Some questions demonstrating the JSXGraph extension, for my talk at the JSXGraph conference 2021.
", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": "0", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], []], "questions": [{"name": "Move point to given coordinates", "extensions": ["jsxgraph", "programming"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "http://localhost:8000/accounts/profile/1/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "tags": [], "metadata": {"description": "The student is shown a diagram containing a single point at the origin. They must move the point to the given integer coordinates.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "The point should be here:
\n{correct_diagram}
\n$\\var[rowvector]{coords}$ means that the horizontal position should be $\\var{x}$, and the vertical position should be $\\var{y}$.
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\n{diagram}
"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Create a line with given gradient", "extensions": ["jsxgraph", "programming"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "http://localhost:8000/accounts/profile/1/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "tags": [], "metadata": {"description": "The student is shown a diagram containing a line between two points. They're given a gradient and $y$-intercept.
\nThey must manipulate the line or the points so that the line has the given gradient and $y$-intercept.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "The line should look like this:
\n{correct_diagram}
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"gradient": {"name": "gradient", "group": "Ungrouped variables", "definition": "dy/dx", "description": "The gradient that the student's line should have. Designed to be a nice fraction.
", "templateType": "anything", "can_override": false}, "diagram": {"name": "diagram", "group": "Ungrouped variables", "definition": "jessiecode(400,400,[-5,10,10,-5],\"\"\"\n a = point(1,0) <The diagram shown to the student.
", "templateType": "anything", "can_override": false}, "dx": {"name": "dx", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "A randomly chosen change in x to get a nice gradient.
", "templateType": "anything", "can_override": false}, "dy": {"name": "dy", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "A randomly chosen change in y to get a nice gradient.
", "templateType": "anything", "can_override": false}, "y_intercept": {"name": "y_intercept", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "The point at which the student's line should cross the y axis.
", "templateType": "anything", "can_override": false}, "x_intercept": {"name": "x_intercept", "group": "Ungrouped variables", "definition": "let([g,y], gradient_intercept_pair,\n -y/g\n)", "description": "The position at which the line crosses the x axis
", "templateType": "anything", "can_override": false}, "gradient_intercept_pair": {"name": "gradient_intercept_pair", "group": "Ungrouped variables", "definition": "[gradient,y_intercept]", "description": "The gradient and y-intercept of the line as a pair, to be replaced by the student's answer for adaptive marking in part b.
", "templateType": "anything", "can_override": false}, "correct_diagram": {"name": "correct_diagram", "group": "Ungrouped variables", "definition": "jessiecode(400,400,[-5,10,10,-5],\"\"\"\n a = point(0,{y_intercept}) <Construct a line with gradient {gradient} and $y$-intercept {y_intercept}.
\nYou can move the line by dragging it, or move the two points individually.
\n{diagram}
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\n$x = $ [[0]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "gradient_intercept_pair", "part": "p0", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": "0.5", "exploreObjective": null, "minValue": "x_intercept", "maxValue": "x_intercept", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Free oscillations of a pendulum", "extensions": ["jsxgraph", "programming"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "http://localhost:8000/accounts/profile/1/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas-editor.mas.ncl.ac.uk/accounts/profile/3/"}, {"name": "Christopher Graham", "profile_url": "https://numbas-editor.mas.ncl.ac.uk/accounts/profile/73/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "tags": [], "metadata": {"description": "Solving 2nd order differential equation for pendulum, with and without damping.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "In the absence of air resistance, the equation of motion of a bob, of mass $m$, at the end of a pendulum of length $l$ is
\n\\[ ml\\frac{\\mathrm{d}^2\\theta}{\\mathrm{d}t^2} = -mg\\sin\\theta\\text{.} \\]
\nIn the small angle approximation, $\\sin\\theta\\approx\\theta$, this reduces to
\n\\[ \\frac{\\mathrm{d}^2\\theta}{\\mathrm{d}t^2}+\\frac{g}{l}\\theta = 0\\text{.} \\]
", "advice": "We have a second order constant coefficient differential equation,
\n\\[ \\frac{\\mathrm{d}^2\\theta}{\\mathrm{d}t^2}+\\frac{g}{l}\\theta = 0 \\text{.} \\]
\nSetting $\\theta = e^{\\lambda t}$, we obtain the auxiliary equation
\n\\[ \\lambda^2 + \\frac{g}{l} = 0 \\implies \\lambda = \\pm i\\sqrt{\\frac{g}{l}} \\text{.} \\]
\nThe general solution for an auxiliary equation with complex roots is
\n\\[ \\theta(t) = A\\sin\\omega t + B\\cos\\omega t \\text{,} \\]
\nwhere $\\omega = \\sqrt{\\frac{g}{l}}$ is the angular frequency of the oscillations, also known as the natural frequency: the frequency with which the pendulum oscillates in the absence of external forcing).
