// Numbas version: exam_results_page_options {"name": "08. Verignon's Theorem", "metadata": {"description": "

Homework set. Find 2-D moments or balance moments using Verignon's theorem: $ M = \\pm F_x d_y \\pm F_y d_x$

", "licence": "Creative Commons Attribution-NonCommercial 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": "Moment Balance: Hoist a pole", "extensions": ["quantities", "weh"], "custom_part_types": [{"source": {"pk": 24, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/24/edit"}, "name": "Angle quantity", "short_name": "angle-quantity-from-reference", "description": "

Angle as a quantity in degrees.

", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "plain_string(settings['correct_quantity'])", "hint": {"static": true, "value": ""}, "allowEmpty": {"static": true, "value": false}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nswitch( \nright and good_units and right_sign and angle_in_range, add_credit(1.0,'Correct.'),\nright and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\nright and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but missing degree symbol.'),\nright and good_units and right_sign and not angle_in_range,add_credit(settings['C1'],'Angle is out of range.'),\nclose and good_units, add_credit(settings['C1'],'Close.'),\nclose and not good_units, add_credit(settings['C3'],'Answer is close, but missing degree symbol.'),\nincorrect('Wrong answer.')\n)\n\ninterpreted_answer:\nqty(student_scalar, student_units)\n\ncorrect_scalar:\nscalar(correct_quantity)\n \n\ncorrect_quantity:\nsettings['correct_quantity']\n\ncorrect_units:\nunits(correct_quantity)\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:\njoin(\nsplit(studentAnswer[len(match_student_number[0])..len(studentAnswer)]\n,\"\u00b0\"),\" deg\")\n\n\n\ngood_units:\ntry(\ncompatible(quantity(1, student_units),correct_quantity),\nmsg,\nfeedback(msg);false)\n\nstudent_quantity:\nswitch(not good_units, \nstudent_scalar * correct_units, \nnot right_sign,\n-quantity(student_scalar, student_units),\nquantity(student_scalar,student_units)\n)\n\nright_sign:\nsign(student_scalar) = sign(correct_quantity)\n\nangle_in_range:\nif(settings['restrict_angle'], abs(student_scalar) <= 90, true)\n\nright:\nwithinTolerance(abs(student_scalar), abs(correct_scalar), settings['right'])\n\nclose:\nwithinTolerance(student_scalar, correct_scalar, settings['close'])", "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( \nright and good_units and right_sign and angle_in_range, add_credit(1.0,'Correct.'),\nright and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\nright and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but missing degree symbol.'),\nright and good_units and right_sign and not angle_in_range,add_credit(settings['C1'],'Angle is out of range.'),\nclose and good_units, add_credit(settings['C1'],'Close.'),\nclose and not good_units, add_credit(settings['C3'],'Answer is close, but missing degree symbol.'),\nincorrect('Wrong answer.')\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "qty(student_scalar, student_units)"}, {"name": "correct_scalar", "description": "", "definition": "scalar(correct_quantity)\n "}, {"name": "correct_quantity", "description": "", "definition": "settings['correct_quantity']"}, {"name": "correct_units", "description": "", "definition": "units(correct_quantity)"}, {"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": "", "definition": "join(\nsplit(studentAnswer[len(match_student_number[0])..len(studentAnswer)]\n,\"\u00b0\"),\" deg\")\n\n"}, {"name": "good_units", "description": "", "definition": "try(\ncompatible(quantity(1, student_units),correct_quantity),\nmsg,\nfeedback(msg);false)"}, {"name": "student_quantity", "description": "", "definition": "switch(not good_units, \nstudent_scalar * correct_units, \nnot right_sign,\n-quantity(student_scalar, student_units),\nquantity(student_scalar,student_units)\n)"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity)"}, {"name": "angle_in_range", "description": "", "definition": "if(settings['restrict_angle'], abs(student_scalar) <= 90, true)"}, {"name": "right", "description": "

Will check for correct sign elswhere.

", "definition": "withinTolerance(abs(student_scalar), abs(correct_scalar), settings['right'])"}, {"name": "close", "description": "

Must have correct sign to be close.

