// Numbas version: exam_results_page_options {"name": "33. Frames and Machines 3", "metadata": {"description": "

Homework set. Frame and machine problems.

", "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": "Crossbuck frame", "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.

", "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": [], "metadata": {"description": "

Find all forces on a two-part, symmetrical frame supporting a load.

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

{geogebra_applet('c3ambsph',[['ab',scalar(ab)],['bc',scalar(bc)],['θ',theta+'°'],['r',scalar(r)]])}

A {scalar(w)} {units[0]} cylinder {2 r} in diameter is lodged between two cross pieces which make an angle of {theta}° with each other as shown. The length of each cross piece is {ab} from A to B and {bc} from B to C. Assume that the weight and width of the cross pieces are negligable and that the floor is smooth.

Determine the tension in horizontal rope C-C'.

", "advice": "

Draw a free body diagram of the entire structure and find the reactions at A and A'. Because of symmetry it is apparent that A = A' = W/2.

Next, draw a free body diagram of the cylinder and determine the reaction forces at D and D' which support it.

$\\Sigma F_y = 0$

$2 D_y = W$

$ 2 D \\sin(\\var{theta/2}°) = \\var{w}$

Next, draw a free body diagram of cross piece ABC. Take moments about point B to determine the tension in the cable. Note that the perpendicular distance from B to D is $d_\\perp = r/\\tan(\\var{theta/2}°) $

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"T": {"name": "T", "group": "Ungrouped variables", "definition": "siground((ay ab sin(radians(theta/2)) + \n FD bd)/(bc cos(radians(theta/2))),4)\n\n\n\n", "description": "", "templateType": "anything", "can_override": false}, "debug": {"name": "debug", "group": "inputs", "definition": "false", "description": "", "templateType": "anything", "can_override": false}, "theta": {"name": "theta", "group": "inputs", "definition": "random(50..120#10)", "description": "", "templateType": "anything", "can_override": false}, "W": {"name": "W", "group": "inputs", "definition": "if(unit_sys='si', \n quantity(random(50..1000#10),'N'), \n quantity(random(10..400#25),'lb'))\n\n\n\n", "description": "", "templateType": "anything", "can_override": false}, "r": {"name": "r", "group": "inputs", "definition": "precround(bc * random(0.25..0.75),1)", "description": "", "templateType": "anything", "can_override": false}, "Ay": {"name": "Ay", "group": "Ungrouped variables", "definition": "W/2", "description": "", "templateType": "anything", "can_override": false}, "FD": {"name": "FD", "group": "Ungrouped variables", "definition": "W/(2 sin(radians(theta/2)))", "description": "", "templateType": "anything", "can_override": false}, "L": {"name": "L", "group": "inputs", "definition": "qty(\n if(unit_sys='si', \n random(2..4#0.2),\n random(6..14#2)),\n units[1])\n\n\n", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "inputs", "definition": "if(unit_sys = 'si',['N','m'],['lb','ft'])", "description": "", "templateType": "anything", "can_override": false}, "bd": {"name": "bd", "group": "Ungrouped variables", "definition": "r/tan(radians(theta/2))", "description": "", "templateType": "anything", "can_override": false}, "ab": {"name": "ab", "group": "inputs", "definition": "qty(\n if(unit_sys='si', \n random(1..3#0.2),\n random(3..8)),\nunits[1])\n\n", "description": "", "templateType": "anything", "can_override": false}, "unit_sys": {"name": "unit_sys", "group": "inputs", "definition": "random('us','si')", "description": "", "templateType": "anything", "can_override": false}, "bc": {"name": "bc", "group": "inputs", "definition": "L-ab", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "ab <= L/3", "maxRuns": 100}, "ungrouped_variables": ["Ay", "FD", "bd", "T"], "variable_groups": [{"name": "inputs", "variables": ["L", "ab", "bc", "theta", "W", "r", "debug", "units", "unit_sys"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_2", "useCustomName": true, "customName": "Implications", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

What can be concluded from the problem statement? (Check all true statements.)

", "minMarks": "-5", "maxMarks": 0, "shuffleChoices": false, "displayType": "checkbox", "displayColumns": "2", "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$A_x = A_y$", "$A_x = A'_x$", "$A_x = 0$", "$A_y =0$", "$A_y = A'_y$", "The object is symmetrical.", "There are no horzontal reactions at the floor."], "matrix": ["-1", "1", "1", "-1", "1", "1", "1"], "distractors": ["", "Symmetry implies $A_x = A'_x$.", "Frictionless surface implies $A_x = A'_x =0$.", "", "Symmetry implies $A_y = A'_y$.", "The object and load are symmetrical.", "'smooth' implies a frictionless surface."]}, {"type": "gapfill", "useCustomName": true, "customName": "Reactions at A", "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 the entire structure and determine the components of the reactions at A.

