// Numbas version: exam_results_page_options {"name": "25. Centroidal Moment of Inertia", "metadata": {"description": "

Homework set. Problems require finding centroid, then Moment of Inertia about a centroidal axis.

", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Parallel Axis Theorem: arbitrary shape", "extensions": ["geogebra", "quantities"], "custom_part_types": [{"source": {"pk": 19, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/19/edit"}, "name": "Engineering Accuracy with units", "short_name": "engineering-answer", "description": "

A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "siground(settings['correctAnswer'],4)", "hint": {"static": true, "value": ""}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nswitch( \n right and good_units and right_sign, add_credit(1.0,'Correct.'),\n right and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\n right and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but wrong or missing units.'),\n close and good_units, add_credit(settings['C1'],'Close.'),\n close and not good_units, add_credit(settings['C3'],'Answer is close, but wrong or missing units.'),\n incorrect('Wrong answer.')\n)\n\n\ninterpreted_answer:\nqty(student_scalar, student_units)\n\n\n\ncorrect_quantity:\nsettings[\"correctAnswer\"]\n\n\n\ncorrect_units:\nunits(correct_quantity)\n\n\nallowed_notation_styles:\n[\"plain\",\"en\"]\n\nmatch_student_number:\nmatchnumber(studentAnswer,allowed_notation_styles)\n\nstudent_scalar:\nmatch_student_number[1]\n\nstudent_units:\nreplace_regex('ohms','ohm',\n replace_regex('\u00b0', ' deg',\n replace_regex('-', ' ' ,\n studentAnswer[len(match_student_number[0])..len(studentAnswer)])),\"i\")\n\ngood_units:\ntry(\ncompatible(quantity(1, student_units),correct_units),\nmsg,\nfeedback(msg);false)\n\n\nstudent_quantity:\nswitch(not good_units, \n student_scalar * correct_units, \n not right_sign,\n -quantity(student_scalar, student_units),\n quantity(student_scalar,student_units)\n)\n \n\n\npercent_error:\ntry(\nscalar(abs((correct_quantity - student_quantity)/correct_quantity))*100 \n,msg,\nif(student_quantity=correct_quantity,0,100))\n \n\nright:\npercent_error <= settings['right']\n\n\nclose:\nright_sign and percent_error <= settings['close']\n\nright_sign:\nsign(student_scalar) = sign(correct_quantity)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "switch( \n right and good_units and right_sign, add_credit(1.0,'Correct.'),\n right and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\n right and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but wrong or missing units.'),\n close and good_units, add_credit(settings['C1'],'Close.'),\n close and not good_units, add_credit(settings['C3'],'Answer is close, but wrong or missing units.'),\n incorrect('Wrong answer.')\n)\n"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "qty(student_scalar, student_units)\n\n"}, {"name": "correct_quantity", "description": "", "definition": "settings[\"correctAnswer\"]\n\n"}, {"name": "correct_units", "description": "", "definition": "units(correct_quantity)\n"}, {"name": "allowed_notation_styles", "description": "", "definition": "[\"plain\",\"en\"]"}, {"name": "match_student_number", "description": "", "definition": "matchnumber(studentAnswer,allowed_notation_styles)"}, {"name": "student_scalar", "description": "", "definition": "match_student_number[1]"}, {"name": "student_units", "description": "

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

", "definition": "replace_regex('ohms','ohm',\n replace_regex('\u00b0', ' deg',\n replace_regex('-', ' ' ,\n studentAnswer[len(match_student_number[0])..len(studentAnswer)])),\"i\")"}, {"name": "good_units", "description": "", "definition": "try(\ncompatible(quantity(1, student_units),correct_units),\nmsg,\nfeedback(msg);false)\n"}, {"name": "student_quantity", "description": "

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

", "definition": "switch(not good_units, \n student_scalar * correct_units, \n not right_sign,\n -quantity(student_scalar, student_units),\n quantity(student_scalar,student_units)\n)\n \n"}, {"name": "percent_error", "description": "", "definition": "try(\nscalar(abs((correct_quantity - student_quantity)/correct_quantity))*100 \n,msg,\nif(student_quantity=correct_quantity,0,100))\n "}, {"name": "right", "description": "", "definition": "percent_error <= settings['right']\n"}, {"name": "close", "description": "

Only marked close if the student actually has the right sign.

