// Numbas version: finer_feedback_settings
{"name": "Dynamics - Forces at an angle", "duration": 0, "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "allQuestions": true, "shuffleQuestions": false, "percentPass": 0, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "feedback": {"showtotalmark": true, "advicethreshold": 0, "allowrevealanswer": true, "feedbackmessages": [], "showactualmark": true, "showanswerstate": true, "intro": "", "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Resolve a force into $x$ and $y$ components", "extensions": [], "custom_part_types": [], "resources": [["question-resources/force_component_image.png", "/srv/numbas/media/question-resources/force_component_image.png"], ["question-resources/force_component_image_PgpiR1U.png", "/srv/numbas/media/question-resources/force_component_image_PgpiR1U.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["force", "angle", "yangle"], "tags": [], "advice": "<h4>a)</h4>\n<p>The component of force in the $x$-direction can be found using $F \\times \\cos\\theta$. Remember to set&nbsp;your calculator to use&nbsp;degrees and not radians.</p>\n<p>\\begin{align} \\text{component in the }x \\text{-direction } &amp; = F \\cos \\theta \\\\<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &amp; = \\var{force} \\times \\cos \\var{angle} \\\\<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp; = \\var{precround(force*cos(radians(angle)),3)}\\end{align}</p>\n<h4>b)</h4>\n<p>Now we need to make sure we find the angle between the force and the direction we are resolving in. Therefore $\\theta = 90 - \\var{angle} = \\var{yangle}$.</p>\n<p>\\begin{align} \\text{component in the y-direction } &amp; = F \\cos \\theta \\\\<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &amp; = \\var{force} \\times \\cos \\var{yangle} \\\\<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp; = \\var{precround(force*cos(radians(yangle)),3)}\\end{align}</p>\n<p>Since $\\sin \\theta = \\cos(90-\\theta)$, we could also use $\\sin \\var{angle}$ in our calculations instead of $\\cos(90 - \\var{angle})$.</p>", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "<p>Find the component of the force in the $x$-direction, in Newtons to 3 decimal places.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "force*cos(radians(angle))", "minValue": "force*cos(radians(angle))", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>Find the component of the force in the $y$-direction,&nbsp;in Newtons to 3 decimal places.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "force*cos(radians(yangle))", "minValue": "force*cos(radians(yangle))", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p style=\"text-align: center;\"><img width=\"552\" height=\"350\" src=\"resources/question-resources/force_component_image_PgpiR1U.png\"/></p>\n<p style=\"text-align: left;\">In the above diagram the force $F=\\var{force} \\ \\mathrm{N}$ and the angle $\\theta = \\var{angle}^{\\circ}$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"yangle": {"definition": "90-angle", "templateType": "anything", "group": "Ungrouped variables", "name": "yangle", "description": ""}, "force": {"definition": "random(3..15#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "force", "description": ""}, "angle": {"definition": "random(1..89#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "angle", "description": ""}}, "metadata": {"description": "<p>Find the $x$ and $y$ components of a force which is&nbsp;applied at an angle to a particle. Resolve using $F \\cos \\theta$. The force acts in the positive $x$ and positive $y$ direction.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Resolve force into $x$ and $y$ components", "extensions": [], "custom_part_types": [], "resources": [["question-resources/force_component_image_2.png", "/srv/numbas/media/question-resources/force_component_image_2.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["force", "theta", "angle", "yangle"], "tags": [], "advice": "<h4>a)</h4>\n<p>We need to find the angle $\\theta_x$ of $F$ relative to the $x$-axis and then use $F \\times cos\\theta_x$.</p>\n<p>We consider the angle between the positive $x$-axis and $F$,&nbsp;i.e.&nbsp;$\\theta_x = 90 + \\var{theta}$.</p>\n<p>\\begin{align} <br/>\\text{component in the } x \\text{-direction} &amp; = F \\cos\\theta_x \\\\<br/>&amp; = \\var{force} \\times \\cos \\var{angle} \\\\<br/>&amp; = \\var{precround(force*cos(radians(angle)),3)} <br/>\\end{align}</p>\n<h4>b)</h4>\n<p>The positive $y$-direction is vertically upwards&nbsp;and&nbsp;we need the angle relative to the positive $y$-direction therefore $\\theta_y = 180 - \\var{theta}$.</p>\n<p>\\begin{align} <br/>\\text{component in the } y \\text{-direction} &amp; = F \\cos\\theta_y \\\\<br/>&amp; = \\var{force} \\times \\cos \\var{yangle} \\\\<br/>&amp; = \\var{precround(force*cos(radians(yangle)),3)} <br/>\\end{align}</p>\n<p>Notice that both these answers are negative as the force acts in the opposite direction to the positive. You could also answer these by resolving in the negative $x$ or $y$ direction and changing the sign of your solution. For example in part <strong>b)</strong> you could use $\\theta_y = \\var{theta}$ which gives $F \\cos \\theta_y =&nbsp;\\var{precround(-force*cos(radians(yangle)),3)}$ and then change the sign.</p>", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "<p>Find the component of the force in the $x$-direction in Newtons.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "force*cos(radians(angle))", "minValue": "force*cos(radians(angle))", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>Find the component of the force in the $y$-direction in Newtons.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "force*cos(radians(yangle))", "minValue": "force*cos(radians(yangle))", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p><img style=\"display: block; margin-left: auto; margin-right: auto;\" src=\"resources/question-resources/force_component_image_2.png\"/></p>\n<p>In the above diagram, $F = \\var{force} \\, \\mathrm{N}$ and $\\theta = \\var{theta}^{\\circ}$.&nbsp;</p>\n<p>Give your answers to the following questions to 3 decimal places.