\nNote that
\n\\[ \\frac{\\mathrm{d}\\theta}{\\mathrm{d}t} = A\\omega\\cos\\omega t - B\\omega\\sin\\omega t \\text{.} \\]
\nApplying the initial conditions:
\n\\begin{align}
\\frac{\\mathrm{d}\\theta}{\\mathrm{d}t} (0) = 0 &=A\\omega\\cos(0) - B\\omega\\sin(0) \\text{,} \\\\
&= A\\omega \\text{,} \\\\
\\therefore A &= 0 \\text{.}
\\end{align}
\\begin{align}
\\theta (0) =\\frac{\\pi}{\\var{theta0_frac}} &=B\\cos(0) \\text{,} \\\\
\\therefore B &=\\frac{\\pi}{\\var{theta0_frac}} \\text{.}
\\end{align}
Therefore our solution is
\n\\[ \\theta(t) = \\frac{\\pi}{\\var{theta0_frac}}\\cos\\left(\\sqrt{\\frac{g}{l}}t\\right)\\text{.} \\]
\nUsing the given values for $g$ and $l$,
\n\\[ \\theta(t) = \\frac{\\pi}{\\var{theta0_frac}}\\cos\\left(\\sqrt{\\frac{\\var{g}}{\\var{l}}}t\\right)\\text{.} \\]
\n\nThe time period of oscillations is given by
\n\\[ T = \\frac{2\\pi}{\\omega} = 2\\pi\\sqrt{\\frac{l}{g}} =2\\pi\\sqrt{\\frac{\\var{l}}{\\var{g}}} = \\var{precround(T,2)} \\text{ s.} \\]
\nDifferentiating our solution from part (a), we have
\n\\[ \\frac{d\\theta}{dt} = -\\frac{\\pi}{\\var{theta0_frac}}\\omega \\sin\\omega t\\text{.} \\]
\nThis is at its maximum when $\\sin \\omega t = -1$,
\n\\[ \\left(\\frac{\\mathrm{d}\\theta}{\\mathrm{d}t}\\right)_{\\text{max}} = \\frac{\\pi}{\\var{theta0_frac}}\\omega =\\frac{\\pi}{\\var{theta0_frac}}\\sqrt{\\frac{\\var{g}}{\\var{l}}} = \\var{precround(theta0*sqrt(g/l),2)}\\text{ rad s}^{-1 }\\text{.} \\]
\nWith the addition of a damping term, our differential equation becomes
\n\\[ \\frac{\\mathrm{d}^2\\theta}{\\mathrm{d}t^2}+\\frac{k}{m}\\frac{\\mathrm{d}\\theta}{\\mathrm{d}t}+\\frac{g}{l}\\theta = 0\\text{.} \\]
\nThis is still a constant-coefficient second-order differential equation, however the extra term changes the auxiliary equation. For convenience we can let $\\beta = \\frac{k}{2m}$ (we will later find that this is $\\beta$ as asked for in the question) and $\\omega^2 = \\frac{g}{l}$, as before, to put our auxiliary equation in the form
\n\\[ \\lambda^2 + 2\\beta \\lambda + \\omega^2 = 0\\text{,} \\]
\nwhich has the solutions
\n\\[ \\begin{align} \\lambda &=-\\frac{2\\beta}{2}\\pm \\frac{\\sqrt{2\\beta^2-4\\omega^2}}{2} \\\\
&= -\\beta\\pm \\sqrt{\\beta^2-\\omega^2} \\\\
&= -\\beta\\pm i\\sqrt{\\omega^2-\\beta^2}\\text{.} \\end{align}\\]
Note that the values in our problem have determined that $\\omega >\\beta$. As in part (a), we have an auxiliary equation with complex roots:
\n\\[\\theta(t) = e^{-\\beta t}\\left[A\\sin\\left(\\sqrt{\\omega^2-\\beta^2}\\; t\\right) + B\\cos\\left(\\sqrt{\\omega^2-\\beta^2}\\; t\\right)\\right]\\text{.} \\]
\nUsing the values and initial conditions from part (a), this becomes
\n\\[\\theta(t) = \\frac{\\pi}{\\var{theta0_frac}}e^{-\\beta t}\\cos\\left(\\sqrt{\\frac{\\var{g}}{\\var{l}}-\\beta^2}\\; t\\right)\\text{,} \\]
\nwith
\n\\[ \\beta = \\frac{k}{2m} = \\frac{\\var{k}}{2\\times\\var{m}} = \\var{precround(beta,2)}\\text{.}\\]
\nThe amplitude of the oscillation is at 5% of the initial release angle when
\n\\[ e^{-\\beta t} = 0.05\\text{.} \\]
\nThe time taken is therefore
\n\\[ t = -\\frac{\\ln{(0.05)}}{\\beta}=-\\frac{\\ln{(0.05)}}{\\var{precround(beta,2)}} = \\var{precround(damping5,2)}\\text{ s.} \\]
