// Numbas version: exam_results_page_options {"name": "Error", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Error", "tags": [], "metadata": {"description": "

Error if torional stresses are calculated incorrectly 

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

An engineer naively calculates (wrongly) the maximum shear stress due to torsion of a member having a rectangular cross-section $a\\times b$ (where $b=\\var{k[j]}a$) by using the expression $\\tau =TR/J$ (where $J$ is the polar moment of area of the section and $R$ is the largest distance from the centre of twist) and assumes that $GJ$ represents the torsional rigidity. 

", "advice": "

Wrong:

\n

\\[\\tau_\\text{incorrect}=\\frac{TR}{J}=\\frac{T(\\var{k[j]/2}a)}{J}.\\]

\n

\\[J=\\iint_A r^2 dA=\\iint_A (z^2+y^2)dz\\,dy=\\int_{-0.5a}^{0.5a}\\int_{-\\var{k[j]/2}a}^{\\var{k[j]/2}a} (z^2+y^2)dz\\,dy=\\var{precround(k[j]*(1+k[j]^2)/12,3)} a^4,\\]

\n

\\[\\tau_\\text{incorrect}=\\frac{T(\\var{k[j]/2}a)}{\\var{precround(k[j]*(1+k[j]^2)/12,3)} a^4}=\\var{tau_a}\\frac{T}{a^3}.\\]

\n

Correct:

\n

\\[\\tau_\\text{correct}=\\frac{T}{a^2 (\\var{k[j]}a)\\gamma}=\\frac{T}{\\var{k[j]}\\times\\var{gamma[j]}a^3}=\\var{tau_b}\\frac{T}{a^3}.\\]

\n

Error:

\n

\\[\\left|\\frac{\\tau_\\text{incorrect}}{\\tau_\\text{correct}}-1\\right|\\times 100\\%=\\left|\\frac{\\var{tau_a}}{\\var{tau_b}}-1\\right|\\times 100\\%\\approx\\var{error}\\%\\]

", "rulesets": {}, "extensions": [], "variables": {"k": {"name": "k", "group": "Ungrouped variables", "definition": "[1.0,1.2,1.5,2.0,2.5,3.0,4.0,5.0]", "description": "", "templateType": "anything"}, "beta": {"name": "beta", "group": "Ungrouped variables", "definition": "[0.141,0.166,0.196,0.229,0.249,0.263,0.281,0.291]", "description": "", "templateType": "anything"}, "j": {"name": "j", "group": "Ungrouped variables", "definition": "random(1..7)", "description": "", "templateType": "anything"}, "tau_a": {"name": "tau_a", "group": "Ungrouped variables", "definition": "precround(6/(1+k[j]^2),3)", "description": "", "templateType": "anything"}, "tau_b": {"name": "tau_b", "group": "Ungrouped variables", "definition": "precround(1/(k[j]*gamma[j]),3)", "description": "", "templateType": "anything"}, "gamma": {"name": "gamma", "group": "Ungrouped variables", "definition": "[0.208,0.219,0.231,0.246,0.258,0.267,0.282,0.291]", "description": "", "templateType": "anything"}, "error": {"name": "error", "group": "Ungrouped variables", "definition": "round(abs(tau_a/tau_b-1)*100)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["k", "beta", "j", "tau_a", "tau_b", "gamma", "error"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "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": "

Making use of the results given in the table of slide 7 (Lecture notes), determine the percentage error in the calculations.

\n

error,%=[[0]]

", "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": "error-1", "maxValue": "error+1", "correctAnswerFraction": false, "allowFractions": false, "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", "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}]}]}], "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}]}