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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=158c0dc9&prod&lang=scss&\"","var render = function render(){var _vm=this,_c=_vm._self._c;return _c('div',[_c('h3',[_vm._v(\"What is center of mass?\")]),_c('p',[_vm._v(\"\\n    Center of mass of a body or a system of particles is a unique point where the entire mass of the system can\\n    be assumed to be concentrated. Its motion is same as if whole mass of the system was concentrated at this\\n    point and all the external forces were applied at this point. The concept of center\\n    of mass helps in solving many complex problems quite simply. Suppose if several forces are acting on a body\\n    of any odd shape, we need not bother about the forces acting on the individual particles, but we can apply the\\n    net force on the center of mass and thus analyze the body's motion.\\n  \")]),_c('h3',[_vm._v(\"Center of Mass of System of Particles\")]),_c('p',[_vm._v(\"\\n    The position of center of mass of a system of particles depends upon the distribution of mass in it. It is\\n    not necessary that the center of mass lies within the material of the body.\\n  \")]),_c('p',[_vm._v(\"\\n    For a system of \\\\(n\\\\) particles with masses \\\\(m_1,m_2, ..., m_n\\\\) and position vectors as \\\\(\\\\vec{r_1},\\\\vec{r_2}, ...,\\\\vec{r_n}\\\\), the\\n    position\\n    vector \\\\(\\\\vec{r}_{com}\\\\) of the center of mass of the system is given by\\n  \")]),_vm._v(\"\\n  $$\\\\vec{r}_{com} =\\\\frac{m_1\\\\vec{r_1}+m_2\\\\vec{r_2}+...+m_n\\\\vec{r_n}}{m_1+m_2+...+m_n}\\n  =\\\\frac{\\\\sum_{i=1}^{n} \\\\vec{r_i}m_i}{\\\\sum_{i=1}^{n} m_i} \\\\;\\\\;\\\\;\\\\;\\\\;\\\\; -(i)\\n\\n  $$\\n  \"),_c('p',[_vm._v(\"Therefore the \\\\(x\\\\) , \\\\(y\\\\) and \\\\(z\\\\) co-ordinates of center of mass is given by \")]),_vm._v(\"\\n  $$ x_{com} =\\\\frac{\\\\sum_{i=1}^{n}x_im_i}{M} \\\\;\\\\;\\\\;\\\\; y_{com} =\\\\frac{\\\\sum_{i=1}^{n}y_im_i}{M} \\\\;\\\\;\\\\;\\\\; z_{com}\\n  =\\\\frac{\\\\sum_{i=1}^{n}z_im_i}{M} $$\\n  \"),_c('p',[_vm._v(\"where \\\\(M\\\\) is the total mass of the body.\")]),_c('h3',[_vm._v(\"Center of Mass of a Continuous Body\")]),_c('p',[_vm._v(\"\\n    In case of extended objects where distribution of mass is continuous, above formula cannot work since\\n    number of particles are very large (tending to infinity) in continuous bodies . Therefore, the summation of\\n    particles in above formula is replaced by integration of small particles of mass dm. The co-ordinates of\\n    center of mass\\n    is now given by\\n  \")]),_vm._v(\"\\n  $$ x_{com}=\\\\frac{\\\\int x\\\\,dm}{M} \\\\;\\\\;\\\\;\\\\; y_{com} =\\\\frac{\\\\int y\\\\,dm}{M} \\\\;\\\\;\\\\;\\\\; z_{com} =\\\\frac{\\\\int z\\\\,dm}{M} $$\\n\\n  \"),_c('p',[_vm._v(\"where \\\\(M\\\\) is the total mass of the body.\")]),_c('p',[_vm._v(\"Center of mass of some common regular objects are given below.\")]),_vm._m(0),_c('h3',[_vm._v(\"Motion of center of mass\")]),_c('p',[_vm._v(\"\\n    Let us now analyze the motion of center of mass of a system where mass of the individual particles are \\\\(m_1, m_2...,m_n\\\\). We\\n    may rewrite Eq. (i) as\\n\\n    $$M\\\\vec{r}_{com}=m_1\\\\vec{r_1}+m_2\\\\vec{r_2}+...+m_n\\\\vec{r_n}$$\\n    On differentiating the above equation with respect to time we get,\\n\\n    $$M\\\\frac{d\\\\vec{r}_{com}}{dt}=m_1\\\\frac{d\\\\vec{r_1}}{dt}+ m_2\\\\frac{d \\\\vec{r_2}}{dt}+...