![](\"resources/question-resources/force_component_image_6.png\"/)
Find the magnitude in Newtons (to 3 d.p.) that $P$ must exceed to achieve movement if $P$ is applied horizontally.
", "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.
", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precisionType": "dp", "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true, "showPrecisionHint": false, "correctAnswerFraction": false, "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showCorrectAnswer": true, "minValue": "max_friction", "strictPrecision": false, "extendBaseMarkingAlgorithm": true, "maxValue": "max_friction", "scripts": {}, "allowFractions": false, "mustBeReduced": false, "customMarkingAlgorithm": ""}, {"precision": "3", "prompt": "Find the magnitude in Newtons (to 3 d.p.) that $P$ must exceed to achieve movement if $P$ is applied at an angle $\\alpha$ above the horizontal, where $ \\tan \\alpha = \\frac{\\var{O}}{\\var{A}}$.
", "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.
", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precisionType": "dp", "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true, "showPrecisionHint": false, "correctAnswerFraction": false, "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showCorrectAnswer": true, "minValue": "(mu*mass*g)/(cos1+mu*sine1)", "strictPrecision": false, "extendBaseMarkingAlgorithm": true, "maxValue": "(mu*mass*g)/(cos1+mu*sine1)", "scripts": {}, "allowFractions": false, "mustBeReduced": false, "customMarkingAlgorithm": ""}, {"precision": "3", "prompt": "Find the magnitude in Newtons (to 3 d.p.) that $P$ must exceed to achieve movement if $P$ is applied at an angle $\\alpha$ below the horizontal, where $\\tan \\alpha = \\frac{\\var{O2}}{\\var{A2}}$.
", "correctAnswerStyle": "plain", "precisionMessage": "You have not given your answer to the correct precision.", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "precisionType": "dp", "variableReplacements": [], "unitTests": [], "showFeedbackIcon": true, "showPrecisionHint": false, "correctAnswerFraction": false, "precisionPartialCredit": 0, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showCorrectAnswer": true, "minValue": "(mu*mass*g)/(cos2-mu*sine2)", "strictPrecision": false, "extendBaseMarkingAlgorithm": true, "maxValue": "(mu*mass*g)/(cos2-mu*sine2)", "scripts": {}, "allowFractions": false, "mustBeReduced": false, "customMarkingAlgorithm": ""}], "variable_groups": [], "ungrouped_variables": ["mass", "g", "R", "mu", "max_friction", "O_raw", "A_raw", "O", "A", "H", "cos1", "sine1", "O2_raw", "A2_raw", "O2", "A2", "H2", "cos2", "sine2"], "variables": {"sine2": {"name": "sine2", "group": "Ungrouped variables", "templateType": "anything", "description": "Sine of the second angle.
", "definition": "O2/H2"}, "mu": {"name": "mu", "group": "Ungrouped variables", "templateType": "randrange", "description": "coefficient of friction
", "definition": "random(0.01..0.99#1)"}, "A2": {"name": "A2", "group": "Ungrouped variables", "templateType": "anything", "description": "Adjacent side in the second angle.
", "definition": "A2_raw/gcd(O2_raw,A2_raw)"}, "mass": {"name": "mass", "group": "Ungrouped variables", "templateType": "randrange", "description": "mass of the block
", "definition": "random(3..10#0.25)"}, "H2": {"name": "H2", "group": "Ungrouped variables", "templateType": "anything", "description": "Hypotenuse in the second angle.
", "definition": "(O2^2+A2^2)^(1/2)"}, "H": {"name": "H", "group": "Ungrouped variables", "templateType": "anything", "description": "Hypotenuse in the first angle.
", "definition": "sqrt(A^2+O^2)"}, "A": {"name": "A", "group": "Ungrouped variables", "templateType": "anything", "description": "Adjacent side in the first angle.
", "definition": "A_raw/gcd(O_raw,A_raw)"}, "cos2": {"name": "cos2", "group": "Ungrouped variables", "templateType": "anything", "description": "Cosine of the second angle.
", "definition": "A2/H2"}, "O": {"name": "O", "group": "Ungrouped variables", "templateType": "anything", "description": "Opposite side in the first angle.
", "definition": "O_raw/gcd(O_raw,A_raw)"}, "O2_raw": {"name": "O2_raw", "group": "Ungrouped variables", "templateType": "randrange", "description": "Opposite side in the second angle.
