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{"name": "Resolve forces into components", "extensions": [], "custom_part_types": [], "resources": [["question-resources/force_component_image_5.png", "/srv/numbas/media/question-resources/force_component_image_5.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"theta2": {"name": "theta2", "group": "Ungrouped variables", "definition": "random(2..88#1)", "description": "", "templateType": "randrange"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(2..15#1)", "description": "<p>force Q</p>", "templateType": "randrange"}, "theta1": {"name": "theta1", "group": "Ungrouped variables", "definition": "random(2..88#1)", "description": "", "templateType": "randrange"}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(5..15#0.25)", "description": "<p>force F</p>", "templateType": "randrange"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(2..18#0.5)", "description": "<p>force P</p>", "templateType": "randrange"}}, "showQuestionGroupNames": false, "ungrouped_variables": ["f", "p", "q", "theta1", "theta2"], "advice": "<h4>a)</h4>\n<p>Resolve each force from the positive $x$-direction (pointing to the right).</p>\n<p>For the force $Q$ this is at $180^{ \\circ}$ to the positive $x$-axis so&nbsp;makes&nbsp;a contribution to the sum of components of $Q \\times \\cos 180^{\\circ} = -\\var{q}$&nbsp;</p>\n<p>The force $F$ is at $(90+\\var{theta1})^{\\circ}=\\var{90 + theta1}^{\\circ}$ to the positive $x$-direction therefore has a contribution of $F \\times \\cos \\var{90+theta1}^{\\circ} = \\var{f} \\times \\cos \\var{90+theta1}^{\\circ} = \\var{precround(f*cos(radians(90+theta1)),3)}$ 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 $P$ is at $\\var{theta2}^{\\circ}$ to the positive $x$-direction therefore has a contribution of $P \\times \\cos&nbsp;\\var{theta2}^{\\circ} = \\var{p} \\times \\cos\\var{theta2}^{\\circ} = \\var{precround(p*cos(radians(theta2)),3)}$. This is positive as it is acting in the same direction as the positive.</p>\n<p>Therefore the sum of component in the $x$-direction is $-\\var{q} - \\var{precround(-f*cos(radians(90+theta1)),3)} +&nbsp; \\var{precround(p*cos(radians(theta2)),3)} = \\var{precround(-q+f*cos(radians(90+theta1)) + p*cos(radians(theta2)),3)}$.</p>\n<h4>b)</h4>\n<p>Resolve each force from the positive $y$-direction (upwards). For the force $Q$ this is acting at a right angle to&nbsp;the positive direction. Therefore it's contribution is $\\var{q} \\times \\cos 90^{\\circ}= 0$. There is no upward or downward pull from force $Q$.</p>\n<p>The force $F$ is at $(180 - \\var{theta1})^{\\circ} = \\var{180 - theta1}^{\\circ}$ to the positive $y$-direction therefore has a contribution of $F \\times \\cos \\var{180 -&nbsp;theta1}^{\\circ} = \\var{f}&nbsp;\\times \\cos \\var{180 - theta1}^{\\circ} =&nbsp;\\var{precround(f*cos(radians(180-theta1)),3)}$ to the sum of components. This is negative&nbsp;as it is acting in the opposite&nbsp;direction to the positive (downwards).</p>\n<p>The force $P$ is at $(90-\\var{theta2})^{\\circ}=\\var{90 - theta2}^{\\circ}$ to the positive $y$-direction therefore has a contribution of $P \\times \\cos&nbsp;\\var{90 - theta2}^{\\circ} = \\var{p} \\times \\cos&nbsp;\\var{90 - theta2}^{\\circ} = \\var{precround(p*cos(radians(90-theta2)),3)}$ to the sum of components. This is positive&nbsp;as it is acting upwards.</p>\n<p>Therefore the sum of components in the $y$-direction is $0&nbsp;- \\var{precround(-f*cos(radians(180-theta1)),3)} + \\var{precround(p*cos(radians(90-theta2)),3)} = \\var{precround(f*cos(radians(180-theta1)) + p*cos(radians(90-theta2)),3)}$.</p>", "extensions": [], "name": "Resolve forces into components", "parts": [{"strictPrecision": false, "allowFractions": false, "marks": 1, "precision": "3", "maxValue": "f*cos(radians(90+theta1))+q*cos(radians(180))+p*cos(radians(theta2))", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacements": [], "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionType": "dp", "prompt": "<p>Find the sum of the components in the $x$-direction.</p>", "showPrecisionHint": false, "scripts": {}, "minValue": "f*cos(radians(90+theta1))+q*cos(radians(180))+p*cos(radians(theta2))", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "precisionPartialCredit": 0}, {"strictPrecision": false, "allowFractions": false, "marks": 1, "precision": "3", "maxValue": "q*cos(radians(90))+f*cos(radians(180-theta1))+p*cos(radians(90-theta2))", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacements": [], "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionType": "dp", "prompt": "<p>Find the sum of the components in the $y$-direction.</p>", "showPrecisionHint": false, "scripts": {}, "minValue": "q*cos(radians(90))+f*cos(radians(180-theta1))+p*cos(radians(90-theta2))", "variableReplacementStrategy": "originalfirst", "type": "numberentry", "precisionPartialCredit": 0}], "rulesets": {}, "variable_groups": [], "statement": "<p><img width=\"564\" height=\"391\" style=\"display: block; margin-left: auto; margin-right: auto;\" src=\"resources/question-resources/force_component_image_5.png\"/></p>\n<p>In the diagram above, $F=\\var{f} \\ \\mathrm{N}, P = \\var{p} \\ \\mathrm{N}$ and $Q=\\var{q} \\ \\mathrm{N}$. The angles are $\\theta = \\var{theta1}^{\\circ}$ and $\\theta^{*}=\\var{theta2}^{\\circ}$.</p>\n<p><em>Give your answers to the following questions in Newtons, to 3 decimal places.</em></p>", "metadata": {"description": "<p>Another example of finding the $x$&nbsp;and $y$&nbsp;components&nbsp;when multiple forces are&nbsp;applied at&nbsp;different angles to a particle.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "tags": [], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "type": "question", "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}