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{"name": "Triangular Truss", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Truss-1.png", "/srv/numbas/media/question-resources/Truss-1.png"], ["question-resources/Truss-1_abNAGhD.png", "/srv/numbas/media/question-resources/Truss-1_abNAGhD.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Triangular Truss", "tags": [], "metadata": {"description": "<p>Tension in simple triangular truss.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Half of mechanics is about balancing forces. Some trusses can be treated like pin-jointed structures and decomposed into axial forces, i.e., without any bending or shear forces. The solution is therefore obtained through trigonometry.</p>", "advice": "<p></p>\n<p>Balancing horizontal forces at B: $P_{AB}\\sin(\\var{theta})+P_{BC}\\sin(90-\\var{theta})=0$, or more simply: $P_{AB}\\sin(\\var{theta})+P_{BC}\\cos(\\var{theta})=0$.</p>\n<p>Balancing vertical forces at B: $P_{AB}\\cos(\\var{theta})=P_{BC}\\cos(90-\\var{theta})+\\var{force}$, or more simply: $P_{AB}\\cos(\\var{theta})=P_{BC}\\sin(\\var{theta})+\\var{force}$,</p>\n<p>where the units are kN. Multiplying the first by $\\cos(\\var{theta})$ and the second by $\\sin(\\var{theta})$, we get:</p>\n<p>$P_{AB}\\sin(\\var{theta})\\cos(\\var{theta})+P_{BC}\\cos(\\var{theta})\\cos(\\var{theta})=0$</p>\n<p>$P_{AB}\\cos(\\var{theta})\\sin(\\var{theta})-P_{BC}\\sin(\\var{theta})\\sin(\\var{theta})=\\var{force}\\sin(\\var{theta})$</p>\n<p>and subtracting the second from the first:</p>\n<p>$P_{BC}\\cos(\\var{theta})\\cos(\\var{theta})+P_{BC}\\sin(\\var{theta})\\sin(\\var{theta})=-\\var{force}\\sin(\\var{theta})$, or more simply: $P_{BC}=-\\var{force}\\sin(\\var{theta})$</p>\n<p>since $\\cos^2 + \\sin^2 = 1$. Balancing vertical forces at C:</p>\n<p>$P_{AC}+P_{BC}\\cos(90-\\var{theta})=0$, or more simply: $P_{AC}+P_{BC}\\sin(\\var{theta})=0$,</p>\n<p>and substituting in the expression for $P_{BC}$ from above:</p>\n<p>$P_{AC}-\\var{force}\\sin(\\var{theta})\\sin(\\var{theta})=0$,</p>\n<p>the final answer simplifies to:</p>\n<p>$P_{AC}=\\var{force}\\sin^2(\\var{theta})=\\var{siground(Pac,3)}$ [units: kN].</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"theta": {"name": "theta", "group": "Ungrouped variables", "definition": "random(35..55#5)", "description": "<p>Angle to vertical of Bar AB.</p>", "templateType": "anything", "can_override": false}, "Force": {"name": "Force", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "<p>Applied vertical (downward) force.</p>", "templateType": "anything", "can_override": false}, "Pac": {"name": "Pac", "group": "Ungrouped variables", "definition": "force*(sin(radians(theta)))^2", "description": "<p>Tension in Bar AC.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Force", "theta", "Pac"], "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": "<p>A pin-jointed truss is shown in the figure below. The pivot at A is fixed, but the pivot at C is free to move vertically. The angle at B is a right angle.</p>\n<p><img src=\"resources/question-resources/Truss-1_abNAGhD.png\" style=\"width: 160px; height: 280px;\" alt=\"Pin-jointed truss\"/></p>\n<p>If the applied force, $F$, is $\\var{force}$ kN vertically down, and the angle of Bar AB to the vertical, $\\theta$, is $\\var{theta}^\\circ$, what is the tension in Bar AC?</p>\n<p>[[0]] [Units: kN]</p>", "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": "Pac", "maxValue": "Pac", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "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", "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}]}]}], "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}]}