+m_n\\\\frac{d \\\\vec{r_n}}{dt}$$\\n    $$ \\\\text{or}\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\n    M \\\\vec{v}_{com}=m_1\\\\vec{v_1}+m_2\\\\vec{v_2}+...+m_n\\\\vec{v_n} \\\\;\\\\;\\\\;\\\\;\\\\;\\\\; -(ii)$$\\n    $$\\\\\\\\\\\\text{or} \\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\;\\\\; \\\\vec{v}_{com}=\\\\frac{\\\\sum_{i=1}^{n}m_iv_i}{M} \\\\;\\\\;\\\\;\\\\;\\\\;\\\\; -(iii)$$\\n  \")]),_c('p',[_vm._v(\"\\n    Thus, Eq. (iii) gives us the velocity of the center of mass of the system where \\\\(\\\\vec{v_1},\\\\vec{v_2},...,\\\\vec{v_r}\\\\) are the\\n    velocities of the\\n    individual particles. Here we have assumed that the total mass of the system remains constant i.e., no mass\\n    enters or leaves the system.\\n  \")]),_c('p',[_vm._v(\"\\n    We know that the product of mass and velocity is equal to the momentum of a particle. Therefore, on rewriting Eq. (ii) we can get the\\n    momentum \\\\(\\\\vec{p}_{com}\\\\) of center of mass of a system of particles. It is given by\\n    $$\\\\vec{p}_{com}=\\\\vec{p_1}+\\\\vec{p_2}+...+\\\\vec{p_n}$$\\n    where \\\\(\\\\vec{p_1},\\\\vec{p_2},...,\\\\vec{p_n}\\\\)\\n    are the\\n    momentum of individual particles of the system.\\n  \")]),_c('p',[_vm._v(\"\\n    Now on differentiating Eq. (ii) with respect to time we get,\\n    $$ M \\\\vec{a}_{com}=m_1\\\\vec{a_1}+m_2\\\\vec{a_2}+...+m_n\\\\vec{a_n} \\\\;\\\\;\\\\;\\\\;\\\\;\\\\; -(iv)$$\\n\\n    The above equation gives the acceleration \\\\(\\\\vec{a}_{com}\\\\) of the center of mass of a system of \\\\(n\\\\) particles having\\n    acceleration\\n    \\\\(\\\\vec{a_1},\\\\vec{a_2},...,\\\\vec{a_n}\\\\).\\n    We know that force applied on a body is equal to the product of its mass and acceleration i.e, \\\\(F=ma\\\\).\\n    Therefore, Eq. (iv) can be rewritten as\\n    $$M\\\\vec{a}_{com}= \\\\vec{F_1}+\\\\vec{F_2}+...+\\\\vec{F_n}$$\\n    Here \\\\(\\\\vec{F}_1\\\\) is the force acting on the first particle, \\\\(\\\\vec{F}_2\\\\) is the force acting on the second particle and\\n    so on.\\n    Since the internal forces are always equal and opposite\\n    in direction and their sum is zero, only external forces contribute in Eq. (iv). Thus, for a system of particles\\n    with total mass \\\\(M\\\\) and \\\\(\\\\vec{F}_{ext}\\\\) as the\\n    vector sum of all the external forces acting on the system, acceleration \\\\(\\\\vec{a}_{com}\\\\) of the center of mass\\n    is given by\\n    $$ \\\\vec{F}_{ext} = M\\\\vec{a}_{com} $$\\n  \")]),_c('h3',[_vm._v(\"MagicGraph | Center of Mass of a System of Particles \")]),_c('h5',[_vm._v(\" To Get Started\")]),_c('p',[_vm._v(\"\\n    Your goal is to balance the structure on the fulcrum.\\n    This will happen if the line of gravitational force acting on the structure passes through the fulcrum.\\n    The gravity acts downward and at the center of mass of the structure.\\n    So, the above task can be accomplished by aligning the center of mass and the fulcrum along a vertical line.\\n    You can shift the center of mass of the system by tapping on the blank squares. Each colored square has a mass of 1 kilogram.