// Numbas version: exam_results_page_options
{"name": "2 masses 2 springs: example 1", "extensions": [], "custom_part_types": [], "resources": [["question-resources/2-springs.jpg", "/srv/numbas/media/question-resources/2-springs.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"d1": {"description": "", "name": "d1", "templateType": "anything", "definition": "-({k2})/{m2}", "group": "Ungrouped variables"}, "m1": {"description": "", "name": "m1", "templateType": "randrange", "definition": "random(2..12#2)", "group": "Ungrouped variables"}, "x2": {"description": "", "name": "x2", "templateType": "anything", "definition": "{k2}", "group": "Ungrouped variables"}, "k2": {"description": "", "name": "k2", "templateType": "anything", "definition": "{k1}/2", "group": "Ungrouped variables"}, "m2": {"description": "", "name": "m2", "templateType": "anything", "definition": "{m1}/2", "group": "Ungrouped variables"}, "n": {"description": "<p>multiplier for m2</p>", "name": "n", "templateType": "randrange", "definition": "random(1..4#1)", "group": "Ungrouped variables"}, "k1": {"description": "", "name": "k1", "templateType": "randrange", "definition": "random(2..20#2)", "group": "Ungrouped variables"}, "y1": {"description": "", "name": "y1", "templateType": "anything", "definition": "{k1}+{k2}+{m1}*{lambda1}", "group": "Ungrouped variables"}, "x1": {"description": "", "name": "x1", "templateType": "anything", "definition": "{k2}", "group": "Ungrouped variables"}, "c1": {"description": "", "name": "c1", "templateType": "anything", "definition": "{k2}/{m2}", "group": "Ungrouped variables"}, "lambda1": {"description": "", "name": "lambda1", "templateType": "anything", "definition": "(-{b}+sqrt({b}^2-4*{c}))/2", "group": "Ungrouped variables"}, "b1": {"description": "", "name": "b1", "templateType": "anything", "definition": "{k2}/{m1}", "group": "Ungrouped variables"}, "matx_eqn": {"description": "", "name": "matx_eqn", "templateType": "anything", "definition": "matrix([\n    [a1,b1],\n    [c1,d1]\n])  ", "group": "Ungrouped variables"}, "y2": {"description": "", "name": "y2", "templateType": "anything", "definition": "{k1}+{k2}+{m1}*{lambda2}", "group": "Ungrouped variables"}, "c": {"description": "", "name": "c", "templateType": "anything", "definition": "({k1}*{k2})/({m1}*{m2})", "group": "Ungrouped variables"}, "lambda2": {"description": "", "name": "lambda2", "templateType": "anything", "definition": "(-{b}-sqrt({b}^2-4*{c}))/2", "group": "Ungrouped variables"}, "a1": {"description": "", "name": "a1", "templateType": "anything", "definition": "-({k1}+{k2})/{m1}", "group": "Ungrouped variables"}, "b": {"description": "", "name": "b", "templateType": "anything", "definition": "({k1}+{k2})/{m1}+({k2})/{m2}", "group": "Ungrouped variables"}}, "ungrouped_variables": ["k1", "k2", "m1", "m2", "b", "c", "lambda1", "lambda2", "x1", "y1", "x2", "y2", "matx_eqn", "n", "a1", "b1", "c1", "d1"], "tags": [], "preamble": {"css": "", "js": ""}, "rulesets": {}, "functions": {}, "advice": "<p>Equations of motion:</p>\n<p>For mass \\(M_1\\): &nbsp; &nbsp; &nbsp; \\(\\var{m1}\\frac{d^2x_a}{dt^2}=-\\var{k1}x_a+\\var{k2}(x_b-x_a)\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;\\(\\var{m1}\\frac{d^2x_a}{dt^2}=-\\simplify{{k1}+{k2}}x_a+\\var{k2}x_b\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;\\(\\frac{d^2x_a}{dt^2}=-\\simplify{({k1}+{k2})/{m1}}x_a+\\simplify{{k2}/{m1}}x_b\\)</p>\n<p>For mass \\(M_2\\): &nbsp; &nbsp; &nbsp; &nbsp;\\(\\var{m2}\\frac{d^2x_b}{dt^2}=-\\var{k2}(x_b-x_a)\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(\\var{m2}\\frac{d^2x_b}{dt^2}=\\var{k2}x_a-\\var{k2}x_b\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(\\frac{d^2x_b}{dt^2}=\\simplify{{k2}/{m2}}x_a-\\simplify{{k2}/{m2}}x_b\\)</p>\n<p>Writing this in matrix format gives:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(\\left( \\begin{array}{cc}\\frac{d^2x_a}{dt^2}\\\\ \\frac{d^2x_b}{dt^2}\\end{array}\\right)=\\left( \\begin{array}{cc}-\\simplify{({k1}+{k2})/{m1}}&amp;\\simplify{{k2}/{m1}}\\\\\\simplify{{k2}/{m2}}&amp;\\simplify{-{k2}/{m2}}\\end{array}\\right)\\left( \\begin{array}{cc}x_a\\\\x_b\\end{array}\\right)\\)</p>\n<p>To find the eigenvalues we must solve:</p>\n<p style=\"padding-left: 180px;\">\\(\\left( \\begin{array}{cc}-\\simplify{({k1}+{k2})/{m1}}-\\lambda&amp;\\simplify{{k2}/{m1}}\\\\\\simplify{{k2}/{m2}}&amp;-\\simplify{{k2}/{m2}}-\\lambda\\end{array}\\right)=0\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\left(-\\simplify{({k1}+{k2})/{m1}}-\\lambda\\right)\\left(-\\simplify{({k2})/{m2}}-\\lambda\\right)-\\left(\\simplify{{k2}/{m1}}\\right)\\left(\\simplify{{k2}/{m2}}\\right)=0\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\lambda^2+\\var{b}\\lambda+\\var{c}=0\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\lambda=\\frac{-\\var{b}\\pm \\sqrt{\\var{b}^2-4*\\var{c}}}{2}\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\lambda=\\frac{-\\var{b}\\pm \\sqrt{\\simplify{{b}^2-4*{c}}}}{2}\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\lambda=\\var{lambda1}\\) &nbsp;and &nbsp;\\(\\lambda=\\var{lambda2}\\)</p>\n<p>For \\(\\lambda=\\var{lambda1}\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\left( \\begin{array}{cc}-\\simplify{({k1}+{k2})/{m1}}-(\\var{lambda1})&amp;\\simplify{{k2}/{m1}}\\\\\\simplify{{k2}/{m2}}&amp;-\\simplify{{k2}/{m2}}-(\\var{lambda1})\\end{array}\\right)\\left( \\begin{array}{cc}x\\\\y\\end{array}\\right)=\\left( \\begin{array}{cc}0\\\\0\\end{array}\\right)\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\left( \\begin{array}{cc}\\simplify{-({k1}+{k2}+{m1}*{lambda1})/{m1}}&amp;\\simplify{{k2}/{m1}}\\\\\\simplify{{k2}/{m2}}&amp;\\simplify{-({k2}+{lambda1}*{m2})/{m2}}\\end{array}\\right)\\left( \\begin{array}{cc}x\\\\y\\end{array}\\right)=\\left( \\begin{array}{cc}0\\\\0\\end{array}\\right)\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\simplify{-({k1}+{k2}+{lambda1}*{m1})/{m1}}x+\\simplify{{k2}/{m1}}y=0\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\simplify{{k2}/{m1}}y=\\simplify{({k1}+{k2}+{lambda1}*{m1})/{m1}}x\\)</p>\n<p style=\"padding-left: 180px;\">When \\(x=\\var{k2}\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\simplify{{k2}/{m1}}y=(\\simplify{({k1}+{k2}+{lambda1}*{m1})/{m1}})\\var{k2}\\)</p>\n<p style=\"padding-left: 180px;\">\\(y=\\var{y1}\\)</p>\n<p>For \\(\\lambda=\\var{lambda2}\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\left( \\begin{array}{cc}-\\simplify{({k1}+{k2})/{m1}}-(\\var{lambda2})&amp;\\simplify{{k2}/{m1}}\\\\\\simplify{{k2}/{m2}}&amp;-\\simplify{{k2}/{m2}}-(\\var{lambda2})\\end{array}\\right)\\left( \\begin{array}{cc}x\\\\y\\end{array}\\right)=\\left( \\begin{array}{cc}0\\\\0\\end{array}\\right)\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\left( \\begin{array}{cc}\\simplify{-({k1}+{k2}+{m1}*{lambda2})/{m1}}&amp;\\simplify{{k2}/{m1}}\\\\\\simplify{{k2}/{m2}}&amp;\\simplify{-({k2}+{lambda2}*{m2})/{m2}}\\end{array}\\right)\\left( \\begin{array}{cc}x\\\\y\\end{array}\\right)=\\left( \\begin{array}{cc}0\\\\0\\end{array}\\right)\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\simplify{-({k1}+{k2}+{lambda2}*{m1})/{m1}}x+\\simplify{{k2}/{m1}}y=0\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\simplify{{k2}/{m1}}y=\\simplify{({k1}+{k2}+{lambda2}*{m1})/{m1}}x\\)</p>\n<p style=\"padding-left: 180px;\">When \\(x=\\var{k2}\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\simplify{{k2}/{m1}}y=(\\simplify{({k1}+{k2}+{lambda2}*{m1})/{m1}})\\var{k2}\\)</p>\n<p style=\"padding-left: 180px;\">\\(y=\\var{y2}\\)</p>", "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>Two masses are connected by two springs. The upper spring is fixed rigidly to a structure.