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

$A_y$ = {Ay}

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "$A_x$", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(0,units[0])", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "$A_y$", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "Ay", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Force at D", "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 the cylinder and determine the contact force at D.

$D = $ [[0]]

$D$ = {FD}

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "D", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "FD", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}, {"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": "

Draw a free body diagram of member ABDC and determine the distance from B to D and the tension in the rope, T.

$\\overline{BD}$ = [[0]] $\\qquad T$ = [[1]]

BD = {bd} T = {T}

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "distance", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "bd", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "Tension", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "T", "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": "Machine: lopping shears", "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": [["question-resources/shearFBD.png", "/srv/numbas/media/question-resources/shearFBD.png"], ["question-resources/shearFBD.gif", "/srv/numbas/media/question-resources/shearFBD.gif"], ["question-resources/shear.gif", "/srv/numbas/media/question-resources/shear.gif"], ["question-resources/shearFBD_XWaIQSM.png", "/srv/numbas/media/question-resources/shearFBD_XWaIQSM.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": ["machine", "Mechanics", "mechanics", "rigid body equilibrium", "Rigid body equilibrium", "Statics", "statics"], "metadata": {"description": "

Determine the mechanical advantage of a pair of compound lopping shears.

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

{geogebra_applet('csq3j7yg', {map(dist(n),n,1..9)} )}

Determine

a) the mechanical advantage of the lopping shears shown, and

b) the {statement[version]}.

Note: The mechanical advantage of the shears is the ratio of the cutting force developed at $A$ to the force $P$ applied to the handles.

", "advice": "

1. Draw and number a neat, consistent, labeled set of free body diagrams for the parts of the shears.

Note link DE is a two force member with a slope of $m = \\dfrac{\\var{d_7}} {\\var{d_9 - d_4}}$, in tension. The free body diagrams must be consistent with each other and internal forces must occur in action-reaction pairs.

2. Find the necessary angle.

Let $\\theta$ be the angle that DE makes with the horizontal.

$\\theta = \\tan^{-1} \\dfrac{\\var{d_7}}{\\var{d_9-d_4}} = \\var{siground(theta,4)}°$

3. Use FBD II and take moments about C to find the tension in DE.

$\\begin{align}\\textrm{II: }\\Sigma M_C &= 0 \\\\ DE_x (\\var{d_6}) + DE_y(\\var{d_3}) &= P \\,(\\var{d_3 + d_4 })\\\\ DE &= \\left ( \\dfrac{\\var{scalar(d_3 + d_4 )}}{\\var{scalar(d_6)} \\cos \\theta + \\var{scalar(d_3)}\\sin \\theta} \\right) P \\\\&= \\left(\\dfrac{\\var{d_3 + d_4 }}{\\var{siground(dp_1,4)}} \\right )P \\\\&= \\var{siground(ma_1,4)} \\,P \\textrm{ (tension) } \\end{align}$

4. FBD III and take moments about B to find the cutting force at A.

$\\begin{align}\\textrm{III: }\\Sigma M_B &= 0 \\\\ A (\\var{d_1}) &=DE_x (\\var{d_5 +d_6 +d_7 }) + DE_y(\\var{d_8-d_1}) \\\\ A &= \\left ( \\dfrac{\\var{scalar(d_5 +d_6 +d_7 )}\\cos \\theta + \\var{scalar(d_8-d_1)}\\sin \\theta}{\\var{scalar(d_1 )}} \\right) DE \\\\&= \\left(\\dfrac{\\var{siground(dp_2,4)}}{\\var{d_1 }} \\right )DE \\\\&= \\var{siground(ma_2,4)} \\,DE \\\\ &= \\var{siground(ma_2,4)}\\, ( \\var{siground(ma_1,4)} \\,P )\\\\ &= \\var{siground(ma_t,4)} P \\end{align}$

5. Find the mechanical advantage

Mechanical Advantage = $\\dfrac{A}{P} = \\var{siground(ma_t,4)}$

6. Find the {statement[version]}.

$A = \\var{siground(ma_t,4)} P$, so for $A$ $P$ = {given[version]}:

$A = \\var{siground(ma_t,4)} (\\var{given[0]}) = \\var{siground(answer[0],4)}$

$P = \\dfrac{\\var{given[1]}}{\\var{siground(ma_t,4)}} = \\var{siground(answer[1],4)}$