\\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('table',{attrs:{\"border\":\"4\",\"bordercolor\":\"black\",\"align\":\"center\"}},[_c('tr',[_c('th',[_vm._v(\"Object\")]),_c('th',[_vm._v(\"Position of center of mass\")])]),_c('tr',[_c('td',[_vm._v(\"Uniform Rod\")]),_c('td',[_vm._v(\"Mid-point of the rod\")])]),_c('tr',[_c('td',[_vm._v(\"Circular plate, circular ring and a solid sphere \")]),_c('td',[_vm._v(\"At their center\")])]),_c('tr',[_c('td',[_vm._v(\" Semi-circular ring \")]),_c('td',[_vm._v(\"At a height equal to \\\\(\\\\frac{2R}{\\\\pi}\\\\) from its center on the axis of symmetry\")])]),_c('tr',[_c('td',[_vm._v(\"Semi-circular disc \")]),_c('td',[_vm._v(\"\\n        At a height equal to \\\\(\\\\frac{4R}{3\\\\pi}\\\\) from its center on the axis of symmetry\\n      \")])]),_c('tr',[_c('td',[_vm._v(\" Hemispherical shell \")]),_c('td',[_vm._v(\"\\n        At a height equal to \\\\(\\\\frac{R}{2}\\\\) from its center on the axis of symmetry\\n      \")])]),_c('tr',[_c('td',[_vm._v(\"Solid hemisphere \")]),_c('td',[_vm._v(\"At a height equal to \\\\( \\\\frac{3R}{8}\\\\) from its center on the axis of symmetry\")])]),_c('tr',[_c('td',[_vm._v(\"Solid cone \")]),_c('td',[_vm._v(\"At a height \\\\(\\\\frac{h}{4} \\\\) from its base on the axis\")])]),_c('tr',[_c('td',[_vm._v(\"Hollow cone \")]),_c('td',[_vm._v(\"At a height \\\\(\\\\frac{h}{3} \\\\) from its base on the axis\")])]),_c('tr',[_c('td',[_vm._v(\"Triangular Lamina\")]),_c('td',[_vm._v(\"At the point of intersection of medians of the triangular lamina\")])])])\n}]\n\nexport { render, staticRenderFns }","import {\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    placeCircle,\r\n    placeAngle,\r\n    placeDash,\r\n    placeLabel,\r\n    placeArc,\r\n    placeLogo\r\n} from '../../../common/edliy_utils-circle';\r\nconst Boxes = {\r\n    //JXG.Options.line.highlight=false;\r\n    box1: function () {\r\n    //Create the Board\r\n    var brd1 = JXG.JSXGraph.initBoard('jxgbox1',{boundingbox: [-2, 24, 24, -1],\r\n        keepaspectratio: true, axis:true, ticks:{visible:true},\r\n        grid:true, showCopyright:false, showNavigation:false,\r\n        pan:{enabled:false}, zoom:{enabled:false}});\r\n\r\n    brd1.suspendUpdate();\r\n    // Layering\r\n    brd1.options.axis.strokeWidth=0;\r\n    brd1.options.layer['image']=15;\r\n    brd1.options.layer['text']=16;\r\n    brd1.options.layer['line']=16;\r\n    brd1.options.layer['point']=17;\r\n    brd1.options.layer['glider']=17;\r\n    brd1.options.layer['angle']=18;\r\n    brd1.options.board.minimizeReflow = 'none';\r\n    brd1.options.point.showInfobox =false;\r\n    brd1.options.line.highlight=false;\r\n    brd1.options.image.highlight=false;\r\n    brd1.options.text.highlight=false;\r\n//    brd1.options.text.display='internal';\r\n    //Make the board responsive\r\n    makeResponsive(brd1);\r\n    // Creat titleStyle\r\n    placeLogo(brd1);\r\n    placeTitle(brd1, 'Center of Mass', '(Tap to add mass until the system balances on the fulcrum)');\r\n    //dragMe(brd1, 5, 5);\r\n    //var stone=placeLine(brd1, [-2,0], [3,5], 10, 'grey');\r\n    //\r\n    brd1.options.layer['image'] =2;\r\n    brd1.options.layer['point'] =3;\r\n  //  placeText(brd1, 5, 10, 'Location of CM is shown by crosshair');\r\n    //Titles and subtitle\r\n    //var title = brd1.create('text', [12, 22, '<b> Center of Mass <b>'],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(28*brd1.canvasWidth/500.)}},);\r\n    brd1.create('image', ['/assets/crosshair.svg', [7, 11], [1,1]], {visible: true, fixed:true});\r\n    //var rectImg = brd1.create('image', ['/assets/orangeSquare.svg', [20,16], [2,2]], {visible: true, fixed:true});\r\n    //var massKg = brd1.create('text', [21, 17, '1 kg'],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd1.canvasWidth/500.)