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"yangle": {"definition": "180-theta", "templateType": "anything", "group": "Ungrouped variables", "name": "yangle", "description": ""}, "theta": {"definition": "random(2..89#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta", "description": ""}, "force": {"definition": "random(3..20#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "force", "description": ""}, "angle": {"definition": "90 + theta", "templateType": "anything", "group": "Ungrouped variables", "name": "angle", "description": ""}}, "metadata": {"description": "<p>Find the $x$ and $y$ components of a force which is&nbsp;applied at an angle to a particle. Resolve using $F \\cos \\theta$. The force is applied in the negative $x$ and negative $y$ direction.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Resolve force into $x$ and $y$ components - negative $x$", "extensions": [], "custom_part_types": [], "resources": [["question-resources/force_component_image_3.png", "/srv/numbas/media/question-resources/force_component_image_3.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["force", "theta", "angle", "yangle"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<h4>a)</h4>\n<p>We resolve in the positive $x$-direction so the answer will be negative. We take $\\theta = \\var{theta}^{\\circ}$ as the angle is already between the force and the $x$-axis.</p>\n<p>\\begin{align} \\text{component in } x \\text{-direction} &amp; = F \\cos \\theta \\\\<br/>&amp; = \\var{force} \\times \\cos \\var{theta} \\\\<br/>&amp; = \\var{precround(force*cos(radians(theta)),3)}<br/> \\end{align}</p>\n<h4>b)</h4>\n<p>We need the angle between the force and the direction we are resolving so take $\\theta = 90 - \\var{theta}= \\var{90-theta}^{\\circ}$.</p>\n<p>\\begin{align} <br/>\\text{component in } y \\text{-direction} &amp; = F \\cos \\theta \\\\<br/>&amp; = \\var{force} \\times \\cos \\var{yangle} \\\\<br/>&amp; = \\var{precround(force*cos(radians(yangle)),3)} <br/>\\end{align}</p>", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "<p>Find the component of the force in the $x$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "force*cos(radians(theta))", "strictPrecision": false, "minValue": "force*cos(radians(theta))", "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 the force in the $y$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "force*cos(radians(yangle))", "strictPrecision": false, "minValue": "force*cos(radians(yangle))", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "<p><img width=\"334\" height=\"304\" style=\"display: block; margin-left: auto; margin-right: auto;\" src=\"resources/question-resources/force_component_image_3.png\"/></p>\n<p>In the diagram above,&nbsp;$F = \\var{force} \\ \\mathrm{N}$ and $\\theta = \\var{theta}^{\\circ}$.</p>\n<p>Give your answers to the following questions in Newtons to 3 decimal places.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"yangle": {"definition": "90-theta", "templateType": "anything", "group": "Ungrouped variables", "name": "yangle", "description": ""}, "theta": {"definition": "random(5..85#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta", "description": ""}, "force": {"definition": "random(2..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "force", "description": ""}, "angle": {"definition": "180-theta", "templateType": "anything", "group": "Ungrouped variables", "name": "angle", "description": ""}}, "metadata": {"description": "<p>Find the $x$ and $y$ components of a force which is&nbsp;applied at an angle to a particle. Resolve using $F \\cos \\theta$. The force is applied in the negative $x$ direction&nbsp;but the positive $y$.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"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}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["force1", "force2", "force3", "theta1", "theta2", "yangle1", "yangle2"], "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}], "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": []}]}, {"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}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["f", "p", "q", "theta1", "theta2"], "tags": [], "preamble": {"css": "", "js": ""}, "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>", "rulesets": {}, "parts": [{"precisionType": "dp", "prompt": "<p>Find the sum of the components in the $x$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "f*cos(radians(90+theta1))+q*cos(radians(180))+p*cos(radians(theta2))", "strictPrecision": false, "minValue": "f*cos(radians(90+theta1))+q*cos(radians(180))+p*cos(radians(theta2))", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "<p>Find the sum of the components in the $y$-direction.</p>", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "q*cos(radians(90))+f*cos(radians(180-theta1))+p*cos(radians(90-theta2))", "strictPrecision": false, "minValue": "q*cos(radians(90))+f*cos(radians(180-theta1))+p*cos(radians(90-theta2))", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "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>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"q": {"definition": "random(2..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "q", "description": "<p>force Q</p>"}, "p": {"definition": "random(2..18#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "p", "description": "<p>force P</p>"}, "theta1": {"definition": "random(2..88#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta1", "description": ""}, "theta2": {"definition": "random(2..88#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta2", "description": ""}, "f": {"definition": "random(5..15#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "f", "description": "<p>force F</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"}, "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/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/force_component_image.png", "/srv/numbas/media/question-resources/force_component_image.png"], ["question-resources/force_component_image_PgpiR1U.png", "/srv/numbas/media/question-resources/force_component_image_PgpiR1U.png"], ["question-resources/force_component_image_2.png", "/srv/numbas/media/question-resources/force_component_image_2.png"], ["question-resources/force_component_image_3.png", "/srv/numbas/media/question-resources/force_component_image_3.png"], ["question-resources/force_component_image_4.png", "/srv/numbas/media/question-resources/force_component_image_4.png"], ["question-resources/force_component_image_5.png", "/srv/numbas/media/question-resources/force_component_image_5.png"]]}