</p>\n<p>The first mass \\(M_1\\) is \\(\\var{m1}kg\\) and the second mass \\(M_2\\) is \\(\\var{m2}kg\\).</p>\n<p></p>\n<p style=\"padding-left: 210px;\"><img src=\"resources/question-resources/2-springs.jpg\" width=\"202\" height=\"326\"/></p>\n<p></p>\n<p>The spring constants are &nbsp;\\(k_1\\)&nbsp;= \\(\\var{k1}\\) &nbsp;and &nbsp; \\(k_2\\)&nbsp;= \\(\\var{k2}\\).</p>\n<p style=\"padding-left: 90px;\"></p>\n<p style=\"padding-left: 90px;\"></p>", "name": "2 masses 2 springs: example 1", "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "parts": [{"type": "gapfill", "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "prompt": "<p>The equations of motion associated with the system can be written in matrix form as:</p>\n<p style=\"padding-left: 60px;\">\\(\\left( \\begin{array}{cc}\\frac{d^2x_a}{dt^2}\\\\ \\frac{d^2x_b}{dt^2}\\end{array}\\right)=\\left( \\begin{array}{cc}A&amp;B\\\\C&amp;D\\end{array}\\right)\\left( \\begin{array}{cc}x_a\\\\x_b\\end{array}\\right)\\)</p>\n<p>Enter the values of the 2x2 matrix as integers or fractions: &nbsp;&nbsp;[[0]]</p>", "gaps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFractions": true, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": true, "variableReplacements": [], "markPerCell": true, "marks": "4", "numRows": "2", "tolerance": 0, "type": "matrix", "allowResize": false, "numColumns": "2", "correctAnswer": "matx_eqn"}]}, {"type": "gapfill", "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "prompt": "<p>The matrix has two eigenvalues.</p>\n<p>The eigenvalue closest zero is \\(\\lambda_1=\\)&nbsp;[[0]]</p>\n<p>The other eigenvalue is \\(\\lambda_2=\\)&nbsp;[[1]]</p>", "gaps": [{"minValue": "{lambda1}", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": true, "variableReplacements": [], "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "maxValue": "lambda1", "mustBeReduced": false}, {"minValue": "lambda2", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": true, "variableReplacements": [], "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "maxValue": "lambda2", "mustBeReduced": false}]}, {"type": "gapfill", "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "prompt": "<p>The eigenvector corresponding to \\(\\lambda_1\\) is given by&nbsp;\\(\\left( \\begin{array}{cc}\\var{k2}\\\\ Y_1\\end{array}\\right)\\)</p>\n<p>Enter the value for \\(Y_1\\)&nbsp;[[0]]</p>\n<p>The eigenvector corresponding to \\(\\lambda_2\\) is given by&nbsp;\\(\\left( \\begin{array}{cc}\\var{k2}\\\\ Y_2\\end{array}\\right)\\)</p>\n<p>Enter the value for \\(Y_2\\)&nbsp;[[1]]</p>", "gaps": [{"minValue": "{k1}+{k2}+{m1}*{lambda1}", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "maxValue": "{k1}+{k2}+{m1}*{lambda1}", "mustBeReduced": false}, {"minValue": "{k1}+{k2}+{m1}*{lambda2}", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "allowFractions": false, "variableReplacements": [], "scripts": {}, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "maxValue": "{k1}+{k2}+{m1}*{lambda2}", "mustBeReduced": false}]}], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}