![](\"resources/question-resources/metac_fig1.PNG\"/)
Numbas representation of the Metacentric Height fluid mechanics laboratory for first year Engineering students.
Students should answer this question after completing the laboratory and the associated Excel template, including the relevant plots.
The first question tests their capacity to correctly calculate metacentric heights, the remaining questions evaluate their graph interpretation skills.
", "licence": "All rights reserved"}, "statement": "Answer this question using your laboratory measurements and the Excel calculations and analysis template. You're expected to have completed all calculations and plots before attempting this question.
", "advice": "We start by calculating the theoretical depth of immersion, dth (m) = Total pontoon mass / (density of water x L x b) = 1.305 / (1000*0.35*0.2) = {dth}
\nFirst let's consider the static case:
\nTherefore can calculate BM (m) as 0.35*(0.2^3)/12/(0.35*0.2*{dth}) = {BM}
\nNow, given an OG of {OG_a} m, we can calculate BG (m) = OG - dth/2 = {OG_a} - {dth}/2 = {BG}
\nFinally, the metacentric height GMstatic (m) = BM - BG = {BM} - {BG} = {GM_a_static}
\n\nNow for the heeling case:
\nGiven a trasversing mass positioned at x (m) = {x_a}
\nWe can calculate the metacentric height heeling as GMheeling (m) = (Mtrasx / W) * cot(Theta) = {GM_a_heeling}
\nWhere the trasversing mass, Mtras is 0.305 kg, the total mass, W, is 1.305 kg and Theta is 10 degrees
\n\nAn example of a correct Figure 1 is shown below, the relationship is clearly LINEAR.
\n
Similarly from Figure 2 we can observe that:
\nFor a given centre of gravity, GM static is always greater than GM heeling.
\nGM heeling and static decrease with OG.
\nGM heeling and static form roughly parallel lines.
\n![](\"resources/question-resources/metac_fig2.PNG\"/)
Measured centre of gravity, Pontton a
", "templateType": "anything"}, "BM": {"name": "BM", "group": "Pontoon A", "definition": "0.35*(0.2^3)/12/(0.35*0.2*dth)", "description": "Metacentric radius, constant, BM (m) = (L b3/12) / (L b dth) where L is 0.35 m, b is 0.2m, and dth is 0.019 m.
", "templateType": "anything"}, "BG": {"name": "BG", "group": "Pontoon A", "definition": "OG_a-(0.019/2)", "description": "Distance between the centre of buoyancy and centre of gravity,
\n| BG (m) = OG - OB | \n
OB is dth/2
", "templateType": "anything"}, "GM_a_static": {"name": "GM_a_static", "group": "Pontoon A", "definition": "BM-BG", "description": "Metacentric height
", "templateType": "anything"}, "dth": {"name": "dth", "group": "Pontoon A", "definition": "1.305/(1000*0.35*0.2)", "description": "Theoretical depth of flotation
", "templateType": "anything"}, "GM_a_heeling": {"name": "GM_a_heeling", "group": "Pontoon A", "definition": "((0.305*x_a)/1.305)*(1/tan(0.175))", "description": "Metacentric height in 10 degree heeling condition. Fixed values are 0.305 kg of the sliding mass, 1.305 kg of the total pontoon mass and 0.175 radians of heel.
", "templateType": "anything"}, "x_a": {"name": "x_a", "group": "Pontoon A", "definition": "0.072", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Pontoon A", "variables": ["BM", "OG_a", "GM_a_static", "BG", "dth", "GM_a_heeling", "x_a"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Write the height of the centre of gravity, OG (m), for Pontoon A in meters, to three decimal places. [[0]]
\nWrite the corresponding metacentric height, GM (m), to three decimal places. [[1]]
\nConsider now the heeling equilibrium, what was the distance x (m) that resulted in a 10 degree inclination of the pontoon? [[2]]
\nFinally, write the corresponding metacentric height,GM [[3]]
", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "0.07", "maxValue": "0.08", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [{"variable": "OG_a", "part": "p0g0", "must_go_first": true}], "variableReplacementStrategy": "alwaysreplace", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "GM_a_static", "maxValue": "GM_a_static", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "0.065", "maxValue": "0.075", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [{"variable": "x_a", "part": "p0g2", "must_go_first": true}], "variableReplacementStrategy": "alwaysreplace", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "GM_a_heeling", "maxValue": "GM_a_heeling", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "According to the first plot in the Excel spreadsheet, the relationship between OG and x is:
", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["Exponential", "Lineal", "Polynomial", "Random"], "matrix": [0, "3", 0, 0], "distractors": ["", "", "", ""]}, {"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Use the second graph in your Excel template to select which of the following statements are correct:
", "minMarks": 0, "maxMarks": "3", "shuffleChoices": true, "displayType": "checkbox", "displayColumns": "1", "minAnswers": "1", "maxAnswers": "5", "warningType": "none", "showCellAnswerState": true, "choices": ["For a given centre of gravity, GM static is always greater than GM heeling.", "GM heeling and static form roughly parallel lines.", "GM heeling and static are the same at OG = 0.100 m", "GM heeling and static decrease with OG", "GM heeling and static increase with OG"], "matrix": ["1", "1", 0, "1", 0], "distractors": ["", "", "", "", ""]}], "contributors": [{"name": "Roberto Villalobos Herrera", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4586/"}]}]}], "contributors": [{"name": "Roberto Villalobos Herrera", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4586/"}]}