", "definition": "right_sign and percent_error <= settings['close']"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity) "}], "settings": [{"name": "correctAnswer", "label": "Correct Quantity.", "help_url": "", "hint": "The correct answer given as a JME quantity.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "right", "label": "% Accuracy for right.", "help_url": "", "hint": "Question will be considered correct if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "0.2", "evaluate": true}, {"name": "close", "label": "% Accuracy for close.", "help_url": "", "hint": "Question will be considered close if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "1.0", "evaluate": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units. if correct answer is 100 N and close is ±1%,
99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
If the correct answer is 100 N, both 100 and -100 N are accepted.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value but forgotten units.
This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["Mechanics", "mechanics", "parallel axis theorem", "Statics", "statics"], "metadata": {"description": "

Given the moment of inertia of an area about an arbitrary axis, find the centroidal moment of inertia and the moment of inertia about a second axis.

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

{geogebra_applet('gy7gyfft', ['d_x': d_1, 'd_m': d_2])}

", "advice": "

Given:

$A = \\var{qty(A,'in^2')}, \\quad I_x = \\var{qty(Ix,'in^4')}, \\quad d_1 = \\var{qty(abs(d_1),\"in\")}, \\quad d_2 = \\var{qty(abs(d_2),\"in\")}$

Find $\\bar{I}_C$

$\\begin{align} I &= \\bar{I} + A d^2 \\Rightarrow &\\bar{I}_C &= I_x - A (d_1)^2 \\\\ && &= \\var{Ix} - \\var{A} (\\var{abs(d_1)})^2\\\\ &&&= \\var{qty(Ibar,'in^4')}\\end{align}$

Find $I_M$

$\\begin{align} I &= \\bar{I} + A d^2 \\Rightarrow &I_M &= \\bar{I}_C + A (d_2)^2 \\\\ && &= \\var{Ibar} + \\var{A} (\\var{abs(d_2)})^2\\\\ &&&= \\var{qty(Im,'in^4')}\\end{align}$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"d_1": {"name": "d_1", "group": "Ungrouped variables", "definition": "random(2..10)random([-1,1])", "description": "", "templateType": "anything", "can_override": false}, "d_2": {"name": "d_2", "group": "Ungrouped variables", "definition": "random(2..10)random([-1,1])\n", "description": "", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "random(500..3000#100)", "description": "", "templateType": "anything", "can_override": false}, "Ibar": {"name": "Ibar", "group": "Ungrouped variables", "definition": "random(1000..6000#50)", "description": "", "templateType": "anything", "can_override": false}, "Ix": {"name": "Ix", "group": "Ungrouped variables", "definition": "Ibar + A d_1^2", "description": "", "templateType": "anything", "can_override": false}, "Im": {"name": "Im", "group": "Ungrouped variables", "definition": "Ibar + A d_2^2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "d_1<>d_2", "maxRuns": 100}, "ungrouped_variables": ["d_1", "d_2", "A", "Ibar", "Ix", "Im"], "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": "

The area of the shape is {qty(A, 'in^2')} and its moment of inertia about the $x$ axis is {qty(Ix,'in^4')}.

Determine the centroidal moment of inertia about the $C$–$C$ axis.

$\\bar{I}_C =$ [[0]]

Determine the moment of inertia about the $M$–$M$ axis.

