\\n    of rotation. Now instead of considering pure rolling motion as a combination of translation and\\n    rotational motion,\\n    it is considered only as pure rotational motion about instantaneous axis of rotation. This concept\\n    simplifies the\\n    study of rolling motion.\\n  \")]),_vm._m(2),_c('h3',{ref:\"playgraph\"},[_vm._v(\"\\n    MagicGraph | Rolling Motion\\n  \")]),_c('v-responsive',[_c('v-layout',{attrs:{\"align-center\":\"\",\"justify-center\":\"\"}},[_c('div',{staticClass:\"edliy-box-about\",attrs:{\"id\":\"jxgbox1\"}})])],1)],1)\n}\nvar staticRenderFns = [function (){var _vm=this,_c=_vm._self._c;return _c('div',{staticClass:\"w3-center\"},[_c('img',{attrs:{\"src\":\"D:\\\\Rolling_motion_1.png\",\"width\":\"400px\"}})])\n},function (){var _vm=this,_c=_vm._self._c;return _c('p',[_vm._v(\"\\n    Now we will calculate kinetic energy of a body in rolling motion. It can be separated into two parts\\n    kinetic energy of translation and kinetic energy of rotation. Thus, total kinetic energy K is given as,\\n    $$\\n    \\\\begin{equation} \\\\label{eq1}\\n    \\\\begin{split}\\n    K&=\\\\text{Translational kinetic energy} + \\\\text{Rotational Kinetic energy}\\n    \\\\\\\\&=\\\\frac{1}{2}mv^2+\\\\frac{1}{2}I{\\\\omega}^2\\n    \\\\end{split}\\n    \\\\end{equation}\\n    $$\\n    Where m is the mass of the body, v is the velocity of center of mass, I is the moment of inertia and w is the\\n    angular velocity (about center) of the body.\\n    \"),_c('br'),_c('br'),_vm._v(\"For the above example of a disc,\\n    $$\\n    \\\\begin{equation} \\\\label{eq2}\\n    \\\\begin{split}\\n    \\\\text{Kinetic Energy} &=\\\\frac{1}{2}mv^2+\\\\frac{1}{2}I{\\\\omega}^2\\n    \\\\\\\\ &=\\\\frac{1}{2}mv^2+\\\\frac{1}{2}(\\\\frac{mr^2}{2})(\\\\frac{v^2}{r^2})\\n    \\\\\\\\ &=\\\\frac{1}{2}mv^2+\\\\frac{1}{4}mv^2\\n    \\\\\\\\ &=\\\\frac{3}{4}mv^2\\n    \\\\end{split}\\n    \\\\end{equation}\\n    $$\\n  \")])\n},function (){var _vm=this,_c=_vm._self._c;return _c('p',[_vm._v(\"\\n    We can calculate the kinetic energy of the body in a much simpler way by using\\n    only a single term as in the equation below.\\n    $$ \\\\text{Kinetic Energy}=\\\\frac{1}{2}I{\\\\omega}^2$$\\n    where I is the moment of inertia of the disc about instantaneous axis of rotation.\\n    \"),_c('br'),_c('br'),_vm._v(\"Also, velocity of any point P can be calculated using instantaneous axis of rotation by a single term\\n    \\\\(v=r{\\\\omega}\\\\) where r is the distance of point P from instantaneous axis of rotation and velocity is\\n    in a direction perpendicular to this line joining the point P and instantaneous axis of rotation in the direction of rotation.\\n\\n    \"),_c('br'),_c('br'),_vm._v(\"In the example of the disc discussed above, line passing through the bottommost point and parallel to \\\\({\\\\omega}\\\\) is\\n    the\\n    instantaneous axis of rotation.\\n  \")])\n}]\n\nexport { render, staticRenderFns }","// Import the edliy_utils\r\nimport {\r\n    makeResponsive,\r\n    placeTitle,\r\n    placeImage,\r\n    placeInput,\r\n    placeSlider,\r\n    hoverMe,\r\n    placeRec,\r\n    hiddenPt,\r\n    fixedPt,\r\n    clearInputFields,\r\n    dragMe,\r\n    placeArrow,\r\n    placeGravity,\r\n    placeText,\r\n    placeLine,\r\n    placePoint,\r\n    placeGlider,\r\n    placeRuler,\r\n    placeLeftText,\r\n    placeSliderSwitch\r\n} from '../../../common/edliy_utils-circle';\r\nconst Boxes = {\r\nbox1: function () {\r\nJXG.Options.point.showInfoBox=false;\r\nJXG.Options.point.highlight=false;\r\nJXG.Options.text.highlight=false;\r\nJXG.Options.text.fixed=true;\r\nJXG.Options.curve.highlight=false;\r\nJXG.Options.text.cssDefaultStyle='fontFamily:Oswald;'\r\nvar brd1 = JXG.JSXGraph.initBoard('jxgbox1',{boundingbox: [-3, 20, 20, -3],\r\n    keepaspectratio: true, axis:true, ticks:{visible:false},\r\n    grid:true, showCopyright:false, showNavigation:false,\r\n    pan:{enabled:false}, zoom:{enabled:false}});\r\nbrd1.options.layer['image']=14;\r\nbrd1.suspendUpdate();\r\n//Title\r\nmakeResponsive(brd1);\r\nplaceTitle(brd1, 'Rolling Motion of a Coin', 'Trajectory of a Point on the Coin');\r\n//Sliders\r\nbrd1.suspendUpdate();\r\n\r\nbrd1.options.layer['image'] =14;\r\n//Title and subtitle\r\n//var title1 = brd1.create('text', [8, 18, '<b> Perimeter of Circle <b>'],{fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(28*brd1.canvasWidth/500.)}},);\r\n//var radius = brd1.create('slider',[[15,12],[20,12],[1,2.5,3]], {snapToGrid: false ,face:'circle', size:3, name:'r', strokeColor:'black', label:{fontSize:function(){return 18*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\n//placeImage(brd1,'/assets/one.svg', 2, 11.7, 0.6, 0);\r\n//placeImage(brd1,'/assets/two.svg', 2, 9.7, 0.6, 0);\r\nvar Go =placeImage(brd1,'/assets/go.svg', 9, 6, 2, 0);\r\nvar radius =placeSlider(brd1, 1, 7, 1, 1.5, 2, 4, '&nbsp; &nbsp;R');\r\nradius.setAttribute({snapWidth:0.25});\r\nplaceText(brd1, 8.5, 16, 20,  '1. Set the Location of Point \\'P\\' 2. Tap on Go button to let the coin roll')\r\n//placeText(brd1, 6, 10, 20,  '1. Set the radius using the slider. 2. Tap on the Go button to roll the circle.')\r\n//placeText(brd1, 3, 6, 20, ()=>'L =2 &pi; R ='+(2*Math.PI*radius.Value()).toFixed(2));\r\nvar segment1 = brd1.create('segment', [[0,function(){return radius.Value()}], [function(){return 4*Math.PI*radius.Value()},function(){return radius.Value()}]], {fixed: true, strokeColor: 'black', visible:false});\r\n//var ptCenter = brd1.create('glider', [0,2, segment1],{face:'o' , name:'Drag Me!', label:{fontSize:function(){return 16*brd1.canvasHeight/800}, cssStyle:'fontFamily:Oswald'}});\r\nvar ptCenter = placeGlider(brd1, segment1, 0, 2, 3,  'red', 'black');\r\n//ptCenter.setLabel('Drag me!');\r\nvar r =placePoint(brd1, ()=>ptCenter.X(), ()=>ptCenter.Y()/2,0, 'red','black');\r\nr.setLabel('R');\r\nr.label.setAttribute({anchorX:'middle', anchorY:'middle'});\r\nvar pt1 =brd1.create('point', [function(){return ptCenter.X()},0.0], {fixed:false, visible:false, face:'circle', size:4, name:' ', label: {offset: [0,25]}, strokeColor: 'black', fillColor: '#e1e1e1'});\r\nradius.on('down', function(){ptCenter.moveTo([0,radius.Value()]); pt1.moveTo([0,0]);});\r\nvar c1 = brd1.create('circle', [ptCenter, pt1], {strokeColor: 'black', strokeWidth: 0.5, fillColor: 'orange', fillOpacity: 0.2});\r\nfunction angle1(){\r\n    return  (2*Math.PI-ptCenter.X()/radius.Value() );\r\n}\r\n\r\nvar pt3 =brd1.create('point',\r\n[function(){return ptCenter.X()+radius.Value()*Math.sin(angle1())},function(){return ptCenter.Y()-radius.Value()*Math.cos(angle1())}], {trace:true, fixed:true, visible:true, face:'circle', size:3, name:'',  strokeColor: 'black', fillColor: 'white'});\r\n\r\nplaceArrow(brd1, [()=>ptCenter.X(), ()=>ptCenter.Y()], [()=>pt3.X(), ()=>pt3.Y()], 2, 'black');\r\n//var txt1 = brd1.create('text', [3, -4,  function(){return 'ANGLE = ' + angle1()*180/Math.PI}],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(12*brd1.canvasWidth/500.)