// Numbas version: finer_feedback_settings
{"name": "Statics - Maths Support Wiki", "duration": 0, "metadata": {"description": "<p>Four questions on statics of a particle.</p>", "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": "Mass resting on a rough plane", "extensions": [], "custom_part_types": [], "resources": [["question-resources/statics6.png", "/srv/numbas/media/question-resources/statics6.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": ["g", "mass", "theta", "R", "mu"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>We can draw a diagram showing the weight, normal reaction $R$ and the force of friction, $F$.</p>\n<p><img src=\"resources/question-resources/statics6.png\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>Here $\\theta = \\var{theta}^{\\circ}$ and $m = \\var{mass}$.&nbsp;</p>\n<h4>a)</h4>\n<p>To find the normal reaction, $R$, we resolve perpendicular to the plane.</p>\n<p>\\begin{align} <br/>R - mg \\cos \\theta &amp; = 0, \\\\<br/>R &amp; = mg \\cos \\theta, \\\\<br/>&amp; = \\var{mass} \\times 9.8 \\cos (\\var{theta}^{\\circ}), \\\\<br/>&amp; = \\var{precround(mass*9.8*cos(radians(theta)),3)} \\ \\mathrm{N}. <br/>\\end{align}</p>\n<h4>b)</h4>\n<p>To find the coefficient of friction we can resolve parallel to the plane and use our 3d.p. value of $R$ from part a) and the fact that friction is limiting ($F = \\mu R$) as we are in equilibrium.</p>\n<p>\\begin{align} <br/>F - mg \\sin \\theta &amp; = 0, \\\\<br/>F &amp; = mg \\sin \\theta, \\\\<br/>\\mu R &amp; = mg \\sin \\theta, \\\\[0.5em]<br/>\\mu &amp; = \\frac{mg \\sin \\theta}{R}, \\\\[0.5em]<br/>&amp; = \\frac{ \\var{mass} \\times 9.8 \\sin (\\var{theta}^{\\circ})}{\\var{R}}, \\\\[0.5em]<br/>&amp; = \\var{precround( (mass*9.8*sin(radians(theta)))/R,3)}. <br/>\\end{align}</p>", "rulesets": {}, "parts": [{"prompt": "<p>Find the normal reaction, $R$, between the ball and the plane, in $\\mathrm{N}$.</p>\n<p>$R = $&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "mass*g*cos(radians(theta))", "strictPrecision": false, "minValue": "mass*g*cos(radians(theta))", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "showPrecisionHint": true, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Using the value of $R$ from part <strong>a)</strong> find the coeffiecient of friction, $\\mu$, between the ball and the plane.</p>\n<p>$\\mu = $&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "mu", "strictPrecision": false, "minValue": "mu", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "showPrecisionHint": true, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>A ball of mass $\\var{mass} \\ \\mathrm{kg}$ rests in limiting equilibrium on a rough plane inclined at $\\var{theta}^{\\circ}$ above the horizontal.</p>\n<p>Suppose that the acceleration due to gravity is $g= 9.8 \\mathrm{ms^{-2}}$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"mu": {"definition": "(mass*g*sin(radians(theta)))/R", "templateType": "anything", "group": "Ungrouped variables", "name": "mu", "description": ""}, "theta": {"definition": "random(10..44#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta", "description": ""}, "R": {"definition": "precround(mass*g*cos(radians(theta)),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "R", "description": "<p>the normal reaction R, 3d.p</p>"}, "mass": {"definition": "random(2..8#0.5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "mass", "description": ""}, "g": {"definition": "9.8", "templateType": "number", "group": "Ungrouped variables", "name": "g", "description": ""}}, "metadata": {"description": "<p>A mass is resting on a rough plane. Given the angle of the plane and the mass of the object, find the normal reaction force&nbsp;and hence the coefficient of friction.