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", "templateType": "number", "can_override": false}, "T": {"name": "T", "group": "Ungrouped variables", "definition": "2*pi*sqrt(l/g)", "description": "Time period of the undamped oscillations (seconds)
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\n\n // Path of the correct solution\n pp2.moveAlong(function() {\n return function(t) {\n if (t >= duration) {\n // Restart the animation\n startAnimation(start_pos);\n }\n t = t/1000;\n var angle1 = start_pos*Math.cos(Math.sqrt(g/l) * t);\n var angled1 = -Math.sqrt(g/l)*start_pos*Math.sin(Math.sqrt(g/l) * t);\n var angle2 = -Math.PI/2+angle1;\n tt2._setUpdateText('t = '+t.toFixed(0)+'s');\n tt3._setUpdateText('θ = '+angle1.toFixed(1)+'rad');\n tt4._setUpdateText('dθ/dt = '+angled1.toFixed(1)+'rads-1');\n return [p1.X()+l*Math.cos(angle2),p1.Y()+l*Math.sin(angle2)];\n }; \n }());\n\n // Path of the student's solution\n pp4.moveAlong(function() { \n return function(t) {\n if (t >= duration) {\n // Restart the animation\n startAnimation(start_pos);\n }\n t = t/1000;\n var utheta = userf(t);\n angle1=-Math.PI/2+utheta;\n angled1=(utheta-userf(t-0.001))/0.001;\n tt5._setUpdateText('θ = '+utheta.toFixed(1)+'rad');\n tt6._setUpdateText('dθ/dt = '+angled1.toFixed(1)+'rads-1'); \n return [p3.X()+l*Math.cos(angle1),p3.Y()+l*Math.sin(angle1)];\n }; \n }());\n}\n\n//pick up the student answer\nquestion.signals.on('HTMLAttached',function(e) {\n startAnimation(theta0); \n ko.computed(function(){\n var expr = question.getPart('p0g0').display.studentAnswer();\n try {\n tree = Numbas.jme.compile(expr,scope);\n }\n catch(e) {\n tree = null;\n }\n\n if(tree){\n startAnimation(theta0);\n tt5.setAttribute({visible: true});\n tt6.setAttribute({visible: true});\n tt9.setAttribute({visible: false});\n board.update();\n } else {\n tt9.setAttribute({visible: true});\n tt5.setAttribute({visible: false});\n tt6.setAttribute({visible: false});\n }\n\n });\n}); \n\n\nreturn div;\n\n"}, "jsx_pendulum2": {"parameters": [], "type": "html", "language": "javascript", "definition": "// Get Numbas variables\nvar g = question.unwrappedVariables.g; // grav acc.\nvar l = question.unwrappedVariables.l; // length of rod\nvar m = question.unwrappedVariables.m; // mass of bob\nvar theta0 = question.unwrappedVariables.theta0; // initial angle\nvar k = question.unwrappedVariables.k; // friction coefficient\nvar damping0p01 = question.unwrappedVariables.damping0p01;\n\n// Set up board\nvar div = Numbas.extensions.jsxgraph.makeBoard('600px','250px',\n {\n boundingBox:[-5, 0.2, 9, -6],\n axis:false,\n showNavigation:false,\n grid:false\n }\n);\n\nvar board2 = div.board; \n\nboard2.options = JXG.merge(board2.options,{\n elements: {\n fixed: true\n },\n point: {\n showInfobox: false\n },\n text: {\n highlightStrokeOpacity: 1\n }\n});\n\n// Pendulum 1: exact solution \nvar p3 = board2.create('point', [0.02, 0], {visible:false, size: 3});\nvar c2 = board2.create('circle', [p3, [0.02,5]], {visible:false, name: 'circle', size: 3});\nvar p4 = board2.create('glider', [0.02,5, c2], {visible: true, name: '', size: 8, color: 'red', highlightFillColor: 'red', strokeWidth: 0, highlightStrokeWidth: 