\nA random number is generated, then we divide by the gcd with A2_raw to make sure they're coprime.
", "definition": "random(1..5#1)"}, "max_friction": {"name": "max_friction", "group": "Ungrouped variables", "templateType": "anything", "description": "Maximum frictional force on the object
", "definition": "mu*R"}, "O2": {"name": "O2", "group": "Ungrouped variables", "templateType": "anything", "description": "Opposite side in the second angle.
", "definition": "O2_raw/gcd(O2_raw,A2_raw)"}, "g": {"name": "g", "group": "Ungrouped variables", "templateType": "number", "description": "acceleration due to gravity
", "definition": "9.8"}, "R": {"name": "R", "group": "Ungrouped variables", "templateType": "anything", "description": "reaction force against the block
", "definition": "mass*g"}, "A2_raw": {"name": "A2_raw", "group": "Ungrouped variables", "templateType": "randrange", "description": "Adjacent side in the second angle.
\nA random number is generated, then we divide by the gcd with O2_raw to make sure they're coprime.
", "definition": "random(6..15#1)"}, "A_raw": {"name": "A_raw", "group": "Ungrouped variables", "templateType": "randrange", "description": "adjacent side in the first angle.
\nA random number is generated, then we divide by the gcd with O_raw to make sure they're coprime.
", "definition": "random(7..15#1)"}, "cos1": {"name": "cos1", "group": "Ungrouped variables", "templateType": "anything", "description": "Cosine of the first angle
", "definition": "A/H"}, "sine1": {"name": "sine1", "group": "Ungrouped variables", "templateType": "anything", "description": "Sine of the first angle
", "definition": "O/H"}, "O_raw": {"name": "O_raw", "group": "Ungrouped variables", "templateType": "randrange", "description": "Opposite side in the first angle.
\nA random number is generated, then we divide by the gcd with A_raw to make sure they're coprime.
", "definition": "random(1..6#1)"}}, "metadata": {"description": "A block lies on rough horizontal ground. The coefficient of friction is given and students find the maximum value of friction $F_{\\text{MAX}} = \\mu R$. This is then used to determine the magnitude of force that must be exceeded to achieve movement when the force is applied horizontally and at two different angles to the horizontal.
", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "statement": "A block of mass $\\var{mass} \\ \\mathrm{kg}$ lies at rest on a rough horizontal floor. The coefficient of friction between the mass and the floor is $\\var{mu}$. A force $P \\ \\mathrm{N}$ is applied to the mass which can push or pull the mass horizontally along.
\nThe acceleration due to gravity is $\\var{g} \\, \\mathrm{ms^{-2}}$.
", "functions": {}, "rulesets": {}, "advice": "If $P$ is applied horizontally to the mass which lies at rest, we can resolve $F = ma$ with acceleration $a=0$ to find the normal reaction $R$, and from this find the maximum frictional force $F_{\\text{MASS}}=\\mu R$.
\nIn the upwards direction we have
\n\\begin{align}
F & = ma \\\\
R - mg & = 0 \\\\
R & = mg \\\\
& = \\var{mass} \\times 9.8 \\\\
& = \\var{mass*9.8}
\\end{align}
The maximum possible frictional force is $F_{\\text{MASS}} = \\mu R = \\var{mu} \\times \\var{R} = \\var{precround(mu*R,3)} \\ \\mathrm{N}$. Therefore to achieve movement $P$ must exceed $\\var{precround(mu*R,3)} \\ \\mathrm{N}$.
\nNote that if $ \\tan \\alpha = \\frac{\\var{O}}{\\var{A}}$ then using SOHCAHTOA where $O = \\var{O}$, $A = \\var{A}$ and $H = \\sqrt{\\var{O}^2 + \\var{A}^2}$ we have that $\\sin \\alpha = \\frac{\\var{O}}{\\sqrt{\\var{O}^2 + \\var{A}^2}}$ and $\\cos \\alpha = \\frac{\\var{A}}{\\sqrt{\\var{O}^2 + \\var{A}^2}}$.
\n$P$ is applied at an angle above the horizontal as shown in the diagram. We can resolve $F=ma$ in the upward direction where acceleration $a=0$.
