// Numbas version: exam_results_page_options
{"name": "Resolve forces into components", "extensions": [], "custom_part_types": [], "resources": [["question-resources/force_component_image_4.png", "/srv/numbas/media/question-resources/force_component_image_4.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["force1", "force2", "force3", "theta1", "theta2", "yangle1", "yangle2"], "name": "Resolve forces into components", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<h4>a) - c)</h4>\n<p>Resolve each force from the positive $x$-direction (pointing to the right). For the force $P$ this is at $90^{ \\circ}$ to the $x$-axis therefore has a contribution of $P \\times \\cos 90^{\\circ} = 0$ to the sum of components.</p>\n<p>The force $F$ is at $(180-\\var{theta1})^{\\circ}=\\var{90 + 90 - theta1}^{\\circ}$ to the positive $x$-direction therefore has a contribution of $F \\times \\cos \\var{180-theta1}^{\\circ} = \\var{force1} \\times \\cos \\var{180-theta1}^{\\circ} = \\var{precround(force1*cos(radians(180-theta1)),3)}$ to the sum of components. This will be negative as you can imagine if you were moving in the positive $x$-direction this force is acting in the opposite direction and pulling you back!</p>\n<p>The force $Q$ is at $\\var{theta2}^{\\circ}$ to the positive $x$-direction therefore has a contribution of $Q \\times cos&nbsp;\\var{theta2}^{\\circ} = \\var{force3} \\times \\cos\\var{theta2}^{\\circ} = \\var{precround(force3*cos(radians(theta2)),3)}$. This is positive as it is acting in the same direction as the positive.</p>\n<p>Therefore the sum of components in the $x$-direction is $0 - \\var{precround(-force1*cos(radians(180-theta1)),3)} +&nbsp; \\var{precround(force3*cos(radians(theta2)),3)} = \\var{precround(force1*cos(radians(180-theta1)) + force3*cos(radians(theta2)),3)}$.</p>\n<h4>d) - g)</h4>\n<p>Resolve each force from the positive $y$-direction (upwards). For the force $P$ this is acting completely in the positive direction, at no angle. Therefore it's contribution is $\\var{force2}$. Note that this is the same as $\\var{force2} \\times \\cos 0^{\\circ}$.&nbsp;</p>\n<p>The force $F$ is at $(90 - \\var{theta1})^{\\circ} = \\var{90 - theta1}^{\\circ}$ to the positive $y$-direction therefore has a contribution of $F \\times \\cos \\var{90 -&nbsp;theta1}^{\\circ} = \\var{force1}&nbsp;\\times \\cos \\var{90 - theta1}^{\\circ} =&nbsp;\\var{precround(force1*cos(radians(90-theta1)),3)}$ to the sum of components. This is positive as it is acting in the same direction to the positive.</p>\n<p>The force $Q$ is at $(90+\\var{theta2})^{\\circ}=\\var{90 + theta2}^{\\circ}$ to the positive $y$-direction therefore has a contribution of $Q \\times \\cos&nbsp;\\var{90 + theta2}^{\\circ} = \\var{force3} \\times \\cos&nbsp;\\var{90 + theta2}^{\\circ} = \\var{precround(force3*cos(radians(90+theta2)),3)}$ to the sum of components. This is negative as it is acting downwards, in the opposite direction to the positive.</p>\n<p>Therefore the sum of components in the $y$-direction is $\\var{force2} +&nbsp;\\var{precround(force1*cos(radians(90-theta1)),3)} -&nbsp;\\var{-precround(force3*cos(radians(90+theta2)),3)} = \\var{precround(force2 + force1*cos(radians(90-theta1)) + force3*cos(radians(90+theta2)),3)}$.</p>\n<p>&nbsp;</p>", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "<p>Find the component of $F$ in the $x$-direction</p>", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "f1x", "strictPrecision": false, "minValue": "f1x", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": "1", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>Find the component of $Q$ in the $x$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "f3x", "strictPrecision": false, "minValue": "f3x", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": "1", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>Find&nbsp;the resultant force in the $x$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [{"variable": "f1x", "part": "p0", "must_go_first": false}, {"variable": "f3x", "part": "p1", "must_go_first": false}], "maxValue": "f1x+f3x", "strictPrecision": false, "minValue": "f1x+f3x", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>Find the component of $P$ in the $y$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "force2", "strictPrecision": false, "minValue": "force2", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": "1", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>Find the component of $F$ in the $y$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "f1y", "strictPrecision": false, "minValue": "f1y", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>Find the component of $Q$ in the $y$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "f3y", "strictPrecision": false, "minValue": "f3y", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": "1", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>Find&nbsp;the resultant force in the $y$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [{"variable": "f1y", "part": "p4", "must_go_first": false}, {"variable": "f3y", "part": "p5", "must_go_first": false}, {"variable": "force2", "part": "p3", "must_go_first": false}], "maxValue": "f1y+force2+f3y", "strictPrecision": false, "minValue": "f1y+force2+f3y", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "extensions": [], "statement": "<p><img width=\"430\" height=\"338\" style=\"display: block; margin-left: auto; margin-right: auto;\" src=\"resources/question-resources/force_component_image_4.png\"/></p>\n<p>In the diagram above, $F = \\var{force1} \\, \\mathrm{N}$, $P = \\var{force2} \\, \\mathrm{N}$ and $Q = \\var{force3} \\, \\mathrm{N}$. The angles are $\\theta = \\var{theta1}^{\\circ}$ and $\\theta^{\\ast} = \\var{theta2}^{\\circ}$.&nbsp;</p>\n<p><em>Give your answers to the following questions in Newtons to 3 decimal places.</em></p>", "variable_groups": [{"variables": ["f1x", "f1y", "f3x", "f3y"], "name": "components"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"theta2": {"definition": "random(5..85#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta2", "description": ""}, "theta1": {"definition": "random(2..88#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta1", "description": ""}, "yangle1": {"definition": "90-theta1", "templateType": "anything", "group": "Ungrouped variables", "name": "yangle1", "description": ""}, "yangle2": {"definition": "90+theta2", "templateType": "anything", "group": "Ungrouped variables", "name": "yangle2", "description": ""}, "f1x": {"definition": "-force1*cos(radians(theta1))", "templateType": "anything", "group": "components", "name": "f1x", "description": ""}, "f1y": {"definition": "force1*sin(radians(theta1))", "templateType": "anything", "group": "components", "name": "f1y", "description": ""}, "force1": {"definition": "random(3..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "force1", "description": ""}, "force3": {"definition": "random(2..10#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "force3", "description": ""}, "force2": {"definition": "random(2..7#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "force2", "description": ""}, "f3x": {"definition": "force3*cos(radians(theta2))", "templateType": "anything", "group": "components", "name": "f3x", "description": ""}, "f3y": {"definition": "-force3*sin(radians(theta2))", "templateType": "anything", "group": "components", "name": "f3y", "description": ""}}, "metadata": {"description": "<p>Find the $x$ and $y$ components of the resultant force on an object, when multiple forces are&nbsp;applied at&nbsp;different angles.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}]}]}], "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}]}