0});\n\n\n// Pendulum 2: student solution\nvar p1 = board2.create('point', [0, 0], {visible:false, size: 3});\nvar c1 = board2.create('circle', [p1, [0,5]], {visible:false, name: 'circle', size: 3});\nvar p2 = board2.create('glider', [0,5, c1], {visible: true, name: '', size: 8, color: '#91bcd8', highlightFillColor: '#91bcd8', strokeWidth: 0, highlightStrokeWidth: 0});\n\n// Decor\nvar l1 = board2.create('line', [[-2,0],[2,0]],{color: 'grey', highlightStrokeColor: 'grey',straightFirst:false, straightLast:false, strokeWidth: 10});\nvar l2 = board2.create('line', [[0.02,0],[0.02,-5]],{color: 'grey', highlightStrokeColor: 'grey',straightFirst:false, straightLast:false, strokeWidth: 1, dash:1});\n//t1 = board2.create('text',[4,-2,\"Enter a solution for θ(t) in the box below to preview your solution\"],{color: 'grey', fontSize: 16});\n\n// Pendulum rods\nvar l3 = board2.create('line', [p3, p4], {visible:true, straightFirst: false,straightLast: false, color: 'red', highlightStrokeColor: 'red', strokeWidth: 3});\nvar l4 = board2.create('line', [p1, p2], {visible:true, straightFirst: false,straightLast: false, color: '#91bcd8', highlightStrokeColor: '#91bcd8', strokeWidth: 3});\n\n// Info panel \nvar t2 = board2.create('text',[5,-0.3,''],{color: '#444', highlightStrokeColor: '#444', fontSize: 14});\nvar t3 = board2.create('text',[5.5,-1.5,''],{color: '#444', highlightStrokeColor: '#444', fontSize: 14});\nvar t4 = board2.create('text',[5.5,-2,''],{color: '#444', highlightStrokeColor: '#444', fontSize: 13});\nvar t5 = board2.create('text',[5.5,-3.5,''],{visible: false, color: '#444', highlightStrokeColor: '#444', fontSize: 14});\nvar t6 = board2.create('text',[5.5,-4,''],{visible: false, color: '#444', highlightStrokeColor: '#444', fontSize: 14});\nvar t7 = board2.create('text',[5.5,-1,'Correct solution'],{color: '#444', highlightStrokeColor: '#444', fontSize: 14});\nvar t8 = board2.create('text',[5.5,-3,'Your solution'],{color: '#444', highlightStrokeColor: '#444', fontSize: 14});\nvar t9 = board2.create('text',[5.5,-4,'Enter an answer below to preview your solution.'],{color: '#AAA', highlightStrokeColor: '#AAA', fontSize: 14});\n\n// Shaded background\nvar sq = board2.create('polygon',[[4.5,-5],[9,-5],[9,0.2],[4.5,0.2]], {color: '#DDD', highlightFillColor: '#DDD', vertices: {visible: false},borders: {strokeWidth: 3,lineCap: 'round',visible: false}});\n\n// Pendulum icons\nvar p5 = board2.create('point', [5, -1.5], {color: '#91bcd8', highlightFillColor: '#91bcd8', size: 8, name: '', strokeWidth: 0, highlightStrokeWidth: 0});\nvar p6 = board2.create('point', [5, -3.5], {color: 'red', highlightFillColor: 'red', size: 8, name: '', strokeWidth: 0, highlightStrokeWidth: 0});\nvar l5 = board2.create('line', [p5, [p5.X(),p5.Y()+0.5]], {straightFirst: false,straightLast: false, color: '#91bcd8', highlightStrokeColor: '#91bcd8', strokeWidth: 3});\nvar l6 = board2.create('line', [p6, [p6.X(),p6.Y()+0.5]], {straightFirst: false,straightLast: false, color: 'red', highlightStrokeColor: 'red', strokeWidth: 3});\n\n// Start button\nbutton1 = board2.create('button', [-4.9,-1, '► Restart', function() {\n startAnimation2(theta0);\n board2.update();\n }], {fontSize: 