// Numbas version: exam_results_page_options {"name": "14. More Equilibrium Problems", "metadata": {"description": "

Homework set. Find reactions for various 2-D rigid bodies in equilibrium.

", "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": "Equilibrium of a rigid body: raise pole", "extensions": ["geogebra", "quantities"], "custom_part_types": [{"source": {"pk": 12, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/12/edit"}, "name": "Angle quantity 2020", "short_name": "angle", "description": "

Adjusts all angles to 0 < $\\theta$ < 360.

Accepts '°' and 'deg' as units.

Penalizes if not close enough or no units.

90° = -270° = 450°

", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "plain_string(settings['expected_answer']) ", "hint": {"static": true, "value": ""}, "allowEmpty": {"static": true, "value": false}}, "can_be_gap": true, "can_be_step": true, "marking_script": "original_student_scalar:\nmatchnumber(studentAnswer,['plain','en'])[1]\n\nstudent_scalar:\nmod(original_student_scalar,360)\n\n\nstudent_unit:\nstudentAnswer[len(matchnumber(studentAnswer,['plain','en'])[0])..len(studentAnswer)]\n\ninterpreted_unit:\nif(trim(student_unit)='\u00b0','deg',student_unit)\n\ninterpreted_answer:\nqty(mod(student_scalar,360),'deg')\n\nclose:\nwithintolerance(student_scalar, correct_scalar,decimal(settings['close_tol']))\n\ncorrect_scalar:\nmod(scalar(settings['expected_answer']),360)\n\nright:\nwithintolerance(student_scalar, correct_scalar, decimal(settings['right_tol']))\n\ngood_unit:\nsame(qty(1,interpreted_unit),qty(1,'deg'))\n\nmark:\nassert(close,incorrect('Incorrect.');end());\nif(right,correct('Correct angle.'), set_credit(1 - settings['close_penalty'],'Angle is close.'));\nassert(good_unit,sub_credit(settings['unit_penalty'], 'Missing or incorrect units.'))", "marking_notes": [{"name": "original_student_scalar", "description": "

Retuns the scalar part of students answer (which is a quantity) as a number.

", "definition": "matchnumber(studentAnswer,['plain','en'])[1]"}, {"name": "student_scalar", "description": "

Normalize angle with mod 360

", "definition": "mod(original_student_scalar,360)\n"}, {"name": "student_unit", "description": "

matchnumber(studentAnswer,['plain','en'])[0] is a string \"12.34\"

", "definition": "studentAnswer[len(matchnumber(studentAnswer,['plain','en'])[0])..len(studentAnswer)]"}, {"name": "interpreted_unit", "description": "

Allows student to use degree symbol or 'deg' for units.

", "definition": "if(trim(student_unit)='\u00b0','deg',student_unit)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "qty(mod(student_scalar,360),'deg')"}, {"name": "close", "description": "", "definition": "withintolerance(student_scalar, correct_scalar,decimal(settings['close_tol']))"}, {"name": "correct_scalar", "description": "

Normalize expected_answer with mod 360

", "definition": "mod(scalar(settings['expected_answer']),360)"}, {"name": "right", "description": "", "definition": "withintolerance(student_scalar, correct_scalar, decimal(settings['right_tol']))"}, {"name": "good_unit", "description": "", "definition": "same(qty(1,interpreted_unit),qty(1,'deg'))"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "assert(close,incorrect('Incorrect.');end());\nif(right,correct('Correct angle.'), set_credit(1 - settings['close_penalty'],'Angle is close.'));\nassert(good_unit,sub_credit(settings['unit_penalty'], 'Missing or incorrect units.'))"}], "settings": [{"name": "expected_answer", "label": "Expected Answer", "help_url": "", "hint": "Expected angle as a quantity.", "input_type": "code", "default_value": "qty(30,'deg')", "evaluate": true}, {"name": "unit_penalty", "label": "Unit penalty", "help_url": "", "hint": "Penalty for not including degree sign or 'deg'.", "input_type": "percent", "default_value": "20"}, {"name": "close_penalty", "label": "Close Penalty", "help_url": "", "hint": "Penalty for close answer.", "input_type": "percent", "default_value": "20"}, {"name": "close_tol", "label": "Close", "help_url": "", "hint": "Angle must be $\\pm$ this many degrees to be marked close. ", "input_type": "code", "default_value": "0.5", "evaluate": false}, {"name": "right_tol", "label": "Right ", "help_url": "", "hint": "Angle must be $\\pm$ this many degrees to be marked correct. ", "input_type": "code", "default_value": "0.1", "evaluate": false}], "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": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["angle from reference", "Equilibrium", "equilibrium", "Mechanics", "mechanics", "Rigid Body", "rigid body", "Statics", "statics"], "metadata": {"description": "

Equilibrium of a rigid body. Find tension to raise a pole.