}},);\r\n\r\nvar segment2 = brd1.create('segment', [[0,0], [function(){return pt1.X()},0]], {fixed: true, strokeWidth:4, strokeColor: 'green'});\r\n\r\nvar arc1 = brd1.create('arc', [ptCenter, pt1, pt3], {strokeColor: 'green', strokeWidth:4});\r\n\r\n\r\n//Arrow\r\nvar ptArrow1 =brd1.create('point', [function(){return pt1.X()} ,-2], {fixed:true, visible:false, face:'circle', size:4, name:' ', label: {offset: [0,25]}, strokeColor: 'black', fillColor: '#e1e1e1'});\r\nvar ptArrow2 =brd1.create('point', [0 ,-2], {fixed:false, visible:false, face:'circle', size:4, name:' ', label: {offset: [0,25]}, strokeColor: 'black', fillColor: '#e1e1e1'});\r\nvar segment2 = brd1.create('segment', [ptArrow1, ptArrow2], {fixed: true, strokeColor: 'black', visible:true, strokeWidth:1,\r\n                                                                                                                highlightStrokeWidth: 2,\r\n                                                                                                                lastArrow: {type: 2, size: 8, highlightSize: 6},\r\n                                                                                                                touchLastPoint: false,\r\n                                                                                                                firstArrow: {type: 2, size: 8}\r\n                                                                                                                });\r\n\r\nvar txtPer = brd1.create('text', [function(){return pt1.X()/2}, -1.5,  function(){return pt1.X().toFixed(2)}],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(12*brd1.canvasWidth/500.)}},);\r\nGo.on('down', function(){ptCenter.moveTo([4*Math.PI*radius.Value(), radius.Value()], 1000)});\r\n\r\n//var ptArrow3 =brd1.create('point', [-5,0], {fixed:true, visible:false, face:'circle', size:4, name:' ', label: {offset: [0,25]}, strokeColor: 'black', fillColor: '#e1e1e1'});\r\n//var ptArrow4 =brd1.create('point', [-5, function(){return radius.Value()}], {fixed:false, visible:false, face:'circle', size:4, name:' ', label: {offset: [0,25]}, strokeColor: 'black', fillColor: '#e1e1e1'});\r\n//var segment3 = brd1.create('segment', [ptArrow3, ptArrow4], {fixed: true, strokeColor: 'black', visible:true, strokeWidth:1,\r\n//                                                                                                                highlightStrokeWidth: 2,\r\n//                                                                                                                lastArrow: {type: 2, size: 8, highlightSize: 6},\r\n                                                                                                                //touchLastPoint: false,\r\n                                                                                                                //firstArrow: {type: 2, size: 8}\r\n                                                                                                                //})\r\n//var txtRad = brd1.create('text', [-4, function(){return radius.Value()/2},  function(){return radius.Value().toFixed(2)}],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(12*brd1.canvasWidth/500.)}},);\r\n\r\nbrd1.unsuspendUpdate();\r\n}\r\n}\r\nexport default Boxes;\r\n","<template>\r\n  <div>\r\n    <h3>Rolling Motion</h3>\r\n    <p>\n      Rolling motion is a combination of translational motion and rotational motion. We see a number of examples\r\n      of rolling motion in our daily life, for example- a ball rolling, wheels of a vehicle rolling on the road\r\n      etc. Rolling motion can occur with or without slipping (i.e., with or without relative motion at the\r\n      point of contact of object with the surface).\n    </p>\r\n    <h3>Pure Rolling Motion</h3>\r\n    <p>\n      In pure rolling motion, there is no relative motion at the point of contact of the object in motion\r\n      with the surface. Therefore, no friction or opposing force acts at the surface in contact with the\r\n      object.\r\n      Center of mass of the object travels with a velocity equal to the velocity of translation and the object\r\n      rotates\r\n      about its center of mass. An example of pure rolling motion of a disc of radius R is shown in the\r\n      figure given below. Let us analyze its motion in detail.