$I_M = $ [[1]]

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "$\\bar{i}$", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(Ibar,'in^4')", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "$I_M$", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(Im,'in^4')", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Centroidal moment of inertia: box beam", "extensions": ["geogebra", "quantities", "weh"], "custom_part_types": [{"source": {"pk": 19, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/19/edit"}, "name": "Engineering Accuracy with units", "short_name": "engineering-answer", "description": "

A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

", "definition": "right_sign and percent_error <= settings['close']"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity) "}], "settings": [{"name": "correctAnswer", "label": "Correct Quantity.", "help_url": "", "hint": "The correct answer given as a JME quantity.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "right", "label": "% Accuracy for right.", "help_url": "", "hint": "Question will be considered correct if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "0.2", "evaluate": true}, {"name": "close", "label": "% Accuracy for close.", "help_url": "", "hint": "Question will be considered close if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "1.0", "evaluate": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units. if correct answer is 100 N and close is ±1%,
99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
If the correct answer is 100 N, both 100 and -100 N are accepted.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value but forgotten units.
This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["Mechanics", "mechanics", "moment of inertia", "parallel axis theorem", "rectangular beam", "Statics", "statics"], "metadata": {"description": "

Calculate the centroidal moment of intertia of a rectangle with a excentric rectangular hole.

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

{geogebra_applet('pv5edgwm',[[\"A\",A],[\"B\",B]])}

The cross-section of a hollow box beam is shown. Grid units are [{units}].

", "advice": "

Part 1: Big Rectangle:

$b_1$ = {b1}, $h_1$ = {h1}, $A_1$ = {A1}, $\\bar{y}_1$ = {ybar1}

Part 2: Small rectangle (removed):

$b_2$ = {b2}, $h_2$ = {h2}, $A_2$ = {A2}, $\\bar{y}_2$ = {ybar2}

Total Area:

$A = \\Sigma A_i = A_1 - A_2 =$ {display(Area)}

Centroid:

$\\bar{y} = \\dfrac{\\Sigma A_i \\bar{y}_i}{\\Sigma A_i} = \\dfrac{A_1 \\bar{y}_1 - A_2 \\bar{y}_2}{ A_1 - A_2}$ = {display(ybar)}

Moments of inerta about the x axis:

$I_{x_1} = \\dfrac{b_1 (h_1)^3}{3} = $ {display(Ix1)}

$I_{x_2}= \\bar{I} + A d^2 = \\dfrac{b_2 (h_2)^3}{12} + A_2 (\\bar{y}_2)^2 = $ {display(Ix2)}

$I_x = I_{x_1} - I_{x_2}= $ {display(Ix)}

Centroidal Moment of inertia:

Using the parallel axis theorem,

$I = \\bar{I} + A d^2 \\implies \\bar{I}_x= I_x - A (\\bar{y})^2$ = {display(Ibar)}

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Ix1": {"name": "Ix1", "group": "Unnamed group", "definition": "b1 h1 h1 h1/3", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Unnamed group", "definition": "qty(A[0],units)", "description": "

base for large rectangle

", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "vector(random(4..20#2),random(8..18))", "description": "

top right corner of large rectangle

", "templateType": "anything", "can_override": false}, "Area": {"name": "Area", "group": "Unnamed group", "definition": "A1-A2", "description": "", "templateType": "anything", "can_override": false}, "ybar": {"name": "ybar", "group": "Unnamed group", "definition": "(A1 ybar1 - A2 ybar2)/(A1-A2)", "description": "", "templateType": "anything", "can_override": false}, "A1": {"name": "A1", "group": "Unnamed group", "definition": "b1 h1", "description": "", "templateType": "anything", "can_override": false}, "debug": {"name": "debug", "group": "Ungrouped variables", "definition": "false", "description": "", "templateType": "anything", "can_override": false}, "Ibar": {"name": "Ibar", "group": "Unnamed group", "definition": "Ix - area ybar ybar", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "

thickness of sides and top

", "templateType": "anything", "can_override": false}, "Ix": {"name": "Ix", "group": "Unnamed group", "definition": "Ix1 - Ix2", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Unnamed group", "definition": "qty(A[0] - 2 t,units)", "description": "", "templateType": "anything", "can_override": false}, "h2": {"name": "h2", "group": "Unnamed group", "definition": "qty(A[1]-t - B[1],units)", "description": "", "templateType": "anything", "can_override": false}, "ybar1": {"name": "ybar1", "group": "Unnamed group", "definition": "h1/2", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "vector(t, random(2..(A[1]-t-1)))", "description": "", "templateType": "anything", "can_override": false}, "h1": {"name": "h1", "group": "Unnamed group", "definition": "qty(A[1],units)", "description": "

