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{"name": " moment of inertia - cylinder-non-uniform", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": " moment of inertia - cylinder-non-uniform", "tags": [], "metadata": {"description": "<p>moment of inertia - cylinder</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Determine the moment of inertia of the cylinder $C$ of density varies as $\\rho = \\var{d}xy$, height $\\var{h}$, and radius $\\var{a}$ about the axis passing through the centres of its circular faces.</p>", "advice": "<p>We can, naturally, consider this problem in cylindricla coordinates.&nbsp;</p>\n<p>We may assume that the cylinder $C$ sits in the space such that the axis rotation described in the question is the $z$-axis.</p>\n<p>Then $C$ can be described by&nbsp;</p>\n<p>\\[0\\leq r \\leq \\var{a} \\quad \\mbox{and } \\quad 0\\leq \\theta\\leq 2\\pi \\quad\\mbox{and}\\quad 0\\leq z\\leq h.\\]</p>\n<p>We also have to turn $\\rho(x,y)$ into polar coordinates.&nbsp;</p>\n<p>\\[\\rho(x,y) = \\var{d}(r\\cos\\theta)(r\\sin\\theta) = \\var{d}\\cdot r^2\\cos\\theta\\sin\\theta.\\]</p>\n<p>Then, in cylindrical coordinates, the mass element becomes $dM = \\rho dV = r^2\\cos\\theta\\sin\\theta \\,r\\,d\\theta\\,dr\\, dz$. Also, observe that each mass element is at the distance of $r$ to the axis of rotation. Then the moment of inertia is&nbsp;</p>\n<p>\\[I_z = \\int_C r^2 \\, dM = \\int_C r^2\\rho\\, dV =\\iiint_C \\var{d}\\cdot r^5 \\cos\\theta\\sin\\theta\\, d\\theta\\,dr\\, dz =\\\\[3mm] \\var{d}\\int_0^{\\var{h}}\\int_0^{2\\pi}\\int_0^{\\var{a}} r^5\\cos\\theta\\sin\\theta\\, dr\\,d\\theta \\, dz = \\var{d}\\int_0^{\\var{h}}\\, dz \\int_0^{2\\pi} \\cos\\theta\\sin\\theta\\,d\\theta \\int_0^{\\var{a}} r^5\\, dr . \\]</p>\n<p>Note that for the last equality we used the fact that boundaries of the integrals are constants.</p>\n<p>We can compute the integrals separately as follows:</p>\n<ul>\n<li>\\[\\int_0^{\\var{h}}\\, dz = [z]_0^{\\var{h}} = \\var{h}\\]</li>\n<li>\\[\\int_0^{2\\pi} \\cos\\theta\\sin\\theta\\,d\\theta =&nbsp; \\left[-\\frac{\\cos^2\\theta}{2}\\right]_0^{2\\pi} = \\frac{1-\\cos^2(2\\pi)}{2} =0.\\]</li>\n<li>\\[\\int_0^{\\var{a}} r^5\\, dr = \\left[\\frac{r^6}{6}\\right]_0^{\\var{a}} = \\frac{\\var{a^6}}{6}.\\]</li>\n</ul>\n<p></p>\n<p>Thus we obtain&nbsp;</p>\n<p>\\[I_z = \\var{d}\\times\\var{h}\\times 0\\times \\frac{\\var{a^6}}{6} = 0.\\]</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 3#1)", "description": "", "templateType": "randrange", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(1 .. 5#0.25)", "description": "", "templateType": "randrange", "can_override": false}, "V": {"name": "V", "group": "Ungrouped variables", "definition": "pi*(a^2)*h", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["d", "a", "h", "V"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "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"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}