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

{geogebra_applet('pf8c3vcn',[['α',alpha +'°'],['β',beta +'°']])}

A {mass} homogeneous pole with a length of {L} is being raised by pulling with a cable at $B$.

", "advice": "

This problem can be solved by the standard method:

Draw a free body diagram of the pole. Include the weight of the pole at the center of gravity. Note that mass must be converted to weight.
Use $\\Sigma M_A = 0$ to find $T$.
Use $\\Sigma F_x = 0$ and $\\Sigma F_y = 0$ to find $A_x$ and $A_y$.
Resolve $A_x$ and $A_y$ to find magnitude and direction of $\\mathbf{A}$.

It can also be solved using the three force body principle.

Find the point of intersection of the lines of action of the weight and the tension.
Determine the direction of force $\\mathbf{A}$. Its line of action must pass through the point.
Draw a force triangle.
Determine the angles in the force triangle.
Use the known weight and the law of sines to solve for the tension and the reaction at $A$.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"L": {"name": "L", "group": "inputs", "definition": "qty(random(3..5#0.2),'m')", "description": "", "templateType": "anything", "can_override": false}, "theta_T": {"name": "theta_T", "group": "Ungrouped variables", "definition": "180 + alpha - beta", "description": "", "templateType": "anything", "can_override": false}, "T": {"name": "T", "group": "Ungrouped variables", "definition": "scalar(weight) cos(radians(alpha))/(2 sin(radians(beta)) )\nvector(cos(radians(theta_t)),sin(radians(theta_t)))", "description": "", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "-(W+T)", "description": "", "templateType": "anything", "can_override": false}, "W": {"name": "W", "group": "Ungrouped variables", "definition": "vector(0,-scalar(weight))", "description": "", "templateType": "anything", "can_override": false}, "theta_A": {"name": "theta_A", "group": "Ungrouped variables", "definition": "mod(degrees(atan2(A[1],A[0])),360)", "description": "", "templateType": "anything", "can_override": false}, "mass": {"name": "mass", "group": "inputs", "definition": "qty(random(10..150#5),'kg')", "description": "", "templateType": "anything", "can_override": false}, "beta": {"name": "beta", "group": "inputs", "definition": "random(20..(90+alpha-20))", "description": "

angle between pole and rope

", "templateType": "anything", "can_override": false}, "weight": {"name": "weight", "group": "inputs", "definition": "mass * qty(9.81,'m/s^2') in 'N'", "description": "", "templateType": "anything", "can_override": false}, "debug": {"name": "debug", "group": "Ungrouped variables", "definition": "false", "description": "", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "inputs", "definition": "random(20..80)", "description": "

angle of pole

", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["W", "T", "theta_T", "A", "theta_A", "debug"], "variable_groups": [{"name": "inputs", "variables": ["alpha", "mass", "beta", "weight", "L"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Tension", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

What is the tension in the cable?