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Mass held in tension by a string", "extensions": [], "custom_part_types": [], "resources": [["question-resources/statics4.png", "/srv/numbas/media/question-resources/statics4.png"], ["question-resources/statics5.png", "/srv/numbas/media/question-resources/statics5.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", "theta", "g", "T", "Tp", "M", "M2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Draw a diagram showing all the forces acting on the bag of mass.</p>\n<p><img src=\"resources/question-resources/statics5.png\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>The tension $T$ is the same in both sections of the string, and the weight of the mass acts downwards.&nbsp;</p>\n<h4>a)</h4>\n<p>We resolve the forces horizontally to get</p>\n<p>\\begin{align} <br/>T \\cos \\theta - F &amp; = 0, \\\\[0.5em]<br/>T &amp; = \\frac{F}{\\cos \\theta}, \\\\[0.5em]<br/>&amp; = \\frac{\\var{F}}{\\cos \\var{theta}^{\\circ}}, \\\\[0.5em]<br/>&amp; = \\var{precround(F/cos(radians(theta)),3)} \\mathrm{N}. <br/>\\end{align}</p>\n<p>The tension in the string is $\\var{precround(F/cos(radians(theta)),3)} \\mathrm{N}.$</p>\n<h4>b)</h4>\n<p>We resolve the forces vertically, and use our 3d.p. value of $T$ found in part <strong>a)</strong> to get</p>\n<p>\\begin{align} <br/>T + T \\sin \\theta - Mg &amp; = 0, \\\\<br/>Mg &amp; = T + T \\sin \\theta, \\\\[0.5em]<br/>M &amp; = \\frac{T + T \\sin \\theta}{g}, \\\\[0.5em]<br/>&amp; = \\frac{ \\var{Tp} + \\var{Tp} \\sin \\var{theta}^{\\circ}}{9.8}, \\\\[0.5em]<br/>&amp; = \\var{precround( (Tp +Tp*sin(radians(theta)))/9.8,3)} \\mathrm{kg}. <br/>\\end{align}</p>\n<p>The mass of the bag is $\\var{precround( (Tp +Tp*sin(radians(theta)))/9.8,3)} \\mathrm{kg}.$</p>", "rulesets": {}, "parts": [{"prompt": "<p>Find the tension, $T$, in the string in Newtons ($\\mathrm{N}$), to 3d.p.</p>\n<p>$T = $&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "T", "strictPrecision": false, "minValue": "T", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Using the value of $T$ found in part <strong>a)</strong> find $M$, the mass of the bag, in $\\mathrm{kg}$ to 3d.p.</p>\n<p>$M = $&nbsp;[[0]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "M", "strictPrecision": false, "minValue": "M", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>A small bag of mass&nbsp;$M \\, \\mathrm{kg}$&nbsp;is held by a light inextensible string. The ends of the string are attached to two fixed points $A$ and $B$ which are horizontally on the same level. The bag is held in equilibrium by a horizontal force of magnitude $F \\ \\mathrm{N} = \\var{F} \\ \\mathrm{N}$ acting parallel to $AB$. The bag is vertically below $A$ and the angle $A\\hat{B}M$ is $\\theta =\\var{theta}^{\\circ}$.</p>\n<p><img src=\"resources/question-resources/statics4.png\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>The acceleration due to gravity is $g = 9.8 \\mathrm{ms^{-2}}$.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"g": {"definition": "9.8", "templateType": "number", "group": "Ungrouped variables", "name": "g", "description": ""}, "F": {"definition": "random(5..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "F", "description": ""}, "M": {"definition": "(Tp+Tp*sin(radians(theta)))/g", "templateType": "anything", "group": "Ungrouped variables", "name": "M", "description": "<p>mass using the tension to 3dp</p>"}, "Tp": {"definition": "precround(T,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "Tp", "description": "<p>T to 3d.p. (answer to part a) students will use this in part b</p>"}, "T": {"definition": "F/cos(radians(theta))", "templateType": "anything", "group": "Ungrouped variables", "name": "T", "description": "<p>tension</p>"}, "M2": {"definition": "(T+T*sin(radians(theta)))/g", "templateType": "anything", "group": "Ungrouped variables", "name": "M2", "description": "<p>mass not using T to 3d.p. to check if part b answer is effected.