// Numbas version: exam_results_page_options {"name": "John's copy of Block sliding down a slope", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["mass", "theta", "a", "R", "mu", "v", "s"], "name": "John's copy of Block sliding down a slope", "tags": [], "advice": "

a)

To find the normal reaction force $R$, we resolve the forces perpendicular to the plane.

\\begin{align}
R - mg \\cos \\theta & = 0, \\\\
R & = mg \\cos \\theta, \\\\
& = (\\var{mass} \\times 9.8) \\cos (\\var{theta}^{\\circ}), \\\\
& = \\var{R} \\ \\mathrm{N}.
\\end{align}

b)

To find the coefficient of friction, $\\mu$, we resolve parallel to the plane and use our $R$ value from part a).

\\begin{align}
mg \\cos (90^{\\circ} - \\theta) - \\mu R & = ma, \\\\
\\mu & = \\frac{mg \\cos(90^{\\circ} - \\theta) - ma}{R}, \\\\
& = \\frac{ (\\var{mass} \\times 9.8) \\cos (\\var{90 - theta}^{\\circ}) - (\\var{mass} \\times \\var{a})}{\\var{R}}, \\\\
& = \\var{precround(mu,3)}. \\end{align}

The coefficient of friction between the block and the plane is $\\var{precround(mu,3)}$.

c)

This question can be solved using the SUVAT equations.

We know that the block was initially at rest, so $u = 0$.

The acceleration $a = \\var{a}$.

The final velocity $v = \\var{v}$.

We can use the equation $v^2 = u^2 + 2as$, and solve for the distance, $s$.

\\begin{align}
v^2 & = u^2 + 2 as, \\\\
\\var{v}^2 & = 0 + (2 \\times \\var{a}s), \\\\
s & = \\simplify[]{{v}/(2*{a})}, \\\\
& = \\var{precround(slide_distance,3)}\\mathrm{m}. \\end{align}

The block slid $\\var{precround(v^2/2*a,3)}\\mathrm{m}$ down the slope before reaching the given speed.

", "rulesets": {}, "parts": [{"prompt": "

Find the normal reaction, $R \\ \\mathrm{N}$ between the block and the plane, to 3 decimal places.

", "allowFractions": false, "variableReplacements": [], "maxValue": "R", "minValue": "R", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "

Using $R$ find the coefficient of friction, $\\mu$, between the block and the plane, to 3 decimal places.

", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "mu", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "mu", "type": "numberentry", "showPrecisionHint": false}, {"precisionType": "dp", "prompt": "

Suppose that the block hits the bottom of the slope at a speed of $\\var{v}\\mathrm{ms^{-1}}$. To 3 decimal places, how far in metres ($\\mathrm{m}$) has the block slid from its initial position?

", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "precision": "3", "maxValue": "v^2/2*a", "variableReplacementStrategy": "originalfirst", "strictPrecision": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "precisionPartialCredit": 0, "scripts": {}, "marks": 1, "minValue": "v^2/2*a", "type": "numberentry", "showPrecisionHint": false}], "extensions": [], "statement": "

A block of mass $\\var{mass}\\mathrm{kg}$ slides down a rough plane which is inclined at an angle $\\theta = \\var{theta}^{\\circ}$ to the horizontal. The mass begins at rest and accelerates at $\\var{a}\\mathrm{ms^{-2}}$.

The acceleration due to gravity is $g=9.8\\mathrm{ms^{-2}}.$

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": "mu<1"}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(2..4.5#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": "

The acceleration of the block down the slope.

"}, "mu": {"definition": "(mass*9.8*cos(radians(90-theta))-mass*a)/R", "templateType": "anything", "group": "Ungrouped variables", "name": "mu", "description": "

The coefficient of friction between the block and the plane.

"}, "s": {"definition": "v^2/(2*a)", "templateType": "anything", "group": "Ungrouped variables", "name": "s", "description": "

The distance the block slides before it reaches the given velocity.

"}, "R": {"definition": "precround(mass*9.8*cos(radians(theta)),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "R", "description": "

The normal reaction force of the plane on the block, rounded to 3 d.p.

"}, "mass": {"definition": "random(0.25..10#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "mass", "description": "

The mass of the block.

"}, "v": {"definition": "random(0.5..5#0.25)", "templateType": "randrange", "group": "Ungrouped variables", "name": "v", "description": ""}, "theta": {"definition": "random(30..50#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "theta", "description": "

The angle of the slope.

"}}, "metadata": {"description": "

A block of given mass is sliding down the plane, with given acceleration. Find the normal reaction force, the coefficient of friction, and the distance travelled before reaching a given speed.

", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "John Bridges", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/913/"}]}]}], "contributors": [{"name": "John Bridges", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/913/"}]}