16, fixed: true}),\n\n\nisInDragMode = false;\n\n// the student's beta\nvar student_beta = null;\n\n// The animation\nfunction startAnimation2(start_pos) {\n\n var duration = 1e5; \n\n // Path of the correct solution\n p2.moveAlong(function() {\n return function(t) {\n if (t >= duration) {\n // Restart the animation\n startAnimation2(start_pos);\n }\n t = t/1000;\n var beta = k/(2*m);\n // Calculate theta \n var angle1= start_pos*Math.exp(-beta*t)*Math.cos(Math.sqrt((g/l)-beta) * t);\n var angle2=-Math.PI/2+angle1;\n t2._setUpdateText('t = '+t.toFixed(0)+'s');\n t3._setUpdateText('θ = '+angle1.toFixed(1)+'rad');\n\n if(t>0.01){\n // Fudge dtheta/dt\n var anglelast = start_pos*Math.exp(-beta*(t-0.01))*Math.cos(Math.sqrt((g/l)-beta) * (t-0.01) );\n var anglediff = (angle1-anglelast)/0.01; \n t4._setUpdateText('dθ/dt = '+anglediff.toFixed(1)+'rads-1');\n }\n return [p1.X()+l*Math.cos(angle2),p1.Y()+l*Math.sin(angle2)];\n }; \n }()\n );\n\n // Path of the student's solution\n p4.moveAlong(function() { \n return function(t) {\n if (t >= duration) {\n // Restart the animation\n startAnimation2(start_pos); \n }\n t = t/1000;\n try {\n if(student_beta===null) {\n throw(new Error(\"beta is not set\"));\n }\n var beta = student_beta || 0;\n var angle1 = start_pos*Math.exp(-beta*t)*Math.cos(Math.sqrt((g/l)-beta) * t);\n var angle2 = -Math.PI/2+angle1;\n\n if(isNaN(angle1) || isNaN(angle2)) {\n throw(new Error(\"angle NaN\"));\n }\n t5._setUpdateText('θ = '+angle1.toFixed(1)+'rad');\n\n if(t>0.01){\n // Fudge dtheta/dt\n anglelast= start_pos*Math.exp(-beta*(t-0.01))*Math.cos(Math.sqrt((g/l)-beta) * (t-0.01) );\n anglediff = (angle1-anglelast)/0.01; \n t6._setUpdateText('dθ/dt = '+anglediff.toFixed(1)+'rads-1');\n } \n\n return [p3.X()+l*Math.cos(angle2),p3.Y()+l*Math.sin(angle2)];\n } catch(e) {\n t5._setUpdateText('θ is not a real number!');\n t6._setUpdateText('');\n return [p3.X(),p3.Y()-l];\n }\n }; \n }());\n\n\n}\n\n// Start the animation with initial condition\nstartAnimation2(theta0);\n\n//pick up the student answer\nquestion.signals.on('HTMLAttached',function(e) {\n var part = question.getPart('p3g0');\n ko.computed(function(){\n var expr = part.display.studentAnswer();\n try {\n student_beta = Numbas.util.parseNumber(expr,part.settings.allowFractions,part.settings.allowFractions);\n if(isNaN(student_beta)) {\n throw(new Error(\"Not a number\"));\n }\n }\n catch(e) {\n student_beta = null;\n }\n console.log(student_beta);\n\n if(student_beta!==null){\n startAnimation2(theta0);\n t5.setAttribute({visible: true});\n t6.setAttribute({visible: true});\n t9.setAttribute({visible: false});\n board2.update();\n } else {\n t5.setAttribute({visible: false});\n t6.setAttribute({visible: false});\n t9.setAttribute({visible: true});\n board2.update();\n }\n\n });\n}); \n\n\nreturn div;\n\n"}}, "preamble": {"js": "", "css": ".JXGtext button{\n color: black;\n background-color: #a2d1f0;\n border: 1px solid #91bcd8;\n border-radius: 6px;\n}\n\n.JXGtext button:hover{\n background-color: #91bcd8;\n}"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "{jsx_pendulum()}
\nYou are told that $g=\\var{g}\\textrm{ms}^{-2}$ and that the length of the pendulum's rod $l = \\var{l}$ m.