</p>"}, "theta": {"definition": "random(10..40#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta", "description": ""}}, "metadata": {"description": "<p>A mass is&nbsp;hanging from&nbsp;a string,&nbsp;tethered to two points. A horizontal force applied to the mass holds it&nbsp;directly underneath one of the points. Given the magnitude of the force, find the tension in the string and the mass of the bag.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "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}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}], "functions": {}, "ungrouped_variables": ["F1", "F3", "F4", "theta1", "tantheta2", "theta2", "F2", "thetadp", "F2check", "F2sig"], "tags": [], "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{siground( (F3 + F4*sin(radians(theta1)))/(cos(radians(thetadp))),3)} \\ \\mathrm{N}. <br/>\\end{align}</p>", "rulesets": {}, "parts": [{"prompt": "<p>By resolving forces in components&nbsp;find, to 3 significant figures, the unknown angle $\\theta_2$.</p>\n<p>$\\theta_2 = $&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "sigfig", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "theta2", "minValue": "theta2", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Using your answer from part <strong>a)</strong> find the magnitude of the force $F_2$, in Newtons to 3 significant figures.</p>\n<p>$F_2 = $&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "sigfig", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "F2+0.01", "minValue": "F2-0.01", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>The diagram below shows a particle in equilibrium under the action of four forces.</p>\n<p><img src=\"resources/question-resources/Particle_in_equilibrium_on_a_plane.png\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>You are given the following information:</p>\n<ul>\n<li>$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>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": "F2>0"}, "preamble": {"css": "", "js": ""}, "variables": {"theta2": {"definition": "degrees(arctan(tantheta2))", "templateType": "anything", "group": "Ungrouped variables", "name": "theta2", "description": ""}, "F1": {"definition": "random(3..4#0.1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "F1", "description": ""}, "F2": {"definition": "(F3 + F4*sin(radians(theta1)))/(cos(radians(thetadp)))", "templateType": "anything", "group": "Ungrouped variables", "name": "F2", "description": "<p>uses theta2 to 3dp</p>"}, "theta1": {"definition": "random(10..40#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta1", "description": ""}, "F4": {"definition": "random(5.1..8#0.1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "F4", "description": ""}, "thetadp": {"definition": "siground(theta2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "thetadp", "description": "<p>theta2 to 3d.p. (part a solution will be used in part b - which will allow for rounding errors)</p>"}, "F2sig": {"definition": "siground(F2,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "F2sig", "description": ""}, "F3": {"definition": "random(0.5..2.8#0.1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "F3", "description": ""}, "tantheta2": {"definition": "(F4*cos(radians(theta1))-F1)/(F3+F4*cos(radians(90-theta1)))", "templateType": "anything", "group": "Ungrouped variables", "name": "tantheta2", "description": ""}, "F2check": {"definition": "(F3 + F4*sin(radians(theta1)))/(cos(radians(theta2)))", "templateType": "anything", "group": "Ungrouped variables", "name": "F2check", "description": "<p>uses theta2</p>"}}, "metadata": {"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>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Resolve forces on particle in equilibrium", "extensions": [], "custom_part_types": [], "resources": [["question-resources/statics1.png", "/srv/numbas/media/question-resources/statics1.