// Numbas version: exam_results_page_options
{"name": "2 masses 3 springs: example 1", "extensions": [], "custom_part_types": [], "resources": [["question-resources/3_Springs.jpg", "/srv/numbas/media/question-resources/3_Springs.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "rulesets": {}, "parts": [{"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>", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"tolerance": 0, "allowResize": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": "4", "type": "matrix", "correctAnswerFractions": true, "numRows": "2", "allowFractions": true, "correctAnswer": "matx_eqn", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true, "markPerCell": true, "numColumns": "2"}]}, {"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>", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"correctAnswerFraction": false, "variableReplacements": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "marks": 1, "type": "numberentry", "minValue": "{lambda1}", "allowFractions": true, "notationStyles": ["plain", "en", "si-en"], "maxValue": "lambda1", "scripts": {}, "showCorrectAnswer": true, "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "variableReplacements": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "marks": 1, "type": "numberentry", "minValue": "lambda2", "allowFractions": true, "notationStyles": ["plain", "en", "si-en"], "maxValue": "lambda2", "scripts": {}, "showCorrectAnswer": true, "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true}]}, {"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>", "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "marks": 0, "type": "gapfill", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"correctAnswerFraction": false, "variableReplacements": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "marks": 1, "type": "numberentry", "minValue": "{k1}+{k2}+{m1}*{lambda1}", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "{k1}+{k2}+{m1}*{lambda1}", "scripts": {}, "showCorrectAnswer": true, "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true}, {"correctAnswerFraction": false, "variableReplacements": [], "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "marks": 1, "type": "numberentry", "minValue": "{k1}+{k2}+{m1}*{lambda2}", "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "maxValue": "{k1}+{k2}+{m1}*{lambda2}", "scripts": {}, "showCorrectAnswer": true, "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true}]}], "preamble": {"js": "", "css": ""}, "name": "2 masses 3 springs: example 1", "functions": {}, "variable_groups": [], "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Two masses are connected by three springs. The spring having spring constant \\({k_1}\\)&nbsp;is rigidly fixed to a wall on the left hand side and the spring having spring constant \\({k_3}\\)&nbsp;is rigidly fixed to a wall on the right hand side.</p>\n<p style=\"padding-left: 90px;\"><img src=\"resources/question-resources/3_Springs.jpg\"/></p>\n<p>The first mass is \\(\\var{m1}kg\\) and the second mass is \\(\\var{m2}kg\\).</p>\n<p>The spring constants are &nbsp;k<sub>1</sub> = \\(\\var{k1}\\), &nbsp; k<sub>2</sub> = \\(\\var{k2}\\) &nbsp; and k<sub>3</sub> = \\(\\var{k3}\\).</p>\n<p style=\"padding-left: 90px;\"></p>\n<p style=\"padding-left: 90px;\"></p>", "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)-\\var{k3}x_b\\)</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-\\simplify{{k3}+{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{({k3}+{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}+{k3})/{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}+{k3})/{m2}}-\\lambda\\end{array}\\right)=0\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\left(-\\simplify{({k1}+{k2})/{m1}}-\\lambda\\right)\\left(-\\simplify{({k2}+{k3})/{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}+{k3})/{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}+{k3}+{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}+{k3})/{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}+{lambda2}*{m1})/{m1}}&amp;\\simplify{{k2}/{m1}}\\\\\\simplify{{k2}/{m2}}&amp;\\simplify{-({k2}+{k3}+{m2}*{lambda2})/{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}+{m1}*{lambda2})/{m1}})x+\\simplify{{k2}/{m1}}y=0\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\simplify{{k2}/{m2}}y=(\\simplify{({k1}+{k2}+{m1}*{lambda2})/{m1}})x\\)</p>\n<p style=\"padding-left: 180px;\">When \\(x=\\var{k2}\\)</p>\n<p style=\"padding-left: 180px;\">\\(\\simplify{{k2}/{m2}}y=(\\simplify{({k1}+{k2}+{m1}*{lambda2})/{m1}})\\var{k2}\\)</p>\n<p style=\"padding-left: 180px;\">\\(y=\\var{y2}\\)</p>", "ungrouped_variables": ["k1", "k2", "k3", "m1", "m2", "b", "c", "lambda1", "lambda2", "x1", "y1", "x2", "y2", "matx_eqn", "n", "a1", "b1", "c1", "d1"], "variables": {"lambda1": {"description": "", "templateType": "anything", "definition": "(-{b}+sqrt({b}^2-4*{c}))/2", "name": "lambda1", "group": "Ungrouped variables"}, "k1": {"description": "", "templateType": "randrange", "definition": "random(1..15#1)", "name": "k1", "group": "Ungrouped variables"}, "b": {"description": "", "templateType": "anything", "definition": "({k1}+{k2})/{m1}+({k2}+{k3})/{m2}", "name": "b", "group": "Ungrouped variables"}, "x1": {"description": "", "templateType": "anything", "definition": "{k2}", "name": "x1", "group": "Ungrouped variables"}, "m2": {"description": "", "templateType": "anything", "definition": "{m1}", "name": "m2", "group": "Ungrouped variables"}, "a1": {"description": "", "templateType": "anything", "definition": "-({k1}+{k2})/{m1}", "name": "a1", "group": "Ungrouped variables"}, "c": {"description": "", "templateType": "anything", "definition": "({k1}*{k2}+{k1}*{k3}+{k2}*{k3})/({m1}*{m2})", "name": "c", "group": "Ungrouped variables"}, "lambda2": {"description": "", "templateType": "anything", "definition": "(-{b}-sqrt({b}^2-4*{c}))/2", "name": "lambda2", "group": "Ungrouped variables"}, "y2": {"description": "", "templateType": "anything", "definition": "{k1}+{k2}+{m1}*{lambda2}", "name": "y2", "group": "Ungrouped variables"}, "x2": {"description": "", "templateType": "anything", "definition": "{k2}", "name": "x2", "group": "Ungrouped variables"}, "y1": {"description": "", "templateType": "anything", "definition": "{k1}+{k2}+{m1}*{lambda1}", "name": "y1", "group": "Ungrouped variables"}, "n": {"description": "<p>multiplier for m2</p>", "templateType": "randrange", "definition": "random(1..4#1)", "name": "n", "group": "Ungrouped variables"}, "matx_eqn": {"description": "", "templateType": "anything", "definition": "matrix([\n    [a1,b1],\n    [c1,d1]\n])  ", "name": "matx_eqn", "group": "Ungrouped variables"}, "b1": {"description": "", "templateType": "anything", "definition": "{k2}/{m1}", "name": "b1", "group": "Ungrouped variables"}, "k3": {"description": "", "templateType": "anything", "definition": "{k1}", "name": "k3", "group": "Ungrouped variables"}, "k2": {"description": "", "templateType": "anything", "definition": "{k1}", "name": "k2", "group": "Ungrouped variables"}, "c1": {"description": "", "templateType": "anything", "definition": "{k2}/{m2}", "name": "c1", "group": "Ungrouped variables"}, "m1": {"description": "", "templateType": "randrange", "definition": "random(1..5#1)", "name": "m1", "group": "Ungrouped variables"}, "d1": {"description": "", "templateType": "anything", "definition": "-({k2}+{k3})/{m2}", "name": "d1", "group": "Ungrouped variables"}}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "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/"}]}