height of large rectangle

", "templateType": "anything", "can_override": false}, "Ix2": {"name": "Ix2", "group": "Unnamed group", "definition": "b2 h2 h2 h2 /12 + A2 ybar2 ybar2", "description": "", "templateType": "anything", "can_override": false}, "ybar2": {"name": "ybar2", "group": "Unnamed group", "definition": "qty(B[1] + scalar(h2/2),units)", "description": "", "templateType": "anything", "can_override": false}, "A2": {"name": "A2", "group": "Unnamed group", "definition": "b2 h2", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Ungrouped variables", "definition": "random('in','mm','cm','ft')", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "B[0]ybar2", "maxRuns": 100}, "ungrouped_variables": ["t", "A", "B", "units", "debug"], "variable_groups": [{"name": "Unnamed group", "variables": ["b1", "h1", "b2", "h2", "A1", "A2", "ybar", "ybar1", "ybar2", "Ix1", "Ix2", "Ix", "Area", "Ibar"]}], "functions": {"display": {"parameters": [["q", "quantity"]], "type": "string", "language": "jme", "definition": "string(siground(q,5))"}}, "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": "

Determine the vertical location of the centroid of the cross section.

$\\bar{y} = $ [[0]] {display(ybar)}

Determine the moment of inertia of the shape about a horizontal ${x'}$ axis (the neutral axis) passing through the centroid.

${I_{x'} = }$ [[1]] {display(Ibar)}

", "gaps": [{"type": "engineering-answer", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "{ybar}", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "{Ibar}", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Centroidal Moment of inertia: T shape", "extensions": ["geogebra", "quantities"], "custom_part_types": [{"source": {"pk": 19, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/19/edit"}, "name": "Engineering Accuracy with units", "short_name": "engineering-answer", "description": "

A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

", "definition": "right_sign and percent_error <= settings['close']"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity) "}], "settings": [{"name": "correctAnswer", "label": "Correct Quantity.", "help_url": "", "hint": "The correct answer given as a JME quantity.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "right", "label": "% Accuracy for right.", "help_url": "", "hint": "Question will be considered correct if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "0.2", "evaluate": true}, {"name": "close", "label": "% Accuracy for close.", "help_url": "", "hint": "Question will be considered close if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "1.0", "evaluate": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units. if correct answer is 100 N and close is ±1%,
99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
If the correct answer is 100 N, both 100 and -100 N are accepted.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value but forgotten units.
This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [["question-resources/Tshape.ggb", "/srv/numbas/media/question-resources/Tshape.ggb"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["centroid", "Mechanics", "mechanics", "moment of inertia", "statics", "Statics"], "metadata": {"description": "

Find the centroidal moment of inertia of a sideways T shape. This requires first locating the centroid, then applying the parallel axis theorem.

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

{geogebra_applet('wfhyktjn',['A':P1, 'B': P2])}

Determine the moment of inertia of the T shape with respect to a vertical axis passing through the shape's centroid. Let this axis be designated $a$. Grid units are {units}.

", "advice": "

Number the parts:

Part 1 horizontal rectangle: $b_1$ = {qty(b1,units)}, $h_1$ = {qty(h1,units)}.

Part 2 vertical rectangle: $b_2$ = {qty(b2,units)}, $h_2$ = {qty(h2,units)}.

Calculate the areas:

$ A_1 = b_1 h_1=\\var{qty(A1,units+'^2')}$

$ A_2 = b_2 h_2 = \\var{qty(A2,units+'^2')}$

Locate the centroids of the parts.

$ \\bar{x}_1 = b_1/2 = \\var{qty(d1,units)} \\qquad \\bar{x}_2 = b_1 + b_2/2 = \\var{qty(d2,units)}$

Locate the centroid of the whole.