", "definition": "withinTolerance(student_scalar, correct_scalar, settings['close'])\n"}], "settings": [{"name": "correct_quantity", "label": "Correct Angle as quantity ", "help_url": "", "hint": "", "input_type": "code", "default_value": "qty(45,'deg')", "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 amount from the correct value.", "input_type": "code", "default_value": "0.1", "evaluate": true}, {"name": "restrict_angle", "label": "Less than 90\u00b0", "help_url": "", "hint": "When checked, angle must be between -90° and +90°.", "input_type": "checkbox", "default_value": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units.", "input_type": "percent", "default_value": "75"}, {"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 amount from the correct value.", "input_type": "code", "default_value": "0.5", "evaluate": true}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value without units.", "input_type": "percent", "default_value": "25"}], "public_availability": "restricted", "published": false, "extensions": ["quantities"]}, {"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": [["question-resources/Hoistingpost.png", "/srv/numbas/media/question-resources/Hoistingpost.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["Equilibrium", "equilibrium", "Mechanics", "mechanics", "moment balance", "Statics", "statics"], "metadata": {"description": "

Find the tension required to hold a pole at a specified angle.

", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "\n\n\n\n\n\n\n

Pole $AC$, which weighs {qty(W,'lb')}, is free to pivot on a frictionless pin at $C$ and is being supported by cable $AB$. The weight of the pole can be considered to act at the center of the pole.

In the position shown, $d_1 = \\var{d1}$ {unit} , $d_2 = \\var{d2}$ {unit}, and $d_3 = \\var{d3}$ {unit}. The diagram is not to scale.

", "advice": "

Solution using the definition of the moment.

$\\angle CAB = \\alpha= \\tan^{-1} \\left(\\frac{d_2 + d_3} {d_1}\\right) - \\tan^{-1} \\left(\\frac{d_2}{d_1}\\right) = \\var{siground(alpha,4)}°$

$L = \\sqrt{d_1^2 + d_2^2} = \\var{qty(siground(l,4),unit)}$

$\\sum M_C = 0$

$ W \\left(\\dfrac{d_2}{2}\\right) = T (L \\sin \\alpha) $

$T= \\var{siground(qty(T,'lb'),4)}$

This can also be solved using Verignon's Theorem

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"version": {"name": "version", "group": "Ungrouped variables", "definition": "random(0..1)", "description": "

Question Version 0 = horizontal, 1 = vertical

", "templateType": "anything", "can_override": false}, "unit": {"name": "unit", "group": "Ungrouped variables", "definition": "'ft'", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(12..24)", "description": "

", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(round(d1/3)..round(2 d1/3)#0.5)", "description": "", "templateType": "anything", "can_override": false}, "d3": {"name": "d3", "group": "Ungrouped variables", "definition": "random(round(1.4 d2)..round(2 D2))", "description": "", "templateType": "anything", "can_override": false}, "W": {"name": "W", "group": "Unnamed group", "definition": "random(50..500#25)", "description": "", "templateType": "anything", "can_override": false}, "T": {"name": "T", "group": "Unnamed group", "definition": "(W d2)/(2 dperp)", "description": "", "templateType": "anything", "can_override": false}, "dperp": {"name": "dperp", "group": "Unnamed group", "definition": "l sin(radians(alpha))", "description": "", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "Unnamed group", "definition": "degrees(arctan((d2+d3)/d1) - arctan(d2/d1))", "description": "", "templateType": "anything", "can_override": false}, "L": {"name": "L", "group": "Unnamed group", "definition": "sqrt(d1^2+d2^2)", "description": "", "templateType": "anything", "can_override": false}, "M_t": {"name": "M_t", "group": "Unnamed group", "definition": "T dperp", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["version", "unit", "d1", "d2", "d3"], "variable_groups": [{"name": "Unnamed group", "variables": ["dperp", "T", "alpha", "W", "L", "M_t"]}], "functions": {}, "preamble": {"js": "\n", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Question", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Solve this problem using the definition of the moment

$(M = F\\; d_\\perp) $

1. Draw an neat labeled free body diagram of the situation.

2. How long is the pole?

$L = $ [[2]]

3. What is the measure of angle $\\angle CAB$? Name this angle $\\alpha$.

$\\alpha = $ [[0]]

4. Write a symbolic equation for the unknown tension $T$ in terms of the weight $W$, length $L$, angle $\\alpha$ and known distances. (Type $d_1$ as d_1, etc.)

$T=$ [[3]]

5. What is the numeric value of the tension in the cable?

$T=$ [[1]]

", "gaps": [{"type": "angle-quantity-from-reference", "useCustomName": true, "customName": "alpha", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_quantity": "siground(qty(alpha,'deg'),4)", "right": "0.2", "restrict_angle": true, "C1": "80", "close": "1.0", "C2": "50", "C3": "50"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "T", "marks": "20", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(T,'lb')", "right": "0.2", "close": "1.0", "C1": "80", "C2": "80", "C3": "60"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "L", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(L,'ft')", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "jme", "useCustomName": true, "customName": "T", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "W d_2/2/L/sin(alpha)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "alpha", "value": ""}, {"name": "d_2", "value": ""}, {"name": "l", "value": ""}, {"name": "w", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Moment of a Force: using Verignon's Theorem", "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": [], "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, statics, moments, verignon's theorem"], "metadata": {"description": "

Calculate the moment of a force about three points using Verignon' theorem. All forces and points are in the same plane.