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"units": {"name": "units", "group": "Ungrouped variables", "definition": "random(['N','mm', \"10 mm\", \"1 mm\" ],['lb','in', \"0.1 in\", \"0.1 in\"])", "description": "

units[2] large rounding base

units[3] small rounding base

", "templateType": "anything", "can_override": false}, "d_6": {"name": "d_6", "group": "Ungrouped variables", "definition": "round((2.59 / 24.4 L random(0.9..1.1#0.05)),units[3])", "description": "", "templateType": "anything", "can_override": false}, "ma_2": {"name": "ma_2", "group": "Unnamed group", "definition": "scalar(dp_2)/scalar(d_1)", "description": "

mechanical advantage 2

", "templateType": "anything", "can_override": false}, "Statement": {"name": "Statement", "group": "Unnamed group", "definition": "[\"shearing force created at A when a \" + \n given[0] + \" force is applied to the handles at P\", \n 'force P that must be applied to the handles in order to create a '\n + given[1] + ' shearing force at A']", "description": "", "templateType": "anything", "can_override": false}, "theta": {"name": "theta", "group": "Unnamed group", "definition": "degrees(atan2(scalar(d_7),scalar(d_9-d_4)))", "description": "", "templateType": "anything", "can_override": false}, "d_9": {"name": "d_9", "group": "Ungrouped variables", "definition": "d_4+ round((0.4 / 24.4 L random(0.9..1.1#0.05)),units[3])", "description": "", "templateType": "anything", "can_override": false}, "dp_1": {"name": "dp_1", "group": "Unnamed group", "definition": "(d_6 cos(radians(theta)) + d_3 sin(radians(theta)))", "description": "

perpendicular distance from DE to C

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mechanical advantage 1

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perpendicular distance from DE to B

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total mechanical advantage

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]]

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Mechanical advantage = [[0]] $\\var{if(version=0,latex('A'),latex('P'))}$ = [[1]]

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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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Determine the forces in the pistons of a hydraulicly operated excavator.

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

{geogebra_applet('cbh4g7zm', {map(dx(n),n,1..6)} + {map(dy(n),n,1..6)} )}

Diagram units are in: [{units[1]}]

", "advice": "

Draw and number neat, labeled free body diagrams.

Given: $W$ = {load}

Find the necessarys angles.

Let $\\theta_B$ and $\\theta_D$ be the angles that BC and DE make with the horizontal.

$\\theta_B = \\tan^{-1} \\left(\\dfrac{\\var{y1+y2}}{\\var{x2}}\\right) = \\var{siground(theta_B,4)}° \\qquad \\theta_D = \\tan^{-1} \\left(\\dfrac{\\var{y4}}{\\var{x4}} \\right)= \\var{siground(theta_D,4)}°$

Use Free Body Diagram I to solve for the force in cylinder $BC$.

$\\begin{align} \\textrm{I: } \\Sigma M_A &= 0 \\\\ BC_x ( \\var{y2}) - BC_y(\\var{x1 + x2}) + W ( \\var{x_t}) &= 0\\\\ BC ( -\\var{scalar(y2)} \\cos \\theta_B + \\var{scalar(x1+x2)} \\sin \\theta_B) &= \\var{scalar(x_t)}\\ W \\\\ BC &= W \\left(\\dfrac{\\var{scalar(x_t)}}{-\\var{scalar(y2)} \\cos \\var{siground(theta_B,4)}° + \\var{scalar(x1+x2)} \\sin \\var{siground(theta_B,4)}°}\\right) \\\\ &= \\var{load} \\left(\\dfrac{\\var{x_t}}{\\var{siground(dperp_B,4)}}\\right) \\\\ &= \\var{siground(BC,4)} \\textrm{ (compression)} \\end{align}$

Use Free Body Diagram II to solve for the force in cylinder $DE$.

$\\begin{align} \\textrm{II: } \\Sigma M_F &= 0 \\\\ -DE_x ( \\var{y5}) - DE_y(\\var{x5}) + W ( \\var{x6}) &= 0\\\\ DE ( \\var{scalar(y5)} \\cos \\theta_D + \\var{scalar(x5)} \\sin \\theta_d) &= \\var{scalar(x6)} W \\\\ DE &= W \\left(\\dfrac{\\var{x6}}{\\var{siground(dperp_D,4)}}\\right) \\\\ &= \\var{siground(DE,4)} \\textrm{ (tension)} \\end{align}$

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units[2] large rounding base

units[3] small rounding base

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scaling value for all lengths, which are also randomly perterbed actual value as scaled.

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perpendicular distance from A to force BC

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The motion of the excavator is controled by three hydraulic cylinders. Determine the force exerted by cylinders $BC$ and $DE$ when supporting a {load} downward vertical load at $G$. Neglect the weights of the parts.

$BC$ = [[0]] [[1]]

$DE$ = [[2]] [[3]]

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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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