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{"name": "Simon's copy of Particle in equilibrium on an incline.", "extensions": [], "custom_part_types": [], "resources": [["question-resources/statics2.png", "/srv/numbas/media/question-resources/statics2.png"], ["question-resources/statics3.png", "/srv/numbas/media/question-resources/statics3.png"], ["question-resources/Particle_in_equilibrium_on_a_plane.png", "/srv/numbas/media/question-resources/Particle_in_equilibrium_on_a_plane.png"], ["question-resources/Particle_in_equilibrium_on_a_plane_solution_gxmuyuG.png", "/srv/numbas/media/question-resources/Particle_in_equilibrium_on_a_plane_solution_gxmuyuG.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Simon's copy of Particle in equilibrium on an incline.", "preamble": {"css": "", "js": ""}, "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>A particle is in equilibrium on an incline. Four forces are acting on it. You're given the angle of the incline and three forces.&nbsp;Resolve the forces to find the angle and magnitude of the other&nbsp;force.</p>"}, "advice": "<p>We can draw our $x$ and $y$ axis as being parallel and perpendicular to the particle, as shown in the diagram below and then resolve the forces.</p>\n<p><img src=\"resources/question-resources/Particle_in_equilibrium_on_a_plane_solution_gxmuyuG.png\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>Resolving in the $x$ direction gives equation <strong>(1)</strong></p>\n<p>\\begin{align}<br/>F_3 + F_4 \\cos(90^{\\circ} - \\theta_1) - F_2 \\cos \\theta_2 &amp; = 0,\\\\[0.5em]<br/>F_2 \\cos \\theta_2 &amp; = F_3 + F_4 \\sin \\theta_1, \\\\<br/>&amp;= \\var{F3} + \\var{F4} \\sin \\var{theta1}^{\\circ}. &amp;&amp;&nbsp;(1)&nbsp;<br/>\\end{align}</p>\n<p>Resolving in the $y$ direction gives equation <strong>(2)</strong></p>\n<p>\\begin{align} <br/>F_1 - F_4 \\cos \\theta_1 + F_2 \\sin \\theta_2 &amp; = 0,\\\\[0.5em]<br/>F_2 \\sin \\theta_2 &amp; = F_4 \\cos \\theta_1 - F_1, \\\\<br/>&amp;= \\var{F4} \\cos \\var{theta1}^{\\circ} - \\var{F1}. &amp;&amp; (2)&nbsp;<br/>\\end{align}</p>\n<p>Dividing equation <strong>(2)</strong> by equation <strong>(1)</strong> gives (to 3 s.f.)</p>\n<p>\\begin{align} <br/>\\frac{F_2 \\sin \\theta_2}{F_2 \\cos \\theta_2} &amp; = \\frac{\\var{F4} \\cos \\var{theta1}^{\\circ} - \\var{F1}}{\\var{F3} + \\var{F4} \\sin \\var{theta1}^{\\circ}}, \\\\[0.5em]<br/>\\tan \\theta_2 &amp; =&nbsp;\\frac{\\var{F4} \\cos \\var{theta1}^{\\circ} - \\var{F1}}{\\var{F3} + \\var{F4} \\sin \\var{theta1}^{\\circ}}, \\\\[0.5em]<br/>\\theta_2 &amp; = \\arctan \\left(\\frac{\\var{F4} \\cos \\var{theta1}^{\\circ} - \\var{F1}}{\\var{F3} + \\var{F4} \\sin \\var{theta1}^{\\circ}}\\right), \\\\[0.5em]<br/>&amp; = \\var{thetadp}^{\\circ}. <br/>\\end{align}&nbsp;</p>\n<p>We can then substitute the value for $\\theta_2$ in either equation <strong>(1)</strong> or <strong>(2)</strong> to find $F_2$.</p>\n<p>Using equation <strong>(1)</strong>, we get</p>\n<p>\\begin{align}&nbsp;<br/>F_2 \\cos \\theta_2 &amp; =&nbsp;\\var{F3} + \\var{F4} \\sin \\var{theta1}^{\\circ}, \\\\[0.5em]<br/>F_2 &amp; = \\frac{\\var{F3} + \\var{F4} \\sin \\var{theta1}^{\\circ}}{\\cos \\var{thetadp}^{\\circ}}, \\\\[0.5em]<br/>&amp; = \\var{precround( (F3 + F4*sin(radians(theta1)))/(cos(radians(thetadp))),3)} \\ \\mathrm{N}. <br/>\\end{align}</p>", "ungrouped_variables": ["F1", "F3", "F4", "theta1", "tantheta2", "theta2", "F2", "thetadp", "F2check", "F2sig"], "variablesTest": {"condition": "F2>0", "maxRuns": 100}, "tags": [], "statement": "<p>The diagram below shows a particle in equilibrium under the action of four forces.