\r\n    </p>\r\n    <div class=\"w3-center\">\r\n      <img src=\"D:\\Rolling_motion_1.png\" width=\"400px\"></img>\r\n    </div>\r\n    <p>\n      The disc translates with a velocity \\(v\\) in forward direction and rotates with an angular velocity\r\n      \\({\\omega}\\) about\r\n      its center. Since translation velocity of the disc is \\(v\\), velocity of its center of mass is also\r\n      \\(v\\).\r\n      Velocity of any point of the disc has two components, one is the translation velocity \\(v\\) in the same\r\n      direction as the velocity of center of mass and other is the velocity on account of rotation of the disc\r\n      in a direction perpendicular to the radius vector of the given point with respect to the center and\r\n      magnitude equal to \\(R{\\omega}\\). Net velocity of any point of the disc is the vector sum of these two\r\n      velocities.\r\n    </p>\r\n    <p>\n      As the given example is of pure rolling motion, the bottom of the disc must be at rest with respect to the\r\n      surface, therefore\r\n      $$v-R \\omega =0\r\n      \\\\ v=R \\omega$$\r\n      The direction of velocity of some other points of the disc are as shown in the figure.\n    </p>\r\n    <h3>Kinetic Energy of Rolling Motion</h3>\r\n    <p>\n      Now we will calculate kinetic energy of a body in rolling motion. It can be separated into two parts\r\n      kinetic energy of translation and kinetic energy of rotation. Thus, total kinetic energy K is given as,\r\n      $$\r\n      \\begin{equation} \\label{eq1}\r\n      \\begin{split}\r\n      K&=\\text{Translational kinetic energy} + \\text{Rotational Kinetic energy}\r\n      \\\\&=\\frac{1}{2}mv^2+\\frac{1}{2}I{\\omega}^2\r\n      \\end{split}\r\n      \\end{equation}\r\n      $$\r\n      Where m is the mass of the body, v is the velocity of center of mass, I is the moment of inertia and w is the\r\n      angular velocity (about center) of the body.\r\n      <br><br>For the above example of a disc,\r\n      $$\r\n      \\begin{equation} \\label{eq2}\r\n      \\begin{split}\r\n      \\text{Kinetic Energy} &=\\frac{1}{2}mv^2+\\frac{1}{2}I{\\omega}^2\r\n      \\\\ &=\\frac{1}{2}mv^2+\\frac{1}{2}(\\frac{mr^2}{2})(\\frac{v^2}{r^2})\r\n      \\\\ &=\\frac{1}{2}mv^2+\\frac{1}{4}mv^2\r\n      \\\\ &=\\frac{3}{4}mv^2\r\n      \\end{split}\r\n      \\end{equation}\r\n      $$\r\n    </p>\r\n    <h3>Instantaneous Axis of Rotation</h3>\r\n    <p>\n      Pure rolling motion can be simplified and assumed as pure rotational motion (with same \\({\\omega}\\)) about\r\n      an axis\r\n      passing through the point of contact of the object and the surface and this axis is called instantaneous\r\n      axis\r\n      of rotation. Now instead of considering pure rolling motion as a combination of translation and\r\n      rotational motion,\r\n      it is considered only as pure rotational motion about instantaneous axis of rotation. This concept\r\n      simplifies the\r\n      study of rolling motion.\n    </p>\r\n    <p>\n      We can calculate the kinetic energy of the body in a much simpler way by using\r\n      only a single term as in the equation below.\r\n      $$ \\text{Kinetic Energy}=\\frac{1}{2}I{\\omega}^2$$\r\n      where I is the moment of inertia of the disc about instantaneous axis of rotation.\r\n      <br><br>Also, velocity of any point P can be calculated using instantaneous axis of rotation by a single term\r\n      \\(v=r{\\omega}\\) where r is the distance of point P from instantaneous axis of rotation and velocity is\r\n      in a direction perpendicular to this line joining the point P and instantaneous axis of rotation in the direction of rotation.