// Numbas version: exam_results_page_options
{"name": "Application 3: Differential equation with an irreducible quadratic", "extensions": [], "custom_part_types": [], "resources": [["question-resources/MSD.jpg", "/srv/numbas/media/question-resources/MSD.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "marks": 0, "variableReplacementStrategy": "originalfirst", "gaps": [{"showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "allowFractions": true, "correctAnswerFraction": true, "maxValue": "{R}/{A_denom}", "variableReplacements": [], "minValue": "{R}/{A_denom}", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "scripts": {}, "marks": 1, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReduced": false}, {"showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "allowFractions": true, "correctAnswerFraction": true, "maxValue": "B_numer/A_denom", "variableReplacements": [], "minValue": "B_numer/A_denom", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "scripts": {}, "marks": "1", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReduced": false}, {"showFeedbackIcon": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "allowFractions": true, "correctAnswerFraction": true, "maxValue": "(C_numer-d1*a1*B_numer)/(d1*a2*A_denom)", "variableReplacements": [], "minValue": "(C_numer-d1*a1*B_numer)/(d1*a2*A_denom)", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "scripts": {}, "marks": "1", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "mustBeReduced": false}], "type": "gapfill", "variableReplacements": [], "prompt": "<p></p>\n<p>Enter the value for \\(A\\) as an exact fraction.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(A=\\)&nbsp;[[0]]</p>\n<p>Enter the value for \\(B\\) as an exact fraction.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(B=\\)&nbsp;[[1]]</p>\n<p>Enter the value for \\(C\\) as an exact fraction.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(C=\\)&nbsp;[[2]]</p>\n<p></p>"}], "statement": "<p>The diagram below shows a typical mass-spring-damper system as might apply to the suspension of a car.</p>\n<p>(Masses have mass M, springs with stiffness k and dampers having damping coefficient B).</p>\n<p>The associated variables are displacement x(t) and force F(t).</p>\n<p style=\"padding-left: 90px;\">&nbsp;<img src=\"resources/question-resources/MSD.jpg\" width=\"422\" height=\"265\"/></p>\n<p>Initially the mass is at a distance \\(\\var{b1}cm\\) from the equilibrium point and is moving at \\(\\var{b2}cm/s\\).</p>\n<p>If \\(B=\\simplify{2*{a1}}\\) and&nbsp; \\(k=\\simplify{{a1}^2+{a2}^2}\\) and the system is subjected to an external applied force \\(F(t)=\\var{R}e^{-\\var{d1}t}\\)</p>\n<p>then from Newton's law we&nbsp;get the differential equation:</p>\n<p></p>\n<p style=\"padding-left: 30px;\">&nbsp;\\(\\frac{d^2x}{dt^2}+\\simplify{2*{a1}}\\frac{dx}{dt}+\\simplify{{a1}*{a1}+{a2}*{a2}}x(t)=\\var{R}e^{-\\var{d1}t}\\)&nbsp; &nbsp;where \\(x(0)=\\var{b1} \\,\\, and \\,\\, &nbsp;x'(0)=\\var{b2}\\)&nbsp;</p>\n<p>&nbsp; &nbsp;</p>\n<p>The solution of the equation is given by:</p>\n<p>&nbsp; &nbsp; &nbsp;&nbsp;\\(x(t)=Ae^{-\\var{d1}t}+Be^{-\\var{a1}t}cos(\\var{a2}t)+Ce^{-\\var{a1}t}sin(\\var{a2}t)\\)</p>\n<p></p>\n<p>.</p>", "rulesets": {}, "preamble": {"js": "", "css": ""}, "variable_groups": [], "functions": {}, "advice": "<p>&nbsp;\\(\\frac{d^2x}{dt^2}+\\simplify{2*{a1}}\\frac{dx}{dt}+\\simplify{{a1}*{a1}+{a2}*{a2}}x(t)=\\var{R}e^{-\\var{d1}t}\\) &nbsp;where \\(x(0)=\\var{b1} \\,\\, and \\,\\, &nbsp;x'(0)=\\var{b2}\\)</p>\n<p>The Laplace transform of this is given by:</p>\n<p>\\(s^2X(s)-\\var{b1}s-\\var{b2}+\\simplify{2*{a1}}(sX(s)-\\var{b1})+\\simplify{{a1}*{a1}+{a2}*{a2}}X(s)=\\frac{\\var{R}}{s+\\var{d1}}\\)</p>\n<p>Gathering only \\(X(s)\\) terms on the left hand side and factoring