</p>\n<p><img src=\"resources/question-resources/Particle_in_equilibrium_on_a_plane.png\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>You are given the following information:</p>\n<ul>\n<li>$F_1 = \\var{F1} \\ \\mathrm{N}$;</li>\n<li>$F_3 = \\var{F3} \\ \\mathrm{N}$;</li>\n<li>$F_4 = \\var{F4} \\ \\mathrm{N}$ acting vertically downwards;</li>\n<li>$\\theta_1 = \\var{theta1}^{\\circ}$.</li>\n</ul>", "rulesets": {}, "functions": {}, "parts": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>By resolving forces in components&nbsp;find, to 3 d.p., the unknown angle $\\theta_2$.</p>\n<p>$\\theta_2 = $&nbsp;[[0]]</p>", "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "scripts": {}, "sortAnswers": false, "marks": 0, "gaps": [{"minValue": "theta2", "showCorrectAnswer": true, "precisionType": "dp", "mustBeReducedPC": 0, "precision": "3", "maxValue": "theta2", "allowFractions": false, "strictPrecision": false, "unitTests": [], "variableReplacements": [], "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "marks": 1, "correctAnswerFraction": false, "precisionPartialCredit": 0, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": ""}], "customMarkingAlgorithm": ""}, {"showCorrectAnswer": true, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "prompt": "<p>Using your answer from part <strong>a)</strong> find the magnitude of the force $F_2$, in Newtons to 3 d.p.</p>\n<p>$F_2 = $&nbsp;[[0]]</p>", "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "scripts": {}, "sortAnswers": false, "marks": 0, "gaps": [{"minValue": "F2", "showCorrectAnswer": true, "precisionType": "dp", "mustBeReducedPC": 0, "precision": "3", "maxValue": "F2", "allowFractions": false, "strictPrecision": false, "unitTests": [], "variableReplacements": [], "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "marks": 1, "correctAnswerFraction": false, "precisionPartialCredit": 0, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "scripts": {}, "mustBeReduced": false, "correctAnswerStyle": "plain", "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": ""}], "customMarkingAlgorithm": ""}], "extensions": [], "variables": {"theta1": {"name": "theta1", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(10..40#1)"}, "F2sig": {"name": "F2sig", "description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "siground(F2,3)"}, "F4": {"name": "F4", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(5.1..8#0.1)"}, "F2": {"name": "F2", "description": "<p>uses theta2 to 3dp</p>", "group": "Ungrouped variables", "templateType": "anything", "definition": "(F3 + F4*sin(radians(theta1)))/(cos(radians(thetadp)))"}, "F3": {"name": "F3", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(0.5..2.8#0.1)"}, "thetadp": {"name": "thetadp", "description": "<p>theta2 to 3d.p. (part a solution will be used in part b - which will allow for rounding errors)</p>", "group": "Ungrouped variables", "templateType": "anything", "definition": "precround(theta2,3)"}, "tantheta2": {"name": "tantheta2", "description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "(F4*cos(radians(theta1))-F1)/(F3+F4*cos(radians(90-theta1)))"}, "F1": {"name": "F1", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "definition": "random(3..4#0.1)"}, "theta2": {"name": "theta2", "description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "degrees(arctan(tantheta2))"}, "F2check": {"name": "F2check", "description": "<p>uses theta2</p>", "group": "Ungrouped variables", "templateType": "anything", "definition": "(F3 + F4*sin(radians(theta1)))/(cos(radians(theta2)))"}}, "type": "question", "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}