}},);\r\n    //FIRST\r\n    var width=2;\r\n    var num_height=5;\r\n    var num_width=11;\r\n    var rects=[];\r\n    var rects2=[];\r\n    var rectsVisibility=[];\r\n    var rectsVisibility2=[];\r\n    var cmx=0,cmy=0;\r\n    for (let x=0; x<num_width; x++)\r\n    {\r\n        let rectx=[];\r\n        let rectx2=[];\r\n        let vx=[];\r\n        let vx2=[];\r\n\r\n        for (let y=0; y< num_height; y++)\r\n        {\r\n            let v = true;\r\n            let v2 = false;\r\n            var rect = brd1.create('image', ['/assets/whiteSquare.svg', [x*width, y*width], [2.01,2.01]], {visible: v, opacity:0.5, fixed:true});\r\n            var rect2 = brd1.create('image', ['/assets/purpleSquare.svg', [x*width, y*width], [2.01,2.01]], {visible: v2, fixed:true});\r\n            rect.on('up', function(){changeRectColorxy(x,y); });\r\n            rect2.on('up', function(){changeRectColorxy(x,y); });\r\n            rectx.push(rect);\r\n            rectx2.push(rect2);\r\n            vx.push(v);\r\n            vx2.push(v2);\r\n        }\r\n        rects.push(rectx);\r\n        rects2.push(rectx2);\r\n        rectsVisibility.push(vx);\r\n        rectsVisibility2.push(vx2);\r\n\r\n    }\r\n    var j=0;\r\n    var array =[7.5, 5.5, 6.5, 4.5];\r\n    var darray=[8, 6, 7, 5];\r\n    var shu =brd1.create('image', ['/assets/shuffle.svg', [16, 16], [2.0, 2.0]], {visible:true, fixed:true, opacity:1});\r\n    hoverMe(brd1, shu, 'Tap to Shuffle', 5, -15);\r\n    shu.on('down', function(){if(j<3){return j=j+1}else{return j=0}});\r\n    brd1.create('image', ['/assets/triangle.svg', [()=>array[j], -1], [1.0, 1.0]], {visible:true, fixed:true, opacity:1});\r\n    var perm =[];\r\n    perm[0] =brd1.create('image', ['/assets/orangeSquare.svg', [0*width, 0*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n    perm[1] =brd1.create('image', ['/assets/orangeSquare.svg', [1*width, 0*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n    perm[2] =brd1.create('image', ['/assets/orangeSquare.svg', [2*width, 0*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n    perm[3] =brd1.create('image', ['/assets/orangeSquare.svg', [3*width, 0*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n    perm[4] =brd1.create('image', ['/assets/orangeSquare.svg', [3*width, 1*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n    perm[5] =brd1.create('image', ['/assets/orangeSquare.svg', [3*width, 2*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n    perm[6] =brd1.create('image', ['/assets/orangeSquare.svg', [3*width, 3*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n    perm[7] =brd1.create('image', ['/assets/orangeSquare.svg', [4*width, 3*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n    perm[8] =brd1.create('image', ['/assets/orangeSquare.svg', [5*width, 3*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n    perm[9] =brd1.create('image', ['/assets/orangeSquare.svg', [6*width, 3*width], [2.01,2.01]], {visible:true, fixed:true, opacity:1});\r\n////////////////////////////////////////////////////////////////\r\nbrd1.create('image', ['/assets/triangle.svg', [7, 12], [1.0, 1.0]], {visible:true, fixed:true, opacity:1});\r\nbrd1.create('text', [8.5, 12.5, function(){return '= Location of Fulcrum = (' + (darray[j]).toFixed(2) + ', 0.00'+')'}],{visible: true, fixed: true, anchorX: 'left', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(18*brd1.canvasWidth/800.)