// Numbas version: exam_results_page_options {"name": "Friction: Range of values 1", "extensions": ["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": [["question-resources/hw5sol1_nKcXJ7J.png", "/srv/numbas/media/question-resources/hw5sol1_nKcXJ7J.png"], ["question-resources/hw5sol2a.png", "/srv/numbas/media/question-resources/hw5sol2a.png"], ["question-resources/hw5sol2b.png", "/srv/numbas/media/question-resources/hw5sol2b.png"], ["question-resources/PNG_image_2021-01-12_12_05_55.png", "/srv/numbas/media/question-resources/PNG_image_2021-01-12_12_05_55.png"], ["question-resources/PNG_image_2021-01-12_12_06_59.png", "/srv/numbas/media/question-resources/PNG_image_2021-01-12_12_06_59.png"], ["question-resources/PNG_image_2021-01-12_12_09_33.png", "/srv/numbas/media/question-resources/PNG_image_2021-01-12_12_09_33.png"], ["question-resources/PNG_image_2021-01-12_12_10_15.png", "/srv/numbas/media/question-resources/PNG_image_2021-01-12_12_10_15.png"], ["question-resources/block_Q19elOW.svg", "/srv/numbas/media/question-resources/block_Q19elOW.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Friction: Range of values 1", "tags": [], "metadata": {"description": "

Friction

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

Knowing that $\\mu_s = 0.20$, $\\mu_k = 0.15$, and $\\theta=\\var{th}^\\circ$, determine the range of values of $P$ for which equilibrium is maintained.

", "advice": "

$+\\!\\!\\nearrow\\sum F_y=0,\\quad\\implies\\quad N-800 \\cos 25^\\circ+P\\sin(\\theta-25^\\circ)=0$

$N=800 \\cos 25^\\circ-P\\sin( \\var{th-25}^\\circ),$

$F_m=\\mu_s N$

To start moving the block up the incline:

$+\\!\\!\\searrow\\sum F_x=0,\\quad\\implies\\quad \\mu_s N-P\\cos(\\theta-25^\\circ)+800\\sin 25^\\circ=0$

$\\mu_s (800 \\cos 25^\\circ-P\\sin\\var{th-25}^\\circ)-P\\cos\\var{th-25}^\\circ+800\\sin 25^\\circ=0$

$\\mu_s 800 \\cos 25^\\circ-\\mu_s P\\sin\\var{th-25}^\\circ-P\\cos\\var{th-25}^\\circ+800\\sin 25^\\circ=0$

$\\mu_s 800 \\cos 25^\\circ+800\\sin 25^\\circ=+P[\\mu_s \\sin\\var{th-25}^\\circ+\\cos\\var{th-25}^\\circ]$

$P=\\dfrac{800(\\sin 25^\\circ+\\mu_s\\cos 25^\\circ)}{\\cos\\var{th-25}^\\circ+\\mu_s \\sin\\var{th-25}^\\circ}=\\var{Pmax}\\,\\text{N}\\quad\\blacktriangleleft$

To prevent the block from moving down:

$+\\!\\!\\searrow\\sum F_x=0,\\quad\\implies\\quad -\\mu_s N-P\\cos(\\theta-25^\\circ)+800\\sin 25^\\circ=0$

$-\\mu_s (800 \\cos 25^\\circ-P\\sin \\var{th-25}^\\circ)-P\\cos\\var{th-25}^\\circ+800\\sin 25^\\circ=0$

$-\\mu_s 800 \\cos 25^\\circ +\\mu_s P\\sin \\var{th-25}^\\circ- P\\cos \\var{th-25}^\\circ+800\\sin 25^\\circ=0$

$\\mu_s P\\sin \\var{th-25}^\\circ- P\\cos\\var{th-25}^\\circ-\\mu_s 800 \\cos 25^\\circ +800\\sin 25^\\circ=0$

$-\\mu_s 800 \\cos 25^\\circ +800\\sin 25^\\circ=P[\\cos\\var{th-25}^\\circ-\\mu_s \\sin\\var{th-25}^\\circ]$

$P=\\dfrac{800(\\sin 25^\\circ-\\mu_s\\cos 25^\\circ)}{\\cos\\var{th-25}^\\circ-\\mu_s \\sin\\var{th-25}^\\circ}=\\var{Pmin}\\,\\text{N}\\quad\\blacktriangleleft$

