// Numbas version: finer_feedback_settings {"name": "06. Multiparticle Equilibrium", "metadata": {"description": "

Homework set. Problems involving two interacting particles.

", "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 particle: two suspended loads", "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": ["2-d", "2-D", "equilibrium", "Equilibrium", "mechanics", "Mechanics", "particle", "Particle", "statics", "Statics"], "metadata": {"description": "

Find the tensions is a system of cables supporting two loads.

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

{geogebra_applet('vzqtfrbq',[['a1',a1+\"°\"],['a2',a2+\"°\"],['a3',a3+\"°\"]])}

Two loads $W_1 = \\var{w1}$, and $W_2 = \\var{w2}$ are suspended from a system of cables as shown. Determine the tensions in the cables.

", "advice": "

Begin by drawing a free body diagram of particle A, then apply the equations of equilibrium.

$\\begin{align}\\\\ \\Sigma F_x = 0 \\\\-{T_B}_x + {T_C}_x = 0 \\end{align} \\\\ T_B \\cos(\\var{a1}°) = T_C \\sin(\\var{90-a2}°) \\qquad(1)$

$\\begin{align}\\\\ \\Sigma F_y = 0 \\\\{T_B}_y + {T_C}_y - W_1 = 0 \\end{align} \\\\ T_B \\sin(\\var{a1}°) + T_C \\cos(\\var{90-a2})° = W_1 \\qquad(2)$

Solve (1) and (2) simultaneously for $T_B$ and $T_C$.

$T_B = \\var{siground(TB,4)}, \\quad T_C = \\var{siground(TC,4)}$

With $T_C$ known, use a similar approach on particle $C$ to find $T_D$ and $T_E$.

Since cable CD is horizontal $T_D$ has no y-component, so the $\\Sigma F_y = 0$ equation only has one unknown and simultaneous equations are unnecessary in this case.

$T_D = \\var{siground(TD,4)}, \\quad T_E = \\var{siground(TE,4)}$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"forceC": {"name": "forceC", "group": "check", "definition": "vector(cos(radians(a2)),sin(radians(a2))) scalar(TC)\n", "description": "", "templateType": "anything", "can_override": false}, "forceE": {"name": "forceE", "group": "check", "definition": "vector(cos(radians(a3)),sin(radians(a3))) scalar(TE)", "description": "", "templateType": "anything", "can_override": false}, "TB": {"name": "TB", "group": "magnitudes", "definition": "w1 cos(radians(a2))/sin(radians(a1+a2))", "description": "", "templateType": "anything", "can_override": false}, "forceD": {"name": "forceD", "group": "check", "definition": "vector(scalar(TD),0)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "input", "definition": "random(10..70#5)", "description": "

a2 is the angle from the horizontal to AC. Note angle shown on diagram is the complement of a2

", "templateType": "anything", "can_override": false}, "debug": {"name": "debug", "group": "input", "definition": "false", "description": "", "templateType": "anything", "can_override": false}, "TE": {"name": "TE", "group": "magnitudes", "definition": "(W2 + TC sin(radians(A2)))/sin(radians(A3))", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "input", "definition": "random(10..80#5)", "description": "

angle of rope AB from horizontal, as shown on diagram.

", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "input", "definition": "random(20..70#5)", "description": "

angle of CE as shown on diagram.

", "templateType": "anything", "can_override": false}, "w2": {"name": "w2", "group": "input", "definition": "qty(random(10..100#5),units)", "description": "", "templateType": "anything", "can_override": false}, "TC": {"name": "TC", "group": "magnitudes", "definition": "w1 cos(radians(a1))/sin(radians(a1+a2))\n\n", "description": "", "templateType": "anything", "can_override": false}, "w1": {"name": "w1", "group": "input", "definition": "qty(random(10..100#5),units)", "description": "", "templateType": "anything", "can_override": false}, "forceB": {"name": "forceB", "group": "check", "definition": "vector(-cos(radians(A1)),sin(radians(A1))) scalar(TB)", "description": "", "templateType": "anything", "can_override": false}, "TD": {"name": "TD", "group": "magnitudes", "definition": "TC cos(radians(a2))-TE cos(radians(a3))", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "input", "definition": "'lb'", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "a1 <> a2 and a3>a2+10 and scalar(TD) > 0", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "input", "variables": ["a1", "a2", "a3", "w1", "w2", "units", "debug"]}, {"name": "magnitudes", "variables": ["TC", "TE", "TD", "TB"]}, {"name": "check", "variables": ["forceB", "forceC", "forceE", "forceD"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "First Particle", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Draw a free body diagram of point A and apply the equations of equilibrium to determine the tensions in cables AB and AC.