}},);\r\n\r\n    var masss =0;\r\n    var cmx0 =0;\r\n    var cmy0 =0;\r\n    for (let i=0;i<=9;i++)\r\n         {\r\n                cmx0 += perm[i].X()+1;\r\n                cmy0 += perm[i].Y()+1;\r\n                masss+=1;\r\n         }\r\n    if (masss>0)\r\n    {\r\n        cmx0 = cmx0/masss;\r\n        cmy0 = cmy0/masss;\r\n    }\r\n    var mass2=0;\r\n////////////////////////////////////////////////////////////////////////////////////\r\n    function changeRectColorxy(x,y){\r\n\r\n        var current_visibility = rectsVisibility[x][y];\r\n        var current_visibility2 = rectsVisibility2[x][y];\r\n\r\n        if (current_visibility==true && current_visibility2==false)\r\n        {\r\n            current_visibility=false;\r\n            current_visibility2=true;\r\n\r\n        }\r\n        else{\r\n            current_visibility=true;\r\n            current_visibility2=false;\r\n        }\r\n\r\n        rects[x][y].setAttribute({visible: current_visibility});\r\n        rects2[x][y].setAttribute({visible: current_visibility2});\r\n        rectsVisibility[x][y]=current_visibility;\r\n        rectsVisibility2[x][y]=current_visibility2;\r\n\r\n        var massrect=1;\r\n        var mass=0;\r\n        x=0;\r\n        y=0;\r\n        cmx=0;\r\n        cmy=0;\r\n        for (let x=0;x<num_width;x++)\r\n        {\r\n             for (let y=0;y< num_height;y++)\r\n             {\r\n                 if (rectsVisibility[x][y]!=true)\r\n                 {\r\n                    cmx += massrect*(x*width + (x+1)*width)/2;\r\n                    cmy += massrect*(y*width + (y+1)*width)/2;\r\n                    mass+=1;\r\n                 }\r\n             }\r\n        }\r\n        if (mass>0)\r\n        {\r\n            cmx = cmx/mass;\r\n            cmy=  cmy/mass;\r\n        }\r\n        mass2 = mass;\r\n\r\n        console.log('cmx=',cmx,'cmy=',cmy, mass2, masss);\r\n    }\r\n    var cgxtxt = brd1.create('text', [1.5, 18, 'X_c_m = '],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd1.canvasWidth/500.)}},);\r\n    var cgxtxt2 = brd1.create('text', [4, 18.5, ' &sum; m_i x_i '],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'bottom', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd1.canvasWidth/500.)}},);\r\n    brd1.create('segment',[[3., 18],[5., 18]],{strokeWidth:2, strokeColor:'black', fixed:true});\r\n    var cgxtxt3 = brd1.create('text', [4, 17.5, '&sum;m_i'],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd1.canvasWidth/500.)}},);\r\n\r\n    var cgytxt = brd1.create('text', [1.5, 15, 'Y_c_m = '],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd1.canvasWidth/500.)}},);\r\n    var cgytxt2 = brd1.create('text', [4, 15.5, ' &sum; m_i y_i'],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'bottom', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd1.canvasWidth/500.)}},);\r\n    brd1.create('segment',[[3., 15],[5., 15]],{strokeWidth:2, strokeColor:'black', fixed:true});\r\n    var cgytxt3 = brd1.create('text', [4, 14.5, '&sum;m_i'],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd1.canvasWidth/500.)