\nSolve for $\\theta(t)$ with initial conditions $\\frac{\\mathrm{d}\\theta}{\\mathrm{d}t}(0)=0$, $\\theta(0)=\\frac{\\pi}{\\var{theta0_frac}}$.
\n$\\theta(t) = $ [[0]]
What is the time period of the undamped oscillations?
\n$T = $ [[0]]s
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "T", "maxValue": "T", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Determine the maximum angular velocity of the bob.
\n$\\left(\\frac{\\mathrm{d}\\theta}{\\mathrm{d}t}\\right)_{\\text{max}} = $ [[0]] rad s$^{-1}$.
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{theta0*sqrt(g/l)}", "maxValue": "{theta0*sqrt(g/l)}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Introducing a friction coefficient, $k$, that is proportional to the speed of the bob, the equation of motion becomes
\n\\[ \\frac{\\mathrm{d}^2\\theta}{\\mathrm{d}t^2}+\\frac{k}{m}\\frac{\\mathrm{d}\\theta}{\\mathrm{d}t}+\\frac{g}{l}\\theta = 0\\text{.} \\]
\nLet $k=\\var{k}\\textrm{kg s}^{-1}$ and $m=\\var{m}$kg. The initial conditions in part (a) yield the solution
\n\\[\\theta(t) = \\frac{\\pi}{\\var{theta0_frac}}e^{-\\beta t}\\cos\\left(t \\;\\sqrt{\\frac{\\var{g}}{\\var{l}}-\\beta^2}\\right)\\text{.} \\]
\nDetermine the value of $\\beta$.
\n$\\beta = $ [[0]]
\n{jsx_pendulum2()}
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{k/(2*m)}", "maxValue": "{k/(2*m)}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "After how long do the amplitude of oscillations decay to 5% of the initial release angle?
\n[[0]]s
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{damping5}", "maxValue": "{damping5}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Graph of a quadratic function", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}], "tags": ["JSXgraph", "jsxgraph", "Jsxgraph", "plot", "quadratic"], "metadata": {"description": "Compute a table of values for a quadratic function. A JSXgraph plot shows the curve going through the entered values.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "You are given the quadratic formula
\n$y=\\simplify[std]{{a}x^2+{c}}$
", "advice": "", "rulesets": {"std": ["all", "fractionNumbers"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-2,-1,-0.5,0.5,1,2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-4..4 except 0)", "description": "", "templateType": "anything", "can_override": false}, "ys": {"name": "ys", "group": "Ungrouped variables", "definition": "map(a*x^2+c,x,-3..3)", "description": "The $y$-coordinates of the points to plot.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "ys"], "variable_groups": [], "functions": {}, "preamble": {"js": "function dragpoint_board() {\n var scope = question.scope;\n \n var a = scope.variables.a.value;\n var c = scope.variables.c.value;\n var maxy = Math.max(Math.abs(a*9+c),Math.abs(c));\n \n var div = Numbas.extensions.jsxgraph.makeBoard('250px','400px',{boundingBox:[-5,maxy+3,5,-maxy-3],grid:true});\n $(question.display.html).find('#dragpoint').append(div);\n \n var board = div.board;\n \n //shorthand to evaluate a mathematical expression to a number\n function evaluate(expression) {\n try {\n var val = Numbas.jme.evaluate(expression,question.scope);\n return Numbas.jme.unwrapValue(val);\n }\n catch(e) {\n // if there's an error, return no number\n return NaN;\n }\n }\n \n // set up points array\n var num_points = 7;\n var points = [];\n \n \n // this function sets up the i^th point\n function make_point(i) {\n \n // calculate initial coordinates\n var x = i-(num_points-1)/2;\n \n // create an invisible vertical line for the point to slide along\n var line = board.create('line',[[x,0],[x,1]],{visible: false});\n \n // create the point\n var point = points[i] = board.create(\n 'glider',\n [i-(num_points-1)/2,0,line],\n {\n name:'',\n size:2,\n snapSizeY: 0.1, // the point will snap to y-coordinates which are multiples of 0.1\n snapToGrid: true\n }\n );\n \n // the contents of the input box for this point\n var studentAnswer = question.parts[1].gaps[i].display.studentAnswer;\n \n // watch the student's input and reposition the point when it changes. \n ko.computed(function() {\n y = evaluate(studentAnswer());\n if(!(isNaN(y)) && board.mode!=board.BOARD_MODE_DRAG) {\n point.moveTo([x,y],100);\n }\n });\n \n // when the student drags the point, update the gapfill input\n point.on('drag',function(){\n var y = Numbas.math.niceNumber(point.Y());\n studentAnswer(y);\n });\n \n }\n \n // create each point\n for(var i=0;iAn upwards-opening parabola
", "A downwards-opening parabola
"], "matrix": ["if(a>0,1,0)", "if(a>0,0,1)"], "distractors": ["", ""]}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Fill in the table of values for $y=\\simplify[std]{{a}x^2+{c}}$:
\n| $x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ |
|---|---|---|---|---|---|---|---|
| $y$ | \n[[0]] | \n[[1]] | \n[[2]] | \n[[3]] | \n[[4]] | \n[[5]] | \n[[6]] | \n
Give the coordinates of the turning point of the parabola.
