![](\"resources/question-resources/MOI_table.png\")
$\\var{if(axis=1,latex(\"\\\\bar\\{I\\}_x\"), latex(\"I_x\"))}= $ [[0]]
\nNote: If your answer produces an error message, try putting an asterisk before parentheses, like so: b h * (h/2)^2
\\[I_x=\\simplify{{expression(Ix)}}\\]
\n", "marks": 0, "useCustomName": false, "type": "gapfill", "showFeedbackIcon": true, "showCorrectAnswer": true, "gaps": [{"failureRate": 1, "customName": "", "unitTests": [], "valuegenerators": [{"name": "c", "value": ""}, {"name": "r", "value": ""}], "variableReplacements": [], "useCustomName": false, "checkingType": "absdiff", "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "{expression(Ix)}", "variableReplacementStrategy": "originalfirst", "marks": "10", "checkingAccuracy": 0.001, "type": "jme", "checkVariableNames": false, "showPreview": true, "vsetRange": [0, 1], "showFeedbackIcon": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5}]}], "statement": "{geogebra_applet('j25dzmbq', [[\"A\",A],[\"shape\",index +1],[\"axis\",axis],[\"reference\",ref]])}
\nFind the moment of inertia with respect to the {if(axis=1,\"centroidal \",\"\")}x axis for the {shape} with {description}.
\n", "rulesets": {}, "extensions": ["geogebra", "weh"], "variables": {"up": {"group": "parts", "definition": "expression(maximums[index])", "name": "up", "templateType": "anything", "description": ""}, "shape": {"group": "parts", "definition": "names[index]", "name": "shape", "templateType": "anything", "description": ""}, "down": {"group": "parts", "definition": "expression(minimums[index])", "name": "down", "templateType": "anything", "description": "distance from centroid to bottom of shape
"}, "Ibars": {"group": "Ungrouped variables", "definition": "[\"b*h^3/12\",\n \"b*h^3/36\", \n \"pi r^4/4\", \n if(even_quadrant,\"0.1098 r^4\",\"pi r^4/8\"),\n \"0.0549 r^4\"]", "name": "Ibars", "templateType": "anything", "description": ""}, "names": {"group": "Ungrouped variables", "definition": "[\"rectangle\",\"triangle\",\"circle\",\"semicircle\",\"quarter circle\"]", "name": "names", "templateType": "anything", "description": ""}, "minimums": {"group": "Ungrouped variables", "definition": "[\"h/2\",\n if(quadrant<2,\"h/3\",\"2 h/3\")\n ,\"r\",\n if(quadrant=0,\"(4 r)/(3 pi)\",if(quadrant=2,\"r-(4 r)/(3 pi)\",\"r\")),\n if(quadrant<=1,\"(4 r)/(3 pi)\", \"r-(4 r)/(3 pi)\")]", "name": "minimums", "templateType": "anything", "description": "distance from centroid to bottom of the shape
"}, "distances": {"group": "Ungrouped variables", "definition": "let(d,string(down),u,string(up),[[\"0\",\"0\",\"0\",\"0\"],\n [d,d,d,d],\n [u,u,u,u],\n [\"0\",\"c\",\"c+(\"+d+\")\",\"c-(\"+u+\")\"],\n [\"0\",\"c\",\"c-\"+d,\"c+\"+u]])", "name": "distances", "templateType": "anything", "description": ""}, "area": {"group": "parts", "definition": "expression(areas[index])", "name": "area", "templateType": "anything", "description": ""}, "axis": {"group": "Ungrouped variables", "definition": "random(1..5)", "name": "axis", "templateType": "anything", "description": "1 through centroid
\n2 bottom of shape
\n3 top of shape
\n4 below shape
\n5 above shape
"}, "Ibar": {"group": "parts", "definition": "expression(Ibars[index])", "name": "Ibar", "templateType": "anything", "description": ""}, "index": {"group": "Ungrouped variables", "definition": "random(0..4)", "name": "index", "templateType": "anything", "description": "rectangle, triangle, circle, semicircle, quarter circle
"}, "quadrant": {"group": "parts", "definition": "if(A[0]>0 and A[1]>0,0,\n if(A[0]<0 and A[1] > 0, 1,\n if(A[0]<0 and A[1]<0,2,3)))", "name": "quadrant", "templateType": "anything", "description": "Calculates which quadrant point A is in. Needed to determine orientation of the triangle, semi and quarter circles.
"}, "d": {"group": "parts", "definition": "expression(distances[axis-1][ref])", "name": "d", "templateType": "anything", "description": ""}, "description": {"group": "parts", "definition": "if(index<2,\"base b and height h\", \"radius r\")", "name": "description", "templateType": "anything", "description": ""}, "ref": {"group": "Ungrouped variables", "definition": "if(axis>3,random(1..3),0)", "name": "ref", "templateType": "anything", "description": "The reference axis, needed for axis 4 and 5.
\n1 centroid
\n2 bottom of shape
\n3 top of shape
"}, "areas": {"group": "Ungrouped variables", "definition": "[\"b*h\", \"b h / 2\", \"pi r^2\", \"pi r^2 /2\" , \"pi r^2 /4\"]", "name": "areas", "templateType": "anything", "description": ""}, "A": {"group": "Ungrouped variables", "definition": "vector(random([2,3],[3,2],[-3,2],[-2,3])) random(-1,1)\n", "name": "A", "templateType": "anything", "description": ""}, "maximums": {"group": "Ungrouped variables", "definition": "[\"h/2\",\n if(quadrant<2,\"2 h/3\",\"h/3\")\n ,\"r\",\n if(quadrant=0,\"(r-(4 r)/(3 pi))\",if(quadrant=2,\"(4 r)/(3 pi)\",\"r\")),\n if(quadrant<=1,\"(r-(4 r)/(3 pi))\", \"(4 r)/(3 pi)\")]", "name": "maximums", "templateType": "anything", "description": "Distance from centroid to the top of the shape.
\n\nmaximums = {abs(y(A) / 2), Max(2 / 3 y(A), (-y(A)) / 3), radius, If(quadrant \u225f 0, radius - ybar, quadrant \u225f 1, radius, quadrant \u225f 2, ybar, quadrant \u225f 3, radius), If(quadrant ≤ 1, radius - ybar, ybar)}
"}, "even_quadrant": {"group": "Ungrouped variables", "definition": "mod(quadrant,2)=0 ", "name": "even_quadrant", "templateType": "anything", "description": "usefull utility
"}, "Ix": {"group": "parts", "definition": "(string(Ibar) + \"+ \" + string(area) + \"*(\"+ string(d) + \")^2\")", "name": "Ix", "templateType": "anything", "description": ""}}, "tags": ["mechanics", "Mechanics", "moment of inertia", "parallel axis theorem", "second moment of area", "statics", "Statics"], "name": "Katrin's copy of Parallel Axis Theorem", "preamble": {"js": "", "css": ""}, "advice": "Some cases are found in the table below. For others use the parallel axis theorem.
\n\\[I = \\bar{I} + A d^2\\]
\nwhere:
\n$\\bar{I}$ is the centroidal moment of inertia, i.e., the moment of inertia of the shape with respect to an axis which is parallel to the desired axis and passes throught the shape's centroid.
\n$A$ is the area of the shape.
\n$d$ is the distance between the desired axis and the parallel centroidal axis.
\n\n\n
Write expressions for the moment of inertia of simple shapes about various axes.
"}, "type": "question", "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}, {"name": "Katrin Thomson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3424/"}]}]}], "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}, {"name": "Katrin Thomson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3424/"}]}