}},);\r\n\r\n    var ok1 = brd1.create('image', ['/assets/check.svg', [10, 17.5], [1 , 1]], {visible: false, fixed: true});\r\n    var ok2 = brd1.create('image', ['/assets/check.svg', [10, 14.5], [1 , 1]], {visible: false, fixed: true});\r\n    var wrong1 = brd1.create('image', ['/assets/cross.svg', [10, 17.5], [1 , 1]], {visible: false, fixed: true});\r\n    var wrong2 = brd1.create('image', ['/assets/cross.svg', [10, 14.5], [1 , 1]], {visible: false, fixed: true});\r\n\r\n    var pointCM = brd1.create('image', ['/assets/crosshair.svg', [function(){return (masss*cmx0 + mass2*cmx)/(mass2+masss)- 0.5}  ,function(){return (masss*cmy0 + mass2*cmy)/(mass2+masss)- 0.5}], [1,1]], {visible: true, fixed:true});\r\n    var cgx = brd1.create('text', [7, 18, function(){return ' = ' + ((masss*cmx0 + mass2*cmx)/(mass2+masss)).toFixed(2)}],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd1.canvasWidth/500.)}},);\r\n    var cgy = brd1.create('text', [7, 15, function(){return ' = ' + ((masss*cmy0 + mass2*cmy)/(mass2+masss)).toFixed(2)}],{visible: true, fixed: true, anchorX: 'middle', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(16*brd1.canvasWidth/500.)}},);\r\n    brd1.create('text', [8.5, 11.5, function(){return '= Center of Mass =' + '('+ ((masss*cmx0 + mass2*cmx)/(mass2+masss)).toFixed(2) +','+ ((masss*cmy0 + mass2*cmy)/(mass2+masss)).toFixed(2) +')' }],{visible: true, fixed: true, anchorX: 'left', anchorY: 'middle', CssStyle:'fontFamily:Oswald',fontSize:function(){return Math.round(18*brd1.canvasWidth/800.)}},);\r\n\r\n    brd1.create('image', ['/assets/check.svg', [12, 15.75], [2 , 2]], {visible: function(){return darray[j]==((masss*cmx0 + mass2*cmx)/(mass2+masss)).toFixed(2)}, fixed: true});\r\n    brd1.unsuspendUpdate();\r\n                      },\r\n              }\r\nexport default Boxes;\r\n","<template>\r\n  <div>\r\n    <h3>What is center of mass?</h3>\r\n    <p>\n      Center of mass of a body or a system of particles is a unique point where the entire mass of the system can\r\n      be assumed to be concentrated. Its motion is same as if whole mass of the system was concentrated at this\r\n      point and all the external forces were applied at this point. The concept of center\r\n      of mass helps in solving many complex problems quite simply. Suppose if several forces are acting on a body\r\n      of any odd shape, we need not bother about the forces acting on the individual particles, but we can apply the\r\n      net force on the center of mass and thus analyze the body's motion.\n    </p>\r\n    <h3>Center of Mass of System of Particles</h3>\r\n    <p>\n      The position of center of mass of a system of particles depends upon the distribution of mass in it. It is\r\n      not necessary that the center of mass lies within the material of the body.\n    </p>\r\n    <p>\n      For a system of \\(n\\) particles with masses \\(m_1,m_2, ..., m_n\\) and position vectors as \\(\\vec{r_1},\\vec{r_2}, ...,\\vec{r_n}\\), the\r\n      position\r\n      vector \\(\\vec{r}_{com}\\) of the center of mass of the system is given by\n    </p>\r\n    $$\\vec{r}_{com} =\\frac{m_1\\vec{r_1}+m_2\\vec{r_2}+...+m_n\\vec{r_n}}{m_1+m_2+...+m_n}\r\n    =\\frac{\\sum_{i=1}^{n} \\vec{r_i}m_i}{\\sum_{i=1}^{n} m_i} \\;\\;\\;\\;\\;\\; -(i)\r\n\r\n    $$\r\n    <p>Therefore the \\(x\\) , \\(y\\) and \\(z\\) co-ordinates of center of mass is given by </p>\r\n    $$ x_{com} =\\frac{\\sum_{i=1}^{n}x_im_i}{M} \\;\\;\\;\\; y_{com} =\\frac{\\sum_{i=1}^{n}y_im_i}{M} \\;\\;\\;\\; z_{com}\r\n    =\\frac{\\sum_{i=1}^{n}z_im_i}{M} $$\r\n    <p>where \\(M\\) is the total mass of the body.