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

{applet}

Force $\\mathbf{F}$, which has a magnitude of {F}, acts along a line passing through points $D$ and $D'$. The grid units are in [{units[0]}] and counter-clockwise moments are positive. Determine the moment of the force about points $A$ and $B$ using Verignon's Theorem

", "advice": "

Verignon's Theorem: $M = \\pm F_x\\ d_y \\pm F_y\\ d_x$

Verigon's theorem says that the moment of a force is the sum of the moments of its components.

Procedure:

Decompose force $\\mathbf{F}$ into its $x$- and $y$-components.
Determine the horizontal and vertical distances from the moment center to a point on the line of action of $\\mathbf{F}$. In this problem:\n
1. $d_x$ is the horizontal distance from $D$ or $D'$ to point $A$ or $B$.
2. $d_y$ is the vertical distance from $D$ or $D'$ to $A$ or $B$.
\n
The moments of the components are $F_x\\ d_y$ and $F_y\\ d_x$.
The signs of these moments are determined by inspection: positive if counter-clockwise, negative if clockwise.
The total moment is the sum of the component moments.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Fy": {"name": "Fy", "group": "Ungrouped variables", "definition": "qty(force[1],units[1])", "description": "", "templateType": "anything", "can_override": false}, "test2": {"name": "test2", "group": "Inputs", "definition": "some(map(p[0]=q[0] or p[1]=q[1],[p,q],combinations([A,B,C,D,D'],2)))", "description": "

map(sqrt(x^2+y^2),[x,y],[ [3,4], [5,12] ])

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position vector

", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Inputs", "definition": "random(['in','lb'],['ft','lb'],['m','N'],['m','kN'])", "description": "", "templateType": "anything", "can_override": false}, "D": {"name": "D", "group": "Inputs", "definition": "vector(random(-8..8),random(-8..8))", "description": "

Point D

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point B

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Vector Force = F λ

λ = (D'-D)/abs(D'-D)

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point A

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replacement variable for direction of moment.

", "templateType": "anything", "can_override": false}, "rc": {"name": "rc", "group": "Ungrouped variables", "definition": "D-C+vector(0,0,0)\n", "description": "

position vector

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position vector

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Magnitude of force F

", "templateType": "anything", "can_override": false}, "debug": {"name": "debug", "group": "Inputs", "definition": "false", "description": "", "templateType": "anything", "can_override": false}, "test": {"name": "test", "group": "Inputs", "definition": "align(A,B)", "description": "", "templateType": "anything", "can_override": false}, "D'": {"name": "D'", "group": "Inputs", "definition": "vector(random(-8..8),random(-8..8))", "description": "

Point D' on loa

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point C

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Moment of F_x about point A

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Determine the magnitudes of the rectangular components of $\\mathbf{F}$.

$F_x$ = [[0]] $F_y$ = [[1]] $F_x$ = {siground(Fx,4)} $F_y$ = {siground(Fy,4)}

Determine the horizontal and vertical distances from $D$ to $A$.

$d_x$ = [[2]] $d_y$ = [[3]] Delta = {D-A}

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "fx", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "abs(Fx)", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "fy", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "abs(Fy)", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "dx", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "abs(qty((D-A)[0],units[0]))", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "dy", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "abs(qty((D-A)[1],units[0]))", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": " Scalar Components of $M_A$", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Determine the moments of the components. Counterclockwise moment are positive.

The moment of $F_x$ about point $A$ is [[0]] = {siground(mx,4)}

The moment of $F_y$ is about point $A$ is [[1]] = {siground(my,4)}

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Add the moments of the components to determine the moment of force $\\mathbf{F}$ about point $A$ .

$\\mathbf{M}_A$ is a [[0]] moment with a magnitude of [[1]]

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In the same way, determine the moment of force $\\mathbf{F}$ about point $B$ .

$\\mathbf{M}_B$ is a [[0]] moment with a magnitude of [[1]]

$M_b$= {siground(Mb,4)}.

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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.

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Find the tension in a rope necessary to prevent a bracket from rotating by applying $\\Sigma M = 0$.

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

An L shaped bracket is supported by a frictionless pin at $A$ and a cable between points $B$ and $D$.