\\[\\var{Pmin}\\,\\text{N} \\le P \\le \\var{Pmax}\\,\\text{N}\\quad\\blacktriangleleft\\]

Alternative solution:

Combine the normal and friction forces $\\vec{R}=\\vec{N}+\\vec{F}_\\mu$. When $F_\\mu=\\mu_sN$, the angle between the resultant $\\vec{R}$ and the normal force is the friction angle:

\\[\\phi_s=\\arctan \\mu_s=\\arctan \\var{mu_s}=\\var{phi}^\\circ.\\]

To prevent the block from moving up:

In this case there are only three forces acting on the block, which in view of equilibrium, form a triangle:

In this triangle, the angle between the force $\\vec{R}$ and weight $\\vec{W}$ is $\\phi_s+25^\\circ$. Using the law of sines:

$\\displaystyle\\frac{P}{\\sin (\\phi_s+25^\\circ)}=\\frac{W}{\\sin \\left[180^\\circ-(\\phi_s+25^\\circ)-(90^\\circ-\\theta)\\right]},$

$\\displaystyle\\frac{P}{\\sin\\var{phi+25}^\\circ}=\\frac{800}{\\sin\\var{180-phi-25-90+th}^\\circ},$

$P=800\\dfrac{\\sin\\var{phi+25}^\\circ}{\\sin\\var{180-phi-25-90+th}^\\circ}=\\var{Pmax}\\,\\text{N}.\\,\\blacktriangleleft$

To prevent the block from moving down send the friction force up the incline:

Form a triangle:

In this triangle, the angle between the force $\\vec{R}$ and weight $\\vec{W}$ is $25^\\circ-\\phi_s$. Using the law of sines:

$\\displaystyle\\frac{P}{\\sin (25^\\circ-\\phi_s)}=\\frac{W}{\\sin \\left[180^\\circ-(25^\\circ-\\phi_s)-(90^\\circ-\\theta)\\right]},$

$\\displaystyle\\frac{P}{\\sin\\var{-phi+25}^\\circ}=\\frac{800}{\\sin\\var{180+phi-25-90+th}^\\circ},$

$P=800\\dfrac{\\sin\\var{-phi+25}^\\circ}{\\sin\\var{180+phi-25-90+th}^\\circ}=\\var{Pmin}\\,\\text{N}\\,\\blacktriangleleft$

\\[\\var{Pmin}\\,\\text{N} \\le P \\le \\var{Pmax}\\,\\text{N}.\\quad\\blacktriangleleft\\]

", "rulesets": {}, "extensions": ["quantities"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Pmin": {"name": "Pmin", "group": "Ungrouped variables", "definition": "precround(800*(sin(radians(25))-mu_s*cos(radians(25)))/(cos(radians(th-25))-mu_s*sin(radians(th-25))),1)", "description": "", "templateType": "anything", "can_override": false}, "th": {"name": "th", "group": "Ungrouped variables", "definition": "random(10..70#5 except 25)", "description": "", "templateType": "anything", "can_override": false}, "mu_k": {"name": "mu_k", "group": "Ungrouped variables", "definition": "0.15", "description": "", "templateType": "anything", "can_override": false}, "Pmax": {"name": "Pmax", "group": "Ungrouped variables", "definition": "precround(800*(sin(radians(25))+mu_s*cos(radians(25)))/(cos(radians(th-25))+mu_s*sin(radians(th-25))),1)", "description": "", "templateType": "anything", "can_override": false}, "mu_s": {"name": "mu_s", "group": "Ungrouped variables", "definition": "0.2", "description": "", "templateType": "number", "can_override": false}, "phi": {"name": "phi", "group": "Ungrouped variables", "definition": "precround(degrees(arctan(mu_s)),2)", "description": "", "templateType": "anything", "can_override": false}, "W": {"name": "W", "group": "Ungrouped variables", "definition": "'800 N'", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["th", "mu_s", "mu_k", "Pmin", "Pmax", "phi", "W"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

The minimum force is [[0]].

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "Pmin", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(Pmin, 'N')", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

The maximum force is [[0]].

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "Pmax", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "qty(Pmax,'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", "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}, {"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}, {"name": "Haoyu Huang", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/17657/"}]}]}], "contributors": [{"name": "Vladimir Vingoradov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1862/"}, {"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}, {"name": "Haoyu Huang", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/17657/"}]}