", "correctAnswer": "matrix([0,c])", "correctAnswerFractions": false, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Use student input in a JSXGraph diagram", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}], "tags": ["graphs", "JSXgraph", "jsxgraph", "Jsxgraph"], "metadata": {"description": "There are copious comments in the definition of the function eqnline about the voodoo needed to have a JSXGraph diagram interact with the input box for a part.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "{eqnline(a,b,x2,y2)}
\nThe above graph shows a line which has an equation of the form $y=ax+b$, where $a$ and $b$ are integers.
\nYou are given two points on the line, $(0,\\var{b})$ and $(\\var{x2},\\var{y2})$, as indicated on the diagram.
", "advice": "\nFirst Method.
\nYou are given that the line goes through $(0,\\var{b})$ and $(-1,\\var{b-a})$ and the equation of the line is of the form $y=ax+b$
\nHence:
\n1) At $x=0$ we have $y=\\var{b}$, and this gives $\\var{b}=a \\times 0 +b =b$ on putting $x=0$ into $y=ax+b$.
\nSo $b=\\var{b}$.
\n2) At $x=-1$ we have $y=\\var{b-a}$, and this gives $\\var{b-a}=a \\times (-1) +b =\\simplify[all,!collectNumbers]{-a+{b}}$ on putting $x=-1$ into $y=ax+b$.
\nOn rearranging we obtain $a=\\simplify[all,!collectNumbers]{{b}-{b-a}}=\\var{a}$.
\nSo $a=\\var{a}$.
\nSo the equation of the line is $\\simplify{y={a}*x+{b}}$.
\nSecond Method.
\nThe equation $y=ax+b$ tells us that the graph crosses the $y$-axis (when $x=0$) at $y=b$.
\nSo looking at the graph we immediately see that $b=\\var{b}$.
\n$a$ is the gradient of the line and is given by the change from $(-1,\\var{b-a})$ to $(0,\\var{b})$:
\n\\[a=\\frac{\\text{Change in y}}{\\text{Change in x}}=\\frac{\\simplify[all,!collectNumbers]{({b-a}-{b})}}{-1-0}=\\var{a}\\]
\n\n", "rulesets": {}, "variables": {"x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(-3..3 except -1..1)", "description": "", "templateType": "anything"}, "y2": {"name": "y2", "group": "Ungrouped variables", "definition": "x2*a+b", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-6..6 except [0,a])", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-4..4 except 0)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x2", "b", "a", "y2"], "variable_groups": [], "functions": {"eqnline": {"parameters": [["a", "number"], ["b", "number"], ["x2", "number"], ["y2", "number"]], "type": "html", "language": "javascript", "definition": "// This function creates the board and sets it up, then returns an\n// HTML div tag containing the board.\n \n// The line is described by the equation \n// y = a*x + b\n\n// This function takes as its parameters the coefficients a and b,\n// and the coordinates (x2,y2) of a point on the line.\n\n// First, make the JSXGraph board.\n// The function provided by the JSXGraph extension wraps the board up in \n// a div tag so that it's easier to embed in the page.\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n{boundingBox: [-13,16,13,-16],\n axis: false,\n showNavigation: false,\n grid: true\n});\n \n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board; 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