$\\bar{x} = \\dfrac{ A_1 \\bar{x}_1 + A_2 \\bar{x}_2} {A_1 + A_2} = \\var{qty(siground(xbar,4),units)}$

$\\bar{y} = \\var{qty(0, units)} $, by symmetry

Calculate the moment of inertia with respect to the y-axis for the parts and the whole.

$\\begin{align} \\\\{I_y}_1 &= \\tfrac{1}{3} h_1 b_1^3 \\\\&= \\tfrac{1}{3}( \\var{h1})(\\var{b1})^3 \\\\&= \\var{qty(display(Iy1),units+'^4')} \\\\\\\\{I_y}_2 &= [\\bar{I} + A d^2]_2\\\\ &= \\frac{1}{12} h_2 b_2^3 + A_2 d_2^2 \\\\&= \\var{display(Ibar2)} + (\\var{display(A2)}) (\\var{display(d2)})^2\\\\&=\\var{qty(display(Iy2),units+'^4')} \\\\\\\\I_y &= {I_y}_1+{I_y}_2\\\\&=\\var{display(Iy1)}+\\var{display(Iy2)}\\\\&=\\var{qty(display(Iy),units+'^4')}\\end{align}$

Use the parallel axis theorem $I = \\bar{I} + A d^2$ backwards to find the centroidal moment of inertia.

$\\begin{align} \\bar{I}_a &= I_y - A (\\bar{x})^2 \\\\&= \\var{display(Iy)} - \\var{display(A)} (\\var{display(xbar)})^2\\\\&=\\var{qty(display(I_a),units+'^4')}\\end{align}$

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width of P1

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true

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Point A in GGB

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height of p1

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width of p2

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height of p2

", "templateType": "anything"}, "ibar1": {"name": "ibar1", "group": "check", "definition": "h1 b1^3/12", "description": "

centroidal moi

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alternate calculation of Iy

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Moment of intertia about y axis

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Centroidal moment of inertia of total shape

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$\\bar{x} = $ [[0]] $\\qquad \\bar{y} = $ [[1]]

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$\\bar{I}_a$ = [[0]] {display(I_a')}

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A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

", "definition": "right_sign and percent_error <= settings['close']"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity) "}], "settings": [{"name": "correctAnswer", "label": "Correct Quantity.", "help_url": "", "hint": "The correct answer given as a JME quantity.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "right", "label": "% Accuracy for right.", "help_url": "", "hint": "Question will be considered correct if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "0.2", "evaluate": true}, {"name": "close", "label": "% Accuracy for close.", "help_url": "", "hint": "Question will be considered close if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "1.0", "evaluate": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units. if correct answer is 100 N and close is ±1%,
99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
If the correct answer is 100 N, both 100 and -100 N are accepted.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value but forgotten units.
This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["first moment of area", "Mechanics", "mechanics", "moment of inertia", "parallel axis theorem", "semicircle", "statics", "Statics"], "metadata": {"description": "

Find moment of inertia of a shape which requires the use of the parallel axis theorem for a semicircle.

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

{geogebra_applet('fqywxznp',{ggbArgs})}

Determine the moment of inertia and radius of gyration of the composite shape with respect to the x- and y-axes knowing that $b$ = {b}, $h$ = {h}, $r_o$ = {r_o}, and $r_i$ = {display(r_i)}.

", "advice": "

Number the parts:

Part 1 rectangle: $b$ = {b}, $h$ = {h}.

Part 2 semicircle: $r_o$ = {r_o}.

Part 3 circular hole: $r_i$ = {display(r_i)}.

Calculate the areas:

$\\begin{align} A_1 &= b h & A_2 &= \\tfrac{1}{2}\\pi r_o^2 &A_3 &= \\pi r_i^2&A&=A_1 +A_2 - A_3\\\\&=\\var{display(A1)} &&=\\var{display(A2)} &&=\\var{display(A3)}&&=\\var{display(A)}\\\\&&\\end{align}$

Calculate the moment of inertia with respect to the x-axis.