\r\n\r\n      <br><br>In the example of the disc discussed above, line passing through the bottommost point and parallel to \\({\\omega}\\) is\r\n      the\r\n      instantaneous axis of rotation.\n    </p>\r\n    <h3 ref=\"playgraph\">\n      MagicGraph | Rolling Motion\n    </h3>\r\n    <v-responsive>\r\n      <v-layout align-center justify-center>\r\n        <div id=\"jxgbox1\" class=\"edliy-box-about\" />\r\n      </v-layout>\r\n    </v-responsive>\r\n  </div>\r\n</template>\r\n<script>\r\nimport Boxes from './Boxes.js'\r\nexport default {\r\n  name: 'FBD',\r\n  created: function () {\r\n    this.$store.commit('navigation/resetState');\r\n    let title = 'Free Body Diagram';\r\n    this.$store.commit('navigation/changeTitle', title);\r\n    this.$store.commit('navigation/changeMenu', title);\r\n    let newTopics = [\r\n      {title: 'What is a Free Body Diagram?', img:'/assets/number-1.svg', action: () => this.goto('ia')},\r\n      {title: 'Purpose of FBD', img:'/assets/number-2.svg', action: () => this.goto('ja')},\r\n      {title: 'Example of FBD', img:'/assets/number-3.svg', action: () => this.goto('ka')},\r\n      {title: 'MagicGraph' ,img:'/assets/touch.svg', action: () => this.goto('graphical')},\r\n    ];\r\n    this.$store.commit('navigation/replaceTopics', newTopics);\r\n    let newshowhome = false;\r\n    this.$store.commit('navigation/toggleshowhome', newshowhome);\r\n    let newPhys =true;\r\n    this.$store.commit('navigation/replacePhys', newPhys);\r\n    let newLeftArrow =true;\r\n    this.$store.commit('navigation/replaceLeftArrow', newLeftArrow);\r\n    let newModule =true;\r\n    this.$store.commit('navigation/replaceModule', newModule);\r\n  },\r\n  mounted () {\r\n    MathJax.Hub.Queue([\"Typeset\", MathJax.Hub]);\r\n    Boxes.box1();\r\n  },\r\n  metaInfo() {\r\n  return{ title: 'Free Body Diagram',\r\n          titleTemplate: '%s - Learn interactively',\r\n          meta: [ {name: 'viewport', content: 'width=device-width, initial-scale=1'},\r\n                  {name: 'description', content: 'Learn interactively how to draw a free body diagram'}\r\n                ]\r\n        }\r\n   },\r\n}\r\n</script>\r\n<style lang=\"scss\">\r\n@import 'src/styles/edliy-box.scss';\r\n@import 'src/styles/subtopic-menu.scss';\r\n@import 'src/styles/edliy-box-about.scss';\r\n@import 'src/styles/screen-sizes.scss';\r\n</style>\r\n","import mod from \"-!../../../../node_modules/cache-loader/dist/cjs.js??ref--12-0!../../../../node_modules/thread-loader/dist/cjs.js!../../../../node_modules/babel-loader/lib/index.js!../../../../node_modules/cache-loader/dist/cjs.js??ref--0-0!../../../../node_modules/vue-loader/lib/index.js??vue-loader-options!./Lesson.vue?vue&type=script&lang=js&\"; export default mod; export * from \"-!../../../../node_modules/cache-loader/dist/cjs.js??ref--12-0!../../../../node_modules/thread-loader/dist/cjs.js!../../../../node_modules/babel-loader/lib/index.js!../../../../node_modules/cache-loader/dist/cjs.js??ref--0-0!../../../../node_modules/vue-loader/lib/index.js??vue-loader-options!./Lesson.vue?vue&type=script&lang=js&\"","import { render, staticRenderFns } from \"./Lesson.vue?vue&type=template&id=7e08f0f1&\"\nimport script from \"./Lesson.vue?vue&type=script&lang=js&\"\nexport * from \"./Lesson.vue?vue&type=script&lang=js&\"\nimport style0 from \"./Lesson.vue?vue&type=style&index=0&id=7e08f0f1&prod&lang=scss&\"\n\n\n/* normalize component */\nimport normalizer from \"!../../../../node_modules/vue-loader/lib/runtime/componentNormalizer.js\"\nvar component = normalizer(\n  script,\n  render,\n  staticRenderFns,\n  false,\n  null,\n  null,\n  null\n  \n)\n\nexport default component.exports"],"sourceRoot":""}