</p>\r\n    <h3>Center of Mass of a Continuous Body</h3>\r\n    <p>\n      In case of extended objects where distribution of mass is continuous, above formula cannot work since\r\n      number of particles are very large (tending to infinity) in continuous bodies . Therefore, the summation of\r\n      particles in above formula is replaced by integration of small particles of mass dm. The co-ordinates of\r\n      center of mass\r\n      is now given by\n    </p>\r\n    $$ x_{com}=\\frac{\\int x\\,dm}{M} \\;\\;\\;\\; y_{com} =\\frac{\\int y\\,dm}{M} \\;\\;\\;\\; z_{com} =\\frac{\\int z\\,dm}{M} $$\r\n\r\n    <p>where \\(M\\) is the total mass of the body.</p>\r\n    <p>Center of mass of some common regular objects are given below.</p>\r\n    <table border=\"4\" bordercolor=\"black\" align=\"center\">\r\n      <tr>\r\n        <th>Object</th>\r\n        <th>Position of center of mass</th>\r\n      </tr>\r\n      <tr>\r\n        <td>Uniform Rod</td>\r\n        <td>Mid-point of the rod</td>\r\n      </tr>\r\n      <tr>\r\n        <td>Circular plate, circular ring and a solid sphere </td>\r\n        <td>At their center</td>\r\n      </tr>\r\n      <tr>\r\n        <td> Semi-circular ring </td>\r\n        <td>At a height equal to \\(\\frac{2R}{\\pi}\\) from its center on the axis of symmetry</td>\r\n      </tr>\r\n      <tr>\r\n        <td>Semi-circular disc </td>\r\n        <td>\n          At a height equal to \\(\\frac{4R}{3\\pi}\\) from its center on the axis of symmetry\n        </td>\r\n      </tr>\r\n      <tr>\r\n        <td> Hemispherical shell </td>\r\n        <td>\n          At a height equal to \\(\\frac{R}{2}\\) from its center on the axis of symmetry\r\n        </td>\n      </tr>\r\n      <tr>\r\n        <td>Solid hemisphere </td>\r\n        <td>At a height equal to \\( \\frac{3R}{8}\\) from its center on the axis of symmetry</td>\r\n      </tr>\r\n      <tr>\r\n        <td>Solid cone </td>\r\n        <td>At a height \\(\\frac{h}{4} \\) from its base on the axis</td>\r\n      </tr>\r\n      <tr>\r\n        <td>Hollow cone </td>\r\n        <td>At a height \\(\\frac{h}{3} \\) from its base on the axis</td>\r\n      </tr>\r\n      <tr>\r\n        <td>Triangular Lamina</td>\r\n        <td>At the point of intersection of medians of the triangular lamina</td>\r\n      </tr>\r\n    </table>\r\n    <h3>Motion of center of mass</h3>\r\n    <p>\n      Let us now analyze the motion of center of mass of a system where mass of the individual particles are \\(m_1, m_2...,m_n\\). We\r\n      may rewrite Eq. (i) as\r\n\r\n      $$M\\vec{r}_{com}=m_1\\vec{r_1}+m_2\\vec{r_2}+...+m_n\\vec{r_n}$$\r\n      On differentiating the above equation with respect to time we get,\r\n\r\n      $$M\\frac{d\\vec{r}_{com}}{dt}=m_1\\frac{d\\vec{r_1}}{dt}+ m_2\\frac{d \\vec{r_2}}{dt}+...+m_n\\frac{d \\vec{r_n}}{dt}$$\r\n      $$ \\text{or}\\;\\;\\;\\;\\;\\;\\;\\;\\;\r\n      M \\vec{v}_{com}=m_1\\vec{v_1}+m_2\\vec{v_2}+...+m_n\\vec{v_n} \\;\\;\\;\\;\\;\\; -(ii)$$\r\n      $$\\\\\\text{or} \\;\\;\\;\\;\\;\\;\\;\\;\\; \\vec{v}_{com}=\\frac{\\sum_{i=1}^{n}m_iv_i}{M} \\;\\;\\;\\;\\;\\; -(iii)$$\r\n    </p>\r\n    <p>\n      Thus, Eq. (iii) gives us the velocity of the center of mass of the system where \\(\\vec{v_1},\\vec{v_2},...,\\vec{v_r}\\) are the\r\n      velocities of the\r\n      individual particles. Here we have assumed that the total mass of the system remains constant i.e., no mass\r\n      enters or leaves the system.