gives:</p>\n<p>\\(s^2X(s)+\\simplify{2*{a1}}sX(s)+\\simplify{{a1}^2+{a2}^2}X(s)=\\frac{\\var{R}}{s+\\var{d1}}+\\var{b1}s+\\simplify{{b2}+2*{a1}*{b1}}\\)</p>\n<p>\\((s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})X(s)=\\frac{\\var{b1}s^2+\\simplify{{b1}*{d1}+{b2}+2*{a1}*{b1}}s+\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}}{s+\\var{d1}}\\)</p>\n<p>\\(X(s)=\\frac{\\var{b1}s^2+\\simplify{{b1}*{d1}+{b2}+2*{a1}*{b1}}s+\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}}{(s+\\var{d1})(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})}\\)</p>\n<p>\\(X(s)=\\frac{A}{s+\\var{d1}}+\\frac{Bs+c}{(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})}\\)</p>\n<p>\\(\\var{b1}s^2+\\simplify{{b1}*{d1}+{b2}+2*{a1}*{b1}}s+\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}=A(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})+Bs(s+\\var{d1})+c(s+\\var{d1})\\)</p>\n<p>Let \\(s=-\\var{d1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{R}=A(\\simplify{{d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2})\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\frac{\\var{R}}{(\\simplify{{d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2})}\\)</p>\n<p>Compare the coefficients of \\(s^2\\) &nbsp;&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(\\var{b1}=A+B\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\var{b1}-\\frac{\\var{R}}{(\\simplify{{d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2})}=\\simplify{({b1}*({d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2)-{R})/({d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2)}\\)</p>\n<p>Let \\(s=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}=\\simplify{({a1}^2+{a2}^2)}A+c(\\var{d1})\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{d1}c=\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}-\\simplify{({a1}^2+{a2}^2)}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{d1}c=\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}-\\simplify{({a1}^2+{a2}^2)}\\left(\\frac{\\var{R}}{\\var{A_denom}}\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(c=\\frac{\\var{C_numer}}{\\simplify{{d1}*{A_denom}}}=\\simplify{{c_numer}/({d1}*{A_denom})}\\)</p>\n<p style=\"padding-left: 30px;\">\\(X(s)=\\frac{\\simplify{{R}/{A_denom}}}{s+\\var{d1}}+\\frac{\\simplify{{B_numer}/{A_denom}}s +\\simplify{{c_numer}/({d1}*{A_denom})}}{ s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(X(s)=\\frac{\\simplify{{R}/{A_denom}}}{s+\\var{d1}}+\\frac{\\simplify{{B_numer}/{A_denom}}(s+\\var{a1}) +\\simplify{({c_numer}-{B_numer}*{a1}*{d1})/({d1}*{A_denom})}}{(s+\\var{a1})^2+\\simplify{{a2}^2}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(X(s)=\\frac{\\simplify{{R}/{A_denom}}}{s+\\var{d1}}+\\frac{\\simplify{{B_numer}/{A_denom}}(s+\\var{a1})}{(s+\\var{a1})^2+\\simplify{{a2}^2}}+\\frac{\\simplify{({c_numer}-{B_numer}*{a1}*{d1})/({d1}*{A_denom})}}{(s+\\var{a1})^2+\\simplify{{a2}^2}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x(t)=\\simplify{{R}/{A_denom}}e^{-\\var{d1}t}+\\simplify{{B_numer}/{A_denom}}e^{-\\var{a1}t}cos(\\var{a2}t)+\\simplify{({c_numer}-{B_numer}*{a1}*{d1})/({d1}*{A_denom})}e^{-\\var{a1}t}\\frac{1}{\\var{a2}}sin(\\var{a2}t)\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "tags": [], "ungrouped_variables": ["a1", "R", "d1", "b1", "b2", "a2", "A_denom", "B_numer", "C_numer", "C_denom"], "name": "Application 3: Differential equation with an irreducible quadratic", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "extensions": [], "variables": {"b1": {"name": "b1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1..10)"}, "b2": {"name": "b2", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1..10) "}, "d1": {"name": "d1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(12..16)"}, "a1": {"name": "a1", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1..5)"}, "A_denom": {"name": "A_denom", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "(d1^2-2*a1*d1+a1^2+a2^2)"}, "C_denom": {"name": "C_denom", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "d1*A_denom"}, "C_numer": {"name": "C_numer", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "((r+b2*d1+2*a1*b1*d1)*(d1^2-2*a1*d1+a1^2+a2^2)-r*(a1^2+a2^2))"}, "R": {"name": "R", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "random(1..10)"}, "a2": {"name": "a2", "group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(2..9#1)"}, "B_numer": {"name": "B_numer", "group": "Ungrouped variables", "description": "", "templateType": "anything", "definition": "b1*A_denom-R"}}, "variablesTest": {"maxRuns": "222", "condition": ""}, "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/"}]}