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": ["F1", "theta1", "F2", "theta2", "F3", "theta3", "check", "check2"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>As the particle is in equilibrium we equate the sum of the forces to zero.&nbsp;</p>\n<p>Resolving horizontally we have&nbsp;</p>\n<p>\\begin{align} <br/>F_1 \\cos \\theta_1 + F_2 \\cos \\theta_2 - F_3 \\cos \\theta_3 &amp; = 0, \\\\<br/>\\var{F1} \\cos \\var{theta1}^{\\circ} + F_2 \\cos \\var{theta2}^{\\circ} - F_3 \\cos \\var{theta3}^{\\circ} &amp; = 0, \\\\<br/>F_2 \\cos \\var{theta2}^{\\circ} &amp; =&nbsp;F_3 \\cos \\var{theta3}^{\\circ} -&nbsp;\\var{F1} \\cos \\var{theta1}^{\\circ}, \\\\<br/>F_2 &amp; = \\frac{F_3 \\cos \\var{theta3}^{\\circ} -&nbsp;\\var{F1} \\cos \\var{theta1}^{\\circ}}{&nbsp;\\cos \\var{theta2}^{\\circ}}. <br/>\\end{align}</p>\n<p>Resolving vertically, we have</p>\n<p>\\begin{align} <br/>F_1 \\sin \\theta_1 - F_2 \\sin \\theta_2 - F_3 \\sin \\theta_3 &amp; = 0, \\\\<br/>\\var{F1} \\sin \\var{theta1}^{\\circ} - F_2 \\sin \\var{theta2}^{\\circ} - F_3 \\sin \\var{theta3}^{\\circ} &amp; = 0, \\\\<br/>F_3 \\sin \\var{theta3}^{\\circ}&amp; = \\var{F1} \\sin \\var{theta1}^{\\circ} - F_2 \\sin \\var{theta2}^{\\circ}. <br/>\\end{align}</p>\n<p>Substituting our value for $F_2$ into the formula $F_3 \\sin \\var{theta3}^{\\circ}$ and rearranging gives</p>\n<p>\\begin{align} <br/>F_3&nbsp;\\sin \\var{theta3}^{\\circ} &amp; =&nbsp;\\var{F1} \\sin \\var{theta1}^{\\circ} - F_2 \\sin \\var{theta2}^{\\circ}&nbsp;\\\\<br/>F_3&nbsp;\\sin \\var{theta3}^{\\circ} &amp; =&nbsp;\\var{F1} \\sin \\var{theta1}^{\\circ} - \\left( \\frac{F_3 \\cos \\var{theta3}^{\\circ} -&nbsp;\\var{F1} \\cos \\var{theta1}^{\\circ}}{\\cos \\var{theta2}^{\\circ}}&nbsp;\\right)&nbsp;\\sin \\var{theta2}^{\\circ}, \\\\<br/>F_3&nbsp;\\sin \\var{theta3}^{\\circ} &amp; =&nbsp;\\var{F1} \\sin \\var{theta1}^{\\circ} - F_3 \\frac{ \\cos&nbsp;\\var{theta3}^{\\circ}}{ \\cos&nbsp;\\var{theta2}^{\\circ}} \\sin&nbsp;\\var{theta2}^{\\circ} + \\var{F1} \\frac{ \\cos&nbsp;\\var{theta1}^{\\circ}}{ \\cos&nbsp;\\var{theta2}^{\\circ}} \\sin&nbsp;\\var{theta2}^{\\circ}, \\\\<br/>F_3 \\left( \\sin \\var{theta3}^{\\circ} + \\frac{ \\cos&nbsp;\\var{theta3}^{\\circ}}{ \\cos&nbsp;\\var{theta2}^{\\circ}} \\sin&nbsp;\\var{theta2}^{\\circ}&nbsp;\\right) &amp; =&nbsp;\\var{F1} \\sin \\var{theta1}^{\\circ} + \\var{F1}&nbsp;\\frac{ \\cos&nbsp;\\var{theta1}^{\\circ}}{ \\cos&nbsp;\\var{theta2}^{\\circ}} \\sin&nbsp;\\var{theta2}^{\\circ} \\\\<br/>F_3 &amp; = \\frac{\\var{F1} \\sin \\var{theta1}^{\\circ} + \\var{F1}&nbsp;\\frac{ \\cos&nbsp;\\var{theta1}^{\\circ}}{ \\cos&nbsp;\\var{theta2}^{\\circ}} \\sin&nbsp;\\var{theta2}^{\\circ}}{&nbsp;\\sin \\var{theta3}^{\\circ} + \\frac{ \\cos&nbsp;\\var{theta3}^{\\circ}}{ \\cos&nbsp;\\var{theta2}^{\\circ}} \\sin&nbsp;\\var{theta2}^{\\circ}}.