\n    </p>\r\n\r\n    <p>\n      We know that the product of mass and velocity is equal to the momentum of a particle. Therefore, on rewriting Eq. (ii) we can get the\r\n      momentum \\(\\vec{p}_{com}\\) of center of mass of a system of particles. It is given by\r\n      $$\\vec{p}_{com}=\\vec{p_1}+\\vec{p_2}+...+\\vec{p_n}$$\r\n      where \\(\\vec{p_1},\\vec{p_2},...,\\vec{p_n}\\)\r\n      are the\r\n      momentum of individual particles of the system.\n    </p>\r\n\r\n    <p>\n      Now on differentiating Eq. (ii) with respect to time we get,\r\n      $$ M \\vec{a}_{com}=m_1\\vec{a_1}+m_2\\vec{a_2}+...+m_n\\vec{a_n} \\;\\;\\;\\;\\;\\; -(iv)$$\r\n\r\n      The above equation gives the acceleration \\(\\vec{a}_{com}\\) of the center of mass of a system of \\(n\\) particles having\r\n      acceleration\r\n      \\(\\vec{a_1},\\vec{a_2},...,\\vec{a_n}\\).\r\n      We know that force applied on a body is equal to the product of its mass and acceleration i.e, \\(F=ma\\).\r\n      Therefore, Eq. (iv) can be rewritten as\r\n      $$M\\vec{a}_{com}= \\vec{F_1}+\\vec{F_2}+...+\\vec{F_n}$$\r\n      Here \\(\\vec{F}_1\\) is the force acting on the first particle, \\(\\vec{F}_2\\) is the force acting on the second particle and\r\n      so on.\r\n      Since the internal forces are always equal and opposite\r\n      in direction and their sum is zero, only external forces contribute in Eq. (iv). Thus, for a system of particles\r\n      with total mass \\(M\\) and \\(\\vec{F}_{ext}\\) as the\r\n      vector sum of all the external forces acting on the system, acceleration \\(\\vec{a}_{com}\\) of the center of mass\r\n      is given by\r\n      $$ \\vec{F}_{ext} = M\\vec{a}_{com} $$\r\n    </p>\r\n    <h3>MagicGraph | Center of Mass of a System of Particles </h3>\r\n    <h5> To Get Started</h5>\r\n    <p>\n      Your goal is to balance the structure on the fulcrum.\r\n      This will happen if the line of gravitational force acting on the structure passes through the fulcrum.\r\n      The gravity acts downward and at the center of mass of the structure.\r\n      So, the above task can be accomplished by aligning the center of mass and the fulcrum along a vertical line.\r\n      You can shift the center of mass of the system by tapping on the blank squares. Each colored square has a mass of 1 kilogram.\r\n    </p>\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: 'CM',\r\n  created: function () {\r\n    this.$store.commit('navigation/resetState');\r\n    let title = 'Center of Mass';\r\n    this.$store.commit('navigation/changeTitle', title);\r\n    this.$store.commit('navigation/changeMenu', title);\r\n    let newTopics = [\r\n      {title: 'What is Center of Mass?', img:'/assets/number-1.svg', action: () => this.goto('ia')},\r\n      {title: 'System of Discrete Particles', img:'/assets/number-2.svg', action: () => this.goto('ja')},\r\n      {title: 'Continuous Bodies', img:'/assets/number-3.svg', action: () => this.goto('ka')},\r\n      {title: 'MagicGraph' ,img:'/assets/touch.svg', action: () => this.goto('graphical')},\r\n    ];\r\n    this.$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          title: 'Center of Mass',\r\n          //titleTemplate: '%s - Learn interactively',\r\n          meta: [\r\n                  {vmid:'description', name: 'description', content: 'Learn interactively how to find center of mass of an object'}\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=158c0dc9&\"\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=158c0dc9&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":""}