$\\begin{align} \\\\{I_1}_x &= \\tfrac{1}{3} b h^3 \\\\&= \\tfrac{1}{3}( \\var{b})(\\var{h})^3 \\\\&= \\var{display(Ix1)} \\\\\\\\{I_x}_2 &= [\\bar{I} + A d^2]_2\\\\ &= 0.1098 r_o^4 + \\left(\\frac{\\pi r_o^2}{2}\\right) \\left(h + \\frac{4 r_o}{3 \\pi}\\right)^2\\\\&= \\var{display(Ibarx2)} + (\\var{display(A2)}) (\\var{display(d2)})^2\\\\&=\\var{display(Ix2)} \\\\\\\\ {I_x}_3 &= [\\bar{I} + A d^2]_3 \\\\ &= \\frac{\\pi r_i^4}{4} + \\left(\\pi r_i^2\\right) \\left(h \\right)^2\\\\&= \\var{display(Ibarx3)} + (\\var{display(A3)}) (\\var{display(h)})^2\\\\&=\\var{display(Ix3)}\\\\\\\\I_x &= {I_x}_1+{I_x}_2-{I_x}_3\\\\&=\\var{display(Ix1)}+\\var{display(Ix2)}-\\var{display(Ix3)}\\\\&=\\var{display(Ix)}\\end{align}$

Calculate the moment of inertia with respect to the y-axis.

$\\begin{align} \\\\{I_y}_1 &= \\tfrac{1}{3} h b^3 \\\\&= \\tfrac{1}{3}( \\var{h})(\\var{b})^3 \\\\&= \\var{display(Iy1)} \\\\\\\\{I_y}_2 &= [\\bar{I} + A d^2]_2\\\\ &= \\frac{\\pi r_o^4}{8} + \\left(\\frac{\\pi r_o^2}{2}\\right) \\left(\\frac{b}{2}\\right)^2\\\\&= \\var{display(Ibary2)} + (\\var{display(A2)}) (\\var{display(b/2)})^2\\\\&=\\var{display(Iy2)} \\\\\\\\ {I_y}_3 &= [\\bar{I} + A d^2]_3 \\\\ &= \\frac{\\pi r_i^4}{4} + \\left(\\pi r_i^2\\right) \\left(\\frac{b}{2} \\right)^2\\\\&= \\var{display(Ibary3)} + (\\var{display(A3)}) (\\var{display(b/2)})^2\\\\&=\\var{display(Iy3)}\\\\\\\\I_y &= {I_y}_1+{I_y}_2-{I_y}_3\\\\&=\\var{display(Iy1)}+\\var{display(Iy2)}-\\var{display(Iy3)}\\\\&=\\var{display(Iy)}\\end{align}$

Calculate the radius of gyration with respect to the x- and y- axes.

$\\begin{align}\\\\k_x &= \\sqrt{\\dfrac{I_x}{A}}&k_y &= \\sqrt{\\dfrac{I_y}{A}}\\\\ &= \\sqrt{\\dfrac{\\var{display(Ix)}}{\\var{display(A)}}}&&= \\sqrt{\\dfrac{\\var{display(Iy)}}{\\var{display(A)}}}\\\\ &= \\var{display(kx)} &&=\\var{display(ky)} \\end{align}$

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rectangle

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$I_x$ = [[0]] $k_x$ = [[1]]

{display(Ix)} {display(kx)}

$I_y$ = [[0]] $k_y$ = [[1]]

{display(Iy)} {display(ky)}

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A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

", "definition": "right_sign and percent_error <= settings['close']"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity) "}], "settings": [{"name": "correctAnswer", "label": "Correct Quantity.", "help_url": "", "hint": "The correct answer given as a JME quantity.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "right", "label": "% Accuracy for right.", "help_url": "", "hint": "Question will be considered correct if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "0.2", "evaluate": true}, {"name": "close", "label": "% Accuracy for close.", "help_url": "", "hint": "Question will be considered close if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "1.0", "evaluate": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units. if correct answer is 100 N and close is ±1%,
99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
If the correct answer is 100 N, both 100 and -100 N are accepted.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value but forgotten units.
This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [["question-resources/Tshape.ggb", "/srv/numbas/media/question-resources/Tshape.ggb"]]}