$T_{AB}$ = [[0]] {siground(TB,4)}

$T_{AC}$ = [[1]] {siground(TC,4)}

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Draw a free body diagram of point C and apply the equations of equilibrium to determine the tensions in cables CE and CD.

$T_{CE}$ = [[0]] {siground(TE,4)}

$T_{CD}$ = [[1]] {siground(TD,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.

", "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", "Free Body Diagrams", "Mechanics", "mechanics", "Particle", "particle", "Statics", "statics"], "metadata": {"description": "

Equilibrium of two interacting particles.

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

<p\">{geogebra_applet('dmn6ddbb', [[\"width\", width], [\"r1\",r1], [\"r2\",r2]] )}

Two cylinders are placed in a {width} {units[0]} wide container. The blue cylinder has a radius $r_1$ = {d1/2} {units[0]} and weighs {w1} {units[1]} and the yellow cylinder has a radius $r_2$ = {d2/2} {units[0] } and weighs {w2} {units[1]}.

", "advice": "

Given:

$d$ = {width} {units[0]} $\\qquad r_1$ = {r1} {units[0]} $\\qquad W_1 = $ {w1} {units[1]} $\\qquad r_2$ = {r2} {units[0]} $\\qquad W_2 = $ {w2} {units[1]}

Draw free body diagrams of the two cylinders.

{geogebra_applet('rchz6eew', [[\"width\", width], [\"r1\",r1], [\"r2\",r2]] )}

Find angle $\\theta$ from the geometry of the problem.

$\\theta =cos^{-1} \\left( \\dfrac{d - r_1- r_2}{r_1+r_2} \\right) = \\var{siground(theta,4)}°$

Find force $A$ using equilibrium of the yellow cylinder

$\\begin{align} \\Sigma F_y &= 0\\\\A_y - W_2 &= 0\\\\ A \\sin \\theta &= W_2\\\\ A &= W_2\\,/\\,\\sin \\theta &=\\var{show(A)}\\end{align}$

Find forces $B$ and $C$ using equilibrium of the blue cylinder

$\\begin{align} \\Sigma F_y &= 0 & \\Sigma F_x &=0\\\\ B - A_y - W_1 &= 0 & C-A_x &=0\\\\ B &= W_1 + A \\sin \\theta & C &= A \\cos\\theta \\\\&= \\var{show(B)} & &=\\var{show(C)} \\end{align}$

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distance from center to center

", "templateType": "anything", "can_override": false}, "width": {"name": "width", "group": "Ungrouped variables", "definition": "random(d1..d1+d2)", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Ungrouped variables", "definition": "['ft','lb']", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(4..12 #2)", "description": "

diameter 1

", "templateType": "anything", "can_override": false}, "r1": {"name": "r1", "group": "calculated", "definition": "d1/2", "description": "", "templateType": "anything", "can_override": false}, "w1": {"name": "w1", "group": "Ungrouped variables", "definition": "random(1000..5000#250)", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(d1/2 .. d1)", "description": "

diameter 2

", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "calculated", "definition": "w2/sin(radians(theta))", "description": "", "templateType": "anything", "can_override": false}, "w2": {"name": "w2", "group": "Ungrouped variables", "definition": "siground(w1 d2^2/d1^2,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "base < r and width > d1 and d1 <> d2", "maxRuns": 100}, "ungrouped_variables": ["width", "d1", "d2", "units", "w1", "w2"], "variable_groups": [{"name": "calculated", "variables": ["base", "r1", "r2", "r", "theta", "A", "B", "C"]}], "functions": {"show": {"parameters": [["f", "number"]], "type": "number", "language": "jme", "definition": "qty(siground(f,4),units[1])"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Answers", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Assuming smooth contact surfaces, determine the magnitudes of Force A, where the two cylinders touch, Force B where the blue cylinder touches the bottom of the trough, and Force C where it touches the left wall.

$A$ = [[0]] $B$ = [[1]] $C$ = [[2]]

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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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Pulley supported by a cable, so the tension in the rope is constant. Advice is a youtube video showing how to solve the problem.

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{geogebra_applet('qbvd5vx9',['L': length, 'W': width, 'd': offset, 'units': '\"' + units[0] + '\"'])}

Cable $ABC$ has a length of {qty(length, units[0])}. Determine the position $x$ and the tension in $ABC$ required for equilibrium when the cable supports a $P = \\var{load}$ load riding on a frictionless pulley of negligable radius.

", "advice": "

See this video

$\\alpha = \\var{siground(alpha,2)}° $

$x = \\var{x}$

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angle ropes make with horzontal

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$x$ = [[1]] $\\qquad T$ = [[0]]

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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.