\n\\begin{align} R + P \\sin \\alpha - mg & = ma \\\\
R + \\frac{\\var{O}}{\\sqrt{\\var{O}^2 + \\var{A}^2}}P - (\\var{mass} \\times 9.8)& = 0 \\\\
R & = \\var{mass*9.8} - \\frac{\\var{O}}{\\sqrt{\\var{O}^2 + \\var{A}^2}}P
\\end{align}
We can now resolve $F=ma$ in the horizontal direction, where acceleration is also zero and $F_{\\text{MAX}} = \\mu R = \\var{mu} \\times \\left( \\var{mass*9.8} - \\frac{\\var{O}}{\\sqrt{\\var{O}^2 + \\var{A}^2}}P \\right)$.
\n\\begin{align} P \\cos \\alpha - F_{\\text{MAX}} &= ma \\\\
P \\cos \\alpha & = F_{\\text{MAX}} \\\\
\\frac{\\var{A}}{\\sqrt{\\var{O}^2 + \\var{A}^2}} P & = \\var{mu} \\times \\left( \\var{mass*9.8} - \\frac{\\var{O}}{\\sqrt{\\var{O}^2 + \\var{A}^2}}P \\right) \\\\
& = \\var{mu*mass*9.8} - \\frac{\\var{mu*O}}{\\sqrt{\\var{O}^2 + \\var{A}^2}} P \\\\
\\left( \\frac{\\var{A}}{\\var{precround(H,3)}} + \\var{precround(mu*O/H,3)} \\right) P & = \\var{mu*mass*9.8} \\\\
P & = \\frac{ \\var{mu*mass*9.8}}{ \\left( \\frac{\\var{A}}{\\var{precround(H,3)}} + \\var{precround(mu*O/H,3)} \\right)} \\\\
& = \\var{precround((mu*mass*9.8)/(A/H + mu*O/H),3)} \\end{align}
Therefore $P$ must exceed $\\var{precround((mu*mass*9.8)/(A/H + mu*O/H),3)} \\ \\mathrm{N}$ in order for the mass to start moving.
\nNote that if $\\tan \\alpha = \\frac{\\var{O2}}{\\var{A2}}$ then using SOHCAHTOA where $O = \\var{O2}$, $A = \\var{A2}$ and $H = \\sqrt{\\var{O2}^2 + \\var{A2}^2}$ we have that $\\sin \\alpha = \\frac{\\var{O2}}{\\sqrt{\\var{O2}^2 + \\var{A2}^2}}$ and $\\cos \\alpha = \\frac{\\var{A2}}{\\sqrt{\\var{O2}^2 + \\var{A2}^2}}$.
\n![](\"resources/question-resources/force_component_image_7.png\"/)
We can resolve $F = ma$ in the upward direction where acceleration is zero.
\n\\begin{align}
R - mg - P \\sin \\alpha & = m \\times 0 \\\\
R & = mg + P \\sin\\alpha \\\\
& = \\var{mass*9.8} + \\frac{\\var{O2}}{\\sqrt{\\var{O2}^2 + \\var{A2}^2}}P
\\end{align}
From this we can calculate $F_{\\text{MAX}} = \\mu \\times R = \\var{mu} \\times \\left(\\var{mass*9.8} + \\frac{\\var{O2}}{\\sqrt{\\var{O2}^2 + \\var{A2}^2}}P\\right)$.
\nWe then resolve $F = ma$ in the horizontal direction, where once again acceleration is zero.
\n\\begin{align} P \\cos \\alpha - F_{\\text{MAX}} & = m \\times 0 \\\\
\\frac{\\var{A2}}{\\sqrt{\\var{O2}^2 + \\var{A2}^2}}P & = \\var{mu}\\left(\\var{mass*9.8} + \\frac{\\var{O2}}{\\sqrt{\\var{O2}^2 + \\var{A2}^2}}P\\right) \\\\
\\left(\\frac{\\var{A2}}{\\sqrt{\\var{O2}^2 + \\var{A2}^2}} - \\frac{\\var{mu} \\times \\var{O2}}{\\sqrt{\\var{O2}^2 + \\var{A2}^2}}\\right)P & = \\var{mu} \\times \\var{mass*9.8} \\\\
\\var{precround((A2/H2) - (mu*O2/H),3)} P & = \\var{mu*9.8*mass} \\\\
P & = \\var{precround((mu*9.8*mass)/(A2/H2 - mu*O2/H),3)} \\end{align}
Therefore $P$ must exceed $\\var{precround((mu*9.8*mass)/(A2/H2 - mu*O2/H),3)}N$ in order for the mass to start moving.
", "extensions": [], "type": "question", "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}