$T$= [[0]] T = {siground(abs(T),4)} N

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "Tension", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(abs(T),'N')", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Reaction at A", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "interpreted_angle: // a qty string corrected to standard angle\n student_angle[2] + student_angle[1] * student_angle[0] + student_units\n\nmagnitude:\n studentAnswer[3]\n\nstudent_angle:\n [mod(matchnumber(studentAnswer[0],['plain','en'])[1],360), // angle\n [1,-1][indices(studentAnswer[1],[true])[0]], // ccw = 1 cw = -1\n [0,90,180,-90][indices(studentAnswer[2],[true])[0]]] // reference axis\n\nstudent_units:\n studentAnswer[0][len(matchnumber(studentAnswer[0],['plain','en'])[0])..len(studentAnswer[0])]\n\ninterpreted_answers:\n [interpreted_angle, studentAnswer[1], studentAnswer[2], studentAnswer[3]]\n\ngap_feedback (Feedback on each of the gaps):\n map(\n try(\n let(\n result, submit_part(gaps[gap_number][\"path\"],answer),\n gap, gaps[gap_number],\n name, gap[\"name\"], \n noFeedbackIcon, not gap[\"settings\"][\"showFeedbackIcon\"],\n assert(name=\"\" or len(gaps)=1,feedback(translate('part.gapfill.feedback header',[\"name\": name])));\n concat_feedback(filter(x[\"op\"]<>\"warning\",x,result[\"feedback\"]), if(marks>0,result[\"marks\"]/marks,1), noFeedbackIcon);\n result\n ),\n err,\n fail(translate(\"part.gapfill.error marking gap\",[\"name\": gaps[gap_number][\"name\"], \"message\": err]))\n ),\n [gap_number,answer,index],\n zip([3,0],[studentAnswer[3], interpreted_angle],[1,2])\n )\n\n", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

What is the magnitude and direction of the reaction force at point A?

$\\mathbfy{A}$ = [[3]]

at an angle of [[0]] measured [[1]] from the [[2]].

A= {siground(abs(A),4)} N @ {siground(theta_A,4)}°

", "gaps": [{"type": "angle", "useCustomName": true, "customName": "angle", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"expected_answer": "precround(qty(theta_A,'deg'),2)", "unit_penalty": "20", "close_penalty": "20", "close_tol": "0.5", "right_tol": "0.2"}}, {"type": "1_n_2", "useCustomName": true, "customName": "sign", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["CCW", "CW"], "matrix": [0, 0], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "ref", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["+x axis", "+y axis", "-x axis", "-yaxis"], "matrix": [0, 0, 0, 0], "distractors": ["", "", "", ""]}, {"type": "engineering-answer", "useCustomName": true, "customName": "Mag A", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(siground(abs(A),4),'N')", "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": "Equilibrium of a rigid body: triangle", "extensions": ["geogebra", "quantities"], "custom_part_types": [{"source": {"pk": 12, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/12/edit"}, "name": "Angle quantity 2020", "short_name": "angle", "description": "

Adjusts all angles to 0 < $\\theta$ < 360.

Accepts '°' and 'deg' as units.

Penalizes if not close enough or no units.

90° = -270° = 450°

Retuns the scalar part of students answer (which is a quantity) as a number.

", "definition": "matchnumber(studentAnswer,['plain','en'])[1]"}, {"name": "student_scalar", "description": "

Normalize angle with mod 360

", "definition": "mod(original_student_scalar,360)\n"}, {"name": "student_unit", "description": "

matchnumber(studentAnswer,['plain','en'])[0] is a string \"12.34\"

", "definition": "studentAnswer[len(matchnumber(studentAnswer,['plain','en'])[0])..len(studentAnswer)]"}, {"name": "interpreted_unit", "description": "

Allows student to use degree symbol or 'deg' for units.

Normalize expected_answer with mod 360

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": ["angle from reference", "equilibrium", "Equilibrium", "Mechanics", "mechanics", "rigid body", "Rigid Body", "Statics", "statics"], "metadata": {"description": "

Find the reactions of a rigid body (a triangular plate) at a pin and roller. The load is at an angle.

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

{applet}

The triangular plate is secured by a pin at $A$ and a roller at $C$. Determine the reactions at the pin and the roller when the plate is subjected to a {FB} load as shown by applying the equations of equilibrium.

B = ({abs(B)};{mod(degrees(atan2(B[1],B[0]))+360,360)}) {units[0]}

", "advice": "

Draw a free body diagram.
Apply $\\Sigma M_A$=0 to find the reaction at $C$. There's no x-component at $C$, because the support there is a roller. This should have been indicated on your free body diagram.
Once $C$ and is known, apply $\\sum F_x = 0$ and $\\sum F_y=0$ to find components $A_x$ and $A_y$.
With $A_x$ and $A_y$ known, use the pathagorean theorem to calculate the magnitude of $A$, and use trig to get the its direction.

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reference direction. Set by student choice.

", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "-(B+C)", "description": "", "templateType": "anything", "can_override": false}, "L": {"name": "L", "group": "Inputs", "definition": "random(2..6#0.2)", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "magB vector(cos(radians(theta)),sin(radians(theta)),0)", "description": "", "templateType": "anything", "can_override": false}, "dirA": {"name": "dirA", "group": "Ungrouped variables", "definition": "mod(degrees(atan2(A[1],A[0])),360)", "description": "", "templateType": "anything", "can_override": false}, "magB": {"name": "magB", "group": "Inputs", "definition": "random(1..15)random(10,20,50)", "description": "", "templateType": "anything", "can_override": false}, "rb": {"name": "rb", "group": "Ungrouped variables", "definition": "L/2 vector(1,tan(radians(alpha)),0)", "description": "

position vector to point B

", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "Inputs", "definition": "65//random(25..65#5)", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Inputs", "definition": "random([['kN','m'],['lb','ft'],['N','cm'],['lb','in']])", "description": "", "templateType": "anything", "can_override": false}, "applet": {"name": "applet", "group": "Inputs", "definition": "geogebra_applet('n3rkawce',params)", "description": "", "templateType": "anything", "can_override": false}, "params": {"name": "params", "group": "Inputs", "definition": "[text1: '\" '+ string(L) +' ' + units[1] + ' \"', alpha: radians(alpha), theta: radians(theta)]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "theta <> 180 + alpha and theta <> 360 + alpha \nand C[1] > 0", "maxRuns": 100}, "ungrouped_variables": ["B", "rb", "C", "A", "dirA", "FB"], "variable_groups": [{"name": "Inputs", "variables": ["theta", "units", "debug", "alpha", "magB", "L", "applet", "params"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Reaction at C", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Draw and label a neat free body diagram of the triangular plate.
Determine any needed distance or angles.
Use $\\Sigma M_A = 0$ to find the reaction at C.

$C = $ [[0]]

C = {abs(C)} {units[0]}

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Use $\\Sigma F_x = 0$ and $\\Sigma F_y=0$ to find the magnitude and direction of the components of the reaction at A.

$A_x = $ [[0]] [[1]]

$A_y = $ [[2]] [[3]]

A= {A} {units[0]}

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Determine the magnitude and direction of the resultant force $\\textbf{A}$ by resolving the components found in part b).

Indicate direction as an angle less than 90° measured from a chosen reference direction. Positive angle = counterclockwise.

Force $\\textbf{A}$ has a magnitude of [[3]]

acting at an angle of [[0]] measured [[1]] from the [[2]].

A= ({abs(A)};{mod(degrees(atan2(A[1],A[0]))+360,360)}) {units[0]}

", "gaps": [{"type": "angle", "useCustomName": true, "customName": "angle", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"expected_answer": "qty(dirA,'deg')", "unit_penalty": "20", "close_penalty": "20", "close_tol": "0.5", "right_tol": "0.2"}}, {"type": "1_n_2", "useCustomName": true, "customName": "sign", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["CCW", "CW"], "matrix": [0, 0], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": true, "customName": "ref", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["+x axis", "+y axis", "-x axis", "-y axis"], "matrix": [0, 0, 0, 0], "distractors": ["", "", "", ""]}, {"type": "engineering-answer", "useCustomName": true, "customName": "Mag A", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(abs(A),units[0])", "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": "Equilibrium of a rigid body: Car on a hill", "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": ["Equilibrium", "equilibrium", "Mechanics", "mechanics", "Rigid Body", "rigid body", "Statics", "statics", "three force body", "Three force body"], "metadata": {"description": "

Classic problem of a vehicle parked on an incline. Best solved by rotating the coordinate system.

Image Credit: https://svgsilh.com/image/34325.html CC-0

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

{geogebra_applet('w4xb6n95',['alpha': radians(alpha), 'fbd': [visible: false] ])}

A car is parked on a {alpha}° slope, with the rear wheels locked by the parking brake. Find the components of the forces acting on each of the front and rear wheels, parallel and perpendicular to the roadbed.

Vehicle Specifications

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

Wheelbase	$L = $ {wb} mm
Curb weight	$m = $ {m} kg
Center of Gravity	$d = $ {d} mm $h = $ {h} mm

", "advice": "

{geogebra_applet('w4xb6n95',['alpha': radians(alpha), 'fbd': [definition: 'true', visible: true] ])}

First, draw a free body diagram of the vehicle.