<br/>\\end{align}&nbsp;</p>\n<p>Therefore, to 3 s.f. we have $F_3 = \\var{siground(F3,3)} \\mathrm{N}. $</p>\n<p>We can then use our $F_3$ value in our equation for $F_2$</p>\n<p>\\begin{align}&nbsp;<br/>F_2 &amp; = \\frac{F_3 \\cos \\var{theta3}^{\\circ} -&nbsp;\\var{F1} \\cos \\var{theta1}^{\\circ}}{\\cos \\var{theta2}^{\\circ}}, \\\\<br/>&amp; =&nbsp;\\frac{\\var{siground(F3,3)} \\cos \\var{theta3}^{\\circ} -&nbsp;\\var{F1} \\cos \\var{theta1}^{\\circ}}{\\cos \\var{theta2}^{\\circ}}, \\\\<br/>&amp; = \\var{siground(F2,3)} \\mathrm{N}. <br/>\\end{align}<br/>&nbsp;&nbsp;<br/>&nbsp;<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<br/><br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</p>", "rulesets": {}, "parts": [{"prompt": "<p>$F_2 = $&nbsp;[[0]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "sigfig", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "F2", "strictPrecision": false, "minValue": "F2", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$F_3 = $&nbsp;[[0]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "sigfig", "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "F3", "strictPrecision": false, "minValue": "F3", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "3", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>The diagram shows a particle in equilibrium under the forces shown.</p>\n<p><img src=\"resources/question-resources/statics1.png\" style=\"display: block; margin-left: auto; margin-right: auto;\"/></p>\n<p>You are told that&nbsp;$F_1 = \\var{F1} \\ \\mathrm{N}$ and that the angles are $\\theta_1 = \\var{theta1}^{\\circ}, \\ \\theta_2 = \\var{theta2}$ and $\\theta_3 = \\var{theta3}^{\\circ}$.</p>\n<p>By resolving horizontally and vertically, find the magnitude of the forces $F_2 $ and $F_3 $, in Newtons, and input them to 3 significant figures below.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"theta2": {"definition": "random(10..70#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta2", "description": ""}, "F1": {"definition": "random(2..4#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "F1", "description": ""}, "F2": {"definition": "(F3*cos(radians(theta3))-F1*cos(radians(theta1)))/(cos(radians(theta2)))", "templateType": "anything", "group": "Ungrouped variables", "name": "F2", "description": ""}, "theta1": {"definition": "random(10..60#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta1", "description": ""}, "theta3": {"definition": "random(2..30#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta3", "description": ""}, "F3": {"definition": "(F1*sin(radians(theta1))+F1*(cos(radians(theta1))/cos(radians(theta2)))*sin(radians(theta2)))/((sin(radians(theta3))+(cos(radians(theta3))*sin(radians(theta2)))/cos(radians(theta2))))", "templateType": "anything", "group": "Ungrouped variables", "name": "F3", "description": "<p>F3 calculated by the student</p>"}, "check2": {"definition": "F1*sin(radians(theta1))-F2*sin(radians(theta2))-F3*sin(radians(theta3))", "templateType": "anything", "group": "Ungrouped variables", "name": "check2", "description": "<p>Total vertical force. Should be zero for equilbrium.</p>"}, "check": {"definition": "F1*cos(radians(theta1))+F2*cos(radians(theta2))-F3*cos(radians(theta3))", "templateType": "anything", "group": "Ungrouped variables", "name": "check", "description": "<p>Total horizontal force. Should be zero for equilibrium</p>"}}, "metadata": {"description": "<p>Three forces act on a particle. You're given the magnitude of one, and the directions of all three. Find the magnitudes of the other two forces.</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/statics6.png", "/srv/numbas/media/question-resources/statics6.png"], ["question-resources/statics4.png", "/srv/numbas/media/question-resources/statics4.png"], ["question-resources/statics5.png", "/srv/numbas/media/question-resources/statics5.png"], ["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"], ["question-resources/statics1.png", "/srv/numbas/media/question-resources/statics1.png"]]}