// Numbas version: exam_results_page_options
{"name": "HW5 Prob3", "extensions": [], "custom_part_types": [], "resources": [["question-resources/hw5prob3.png", "/srv/numbas/media/question-resources/hw5prob3.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "HW5 Prob3", "tags": [], "metadata": {"description": "<p>Friction</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p><img src=\"resources/question-resources/hw5prob3.png\" width=\"220\" height=\"246\"/></p>\n<p>The coefficient of friction between the block and the rail is $\\mu_s=\\var{mu_s}$. Knowing that $\\theta=\\var{th}^\\circ$, determine the smallest value of <em>P</em>&nbsp;(in N) required&nbsp;</p>", "advice": "<p>$+\\!\\!\\nwarrow\\sum F_y=0,\\quad\\implies\\quad N-500 \\cos 35^\\circ+P\\sin(\\theta-35^\\circ)=0$</p>\n<p style=\"padding-left: 30px;\">$N=500 \\cos 35^\\circ-P\\sin( \\var{th-35}^\\circ),$</p>\n<p style=\"padding-left: 30px;\">$F_m=\\mu_s N$</p>\n<p>To start moving the block up the incline:</p>\n<p></p>\n<p>$+\\!\\!\\swarrow\\sum F_x=0,\\quad\\implies\\quad \\mu_s N-P\\cos(\\theta-35^\\circ)+500\\sin 35^\\circ=0$</p>\n<p style=\"padding-left: 30px;\">$\\mu_s (500 \\cos 35^\\circ-P\\sin \\var{th-35}^\\circ)-P\\cos \\var{th-35}^\\circ+500\\sin 35^\\circ=0$</p>\n<p style=\"padding-left: 30px;\">$P=\\dfrac{500\\,(\\sin 35^\\circ+\\mu_s\\cos 35^\\circ)}{\\cos \\var{th-35}^\\circ+\\mu_s \\sin \\var{th-35}^\\circ}=\\var{Pmax}\\,\\text{N}\\quad\\blacktriangleleft$</p>\n<p></p>\n<p>To prevent the block from moving down:</p>\n<p>$+\\!\\!\\swarrow\\sum F_x=0,\\quad\\implies\\quad -\\mu_s N-P\\cos(\\theta-35^\\circ)+500\\sin 35^\\circ=0$</p>\n<p style=\"padding-left: 30px;\">$-\\mu_s&nbsp;(500 \\cos 35^\\circ-P\\sin \\var{th-35}^\\circ)-P\\cos \\var{th-35}^\\circ+500\\sin 35^\\circ=0$</p>\n<p style=\"padding-left: 30px;\">$P=\\dfrac{500\\,(\\sin 35^\\circ-\\mu_s\\cos 35^\\circ)}{\\cos\\var{th-35}^\\circ-\\mu_s \\sin \\var{th-35}^\\circ}=\\var{Pmin}\\,\\text{N}\\quad\\blacktriangleleft$</p>\n<p style=\"padding-left: 30px;\">\\[\\var{Pmin}\\,\\text{N} \\le P \\le \\var{Pmax}\\,\\text{N}\\quad\\blacktriangleleft\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"th": {"name": "th", "group": "Ungrouped variables", "definition": "random(40 .. 80#5)", "description": "", "templateType": "randrange", "can_override": false}, "mu_s": {"name": "mu_s", "group": "Ungrouped variables", "definition": "random(0.15 .. 0.4#0.05)", "description": "", "templateType": "randrange", "can_override": false}, "Pmax": {"name": "Pmax", "group": "Ungrouped variables", "definition": "precround(500*(sin(radians(35))+mu_S*cos(radians(35)))/(cos(radians(th-35))+mu_s*sin(radians(th-35))),1)", "description": "", "templateType": "anything", "can_override": false}, "Pmin": {"name": "Pmin", "group": "Ungrouped variables", "definition": "precround(500*(sin(radians(35))-mu_S*cos(radians(35)))/(cos(radians(th-35))-mu_s*sin(radians(th-35))),1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["mu_s", "th", "Pmin", "Pmax"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>to start the block moving up the rail,&nbsp;</p>", "minValue": "Pmax-1", "maxValue": "Pmax+1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>to keep it from moving down.</p>", "minValue": "Pmin-1", "maxValue": "Pmin+1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "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": "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": "Haoyu Huang", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/17657/"}]}