{applet}

Determine the tension in the cable required for equilibrium when a {F} force $\\mathbf{F}$ is applied to the bracket as shown.

", "advice": "

Since the object is in equilibrium $\\Sigma M_A = 0$ implies that the opposing moments must have equal magnitudes. This is the principle we use to solve the problem. At the start we know the magnitude of $\\mathbf{F}$ and the geometry of the system.

1. Begin by drawing a sketch, defining your symbols and determining geometric values.

Let $\\alpha$ be the angle force $\\mathbf{F}$ makes with the horizontal and $\\beta$ be the angle the cable makes with the horizontal.

$\\alpha = \\var{abs(alpha)}°$, $\\beta = \\var{siground(if(abs(beta)>90, 180-abs(beta), abs(beta)),4)}°$

Let $d_x$ and $d_y$ represent the horizontal and vertical distances from $A$ to $B$.

$d_x = \\var{d_x}$, $d_y = \\var{d_y}$

2. Develop expressions for the moments about point $A$ caused by the force $\\mathbf{F}$ and the cable. The moment caused by the force can be found completely, but the moment caused by the cable tension can only be expressed as a function of the unknown tension $T$.

$M_F = F_y \\cdot d_x = F \\sin \\alpha \\cdot d_x $

$M_T = T_x \\cdot d_y \\pm T_y \\cdot d_x = (T \\cos \\beta) \\, d_y \\pm (T \\sin \\beta) \\, d_x = T (\\var{scalar(d_y)} \\cos \\beta \\pm \\var{scalar(d_x)} \\sin \\beta) $

When calculating the moment due to the cable $M_T$, if the two component moments act in the same direction, add the terms. If they twist in opposite directions, take their difference. Either way, make the final result positive, because we're comparing magnitudes.

3. After substituting the known values and simplifying, equate the moments to get:

$M_T = M_F$

$ T (\\var{format(d_perp_t)}) = \\var{format(M_F)} $

$T = \\dfrac{\\var{format(M_F)}}{\\var{format(d_perp_t)}}= \\var{format(T)} $

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Position of point B

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Position of point A

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Magnitude of force A

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force A as a vector

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direction of force A

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Position of point C

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Determine the magnitude of the moment that the force $\\mathbf{F}$ produces about point $A$.

$M_F =$ [[0]] {M_F}

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Determine the tension required to hold the bracket in position.

$T$ = [[0]] {format(T)}

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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.

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A hand truck on wheels. Easiest to solve by rotating coordinate system.

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A hand truck is used to move a{random(' nitrogen', 'n oxygen', 'n R-134a', ' compressed air', ' refrigerant')} cylinder. Knowing that the combined weight of the truck and cylinder is {W} acting at the center of gravity $G$, determine the vertical force $A$ which must be applied to the handle to maintain the cylinder at this {alpha}° angle, and also the corresponding reaction at each of the two wheels.

", "advice": "

Start by drawing a free body diagram and gathering the known values:

$W = \\var{W}, \\alpha = \\var{alpha}°, d_1 = \\var{d1}, d_2 = \\var{d2}$ and $d_3 = \\var{d3}$.

It's easiest to solve this problem if you resolve the weight into components parallel and perpendicular to the axis of the cylinder, since the dimensions in these directions are given, so:

$W_{\\perp} = W \\cos \\alpha = \\var{display(W * cos(radians(alpha)))}$ and

$W_{\\|} = W \\sin \\alpha = \\var{display(W * sin(radians(alpha)))}$

To find $A$, apply the sum of the moments about point $B$:

$\\begin{align} \\Sigma M_B &= 0\\\\ A d_3 + W_{\\|} d_1 &= W_{\\perp} d_2 \\\\ A &= \\dfrac{ W_{\\perp} d_2 -W_{\\|} d_1 }{d_3}\\\\ &= \\frac{\\var{display(W_perp)}(\\var{d2}) -\\var{display(W_par)}(\\var{d1}) }{\\var{d3}}\\\\&= \\var{display(A)}\\end{align}$

To find B, the force on one wheel, apply the sum of the forces in the y-direction:

$\\begin{align}\\Sigma F_y &= 0\\\\ A - W + 2 B &= 0\\\\B &= \\frac{W-A}{2} \\\\&=\\frac{\\var{W} - \\var{display(A)}}{2}\\\\& = \\var{display(B)}\\end{align}$

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Determine the force on the handle:

$A =$ [[0]]

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Determine the force acting on each wheel:

$B =$ [[0]]

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Angle as a quantity in degrees.

Will check for correct sign elswhere.

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Must have correct sign to be close.

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.

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