For convenience, let the x- and y- directions be parallel and perpendicular to the roadbed and define $\\alpha$ as the the angle the road makes with the horizontal. Note that $\\alpha$ is also the angle between the weight and the y-direction. Treat the front wheels like a roller, the rear wheels like a rough surface.

$\\alpha = \\var{alpha}°$

Define $L$ as the distance between the front and rear wheels:

$L= \\var{qty(wb,'mm')}$

Note that the curb 'weight' is actually a mass, so let:

$\\begin{align}W &= m g\\\\ &= (\\var{qty(m,'kg')}) \\cdot (\\var{qty(g, 'm/s^2')}) =\\var{show(w)}\\\\ W_x& = W \\sin \\alpha = \\var{show(w_x)}\\\\W_y& = W \\cos \\alpha = \\var{show(w_y)}\\end{align}$

Apply the equations of equilibrium, starting with the sum of the moments about either A or B, since two unknown forces intersect there.

$\\begin{align}\\Sigma M_A &= 0\\\\ B_y (L) &= W_y(d) + W_x (h)\\\\ B_y &= W \\left(\\dfrac{d \\cos \\alpha + h \\sin \\alpha}{L}\\right)\\\\&= \\var{show(B_y)}\\\\\\\\ \\Sigma F_x &= 0\\\\ B_x &= W_x\\\\ B_x &= \\var{show(w_x)}\\\\\\\\ \\Sigma F_y &= 0\\\\A + B_y &= W_y \\\\A &= W_y-B_y\\\\&=\\var{show(A)}\\end{align}$

Since there are two front wheels and two rear wheels, the forces acting on each wheel are half of these values:

$A_{||} = 0 \\text{ N} \\qquad A_\\perp = A/2 = \\var{show(A/2)} \\qquad B_{||} = B_x/2 = \\var{show(B_x/2)} \\qquad B_\\perp = B_y/2 = \\var{show(B_y/2)}$

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curb weight in kg

", "templateType": "anything", "can_override": false}, "W": {"name": "W", "group": "answers", "definition": "m g ", "description": "

weight of vechicle

", "templateType": "anything", "can_override": false}, "w_x": {"name": "w_x", "group": "answers", "definition": "W sin(radians(alpha))", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "9.81", "description": "", "templateType": "anything", "can_override": false}, "wb": {"name": "wb", "group": "Ungrouped variables", "definition": "random(2450..2580)", "description": "

wheelbase in mm

", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "Ungrouped variables", "definition": "random([10,15,20,25,30])", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Ungrouped variables", "definition": "['N','mm']", "description": "", "templateType": "anything", "can_override": false}, "debug": {"name": "debug", "group": "answers", "definition": "false", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alpha", "m", "wb", "units", "d", "h", "g"], "variable_groups": [{"name": "answers", "variables": ["w_y", "A", "w_x", "B_x", "B_y", "W", "debug"]}], "functions": {"show": {"parameters": [["f", "number"]], "type": "number", "language": "jme", "definition": "siground(qty(f,'N'),4)"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {"mark": {"script": "//numbasGGBApplet0.setValue('fbd',true)\n//numbasGGBApplet0.setVisible('fbd',true)", "order": "after"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$A_{||} =$ [[0]] $\\qquad A_\\perp = $ [[3]] $0 B_{\\perp} = \\var{qty(A/2 ,'N')}$

$B_{||} =$ [[1]] $\\qquad B_\\perp = $ [[2]] $B_{||}= \\var{qty(B_x/2,'N')} B_{\\perp} = \\var{qty(B_y/2,'N')}$

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Adjusts all angles to 0 < $\\theta$ < 360.

Accepts '°' and 'deg' as units.

Penalizes if not close enough or no units.

90° = -270° = 450°

Retuns the scalar part of students answer (which is a quantity) as a number.

", "definition": "matchnumber(studentAnswer,['plain','en'])[1]"}, {"name": "student_scalar", "description": "

Normalize angle with mod 360

", "definition": "mod(original_student_scalar,360)\n"}, {"name": "student_unit", "description": "

matchnumber(studentAnswer,['plain','en'])[0] is a string \"12.34\"

", "definition": "studentAnswer[len(matchnumber(studentAnswer,['plain','en'])[0])..len(studentAnswer)]"}, {"name": "interpreted_unit", "description": "

Allows student to use degree symbol or 'deg' for units.

Normalize expected_answer with mod 360

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.