![](\"resources/question-resources/Connected_particles.png\")
Two masses in a box which is attached to a string. Finding acceleration by modelling the two masses as the whole system. Find forces exerted by the two masses.
"}, "variablesTest": {"condition": "a>0", "maxRuns": 100}, "statement": "Suppose that there is a light box attached to a vertical light inextensible string. The box holds two masses $A$ and $B$ as shown in the diagram below, where $A$ rests upon $B$.
\n
Suppose that the mass of $A$ is $m_1 = \\var{massa} \\, \\mathrm{kg}$ and the mass of $B$ is $m_2 = \\var{massb} \\, \\mathrm{kg}$. The acceleration due to gravity is $g=9.8 \\, \\mathrm{ms^{-2}}$.
\nGive your answers to the following questions to 3 decimal places.
", "extensions": [], "preamble": {"css": "", "js": ""}, "parts": [{"notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "strictPrecision": false, "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "scripts": {}, "correctAnswerFraction": false, "mustBeReduced": false, "variableReplacements": [], "precision": "3", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "marks": 1, "maxValue": "(T-massa*g-massb*g)/mass", "unitTests": [], "prompt": "Suppose that the tension in the string is $\\var{T} \\, \\mathrm{N}$ and that the box is raised vertically, using the string. With what acceleration, in $\\mathrm{ms^{-2}}$, is the box raised?
", "customMarkingAlgorithm": "", "minValue": "(T-massa*g-massb*g)/mass", "precisionPartialCredit": 0}, {"notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "strictPrecision": false, "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "scripts": {}, "correctAnswerFraction": false, "mustBeReduced": false, "variableReplacements": [], "precision": "3", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "marks": 1, "maxValue": "massa*g+a*massa", "unitTests": [], "prompt": "Using the acceleration in $\\mathrm{ms^{-2}}$ found in part a), find the force in Newtons ($\\mathrm{N}$) exerted on mass $B$ by mass $A$.
", "customMarkingAlgorithm": "", "minValue": "massa*g+a*massa", "precisionPartialCredit": 0}, {"notationStyles": ["plain", "en", "si-en"], "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "strictPrecision": false, "precisionType": "dp", "showPrecisionHint": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "allowFractions": false, "type": "numberentry", "correctAnswerStyle": "plain", "scripts": {}, "correctAnswerFraction": false, "mustBeReduced": false, "variableReplacements": [], "precision": "3", "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "marks": 1, "maxValue": "T+0.001", "unitTests": [], "prompt": "Find the force in Newtons ($\\mathrm{N}$) exerted on mass $B$ by the box.
", "customMarkingAlgorithm": "", "minValue": "T-0.001", "precisionPartialCredit": 0}], "tags": [], "variables": {"a": {"templateType": "anything", "description": "acceleration 3d.p.
", "definition": "precround((T-massa*g-massb*g)/mass,3)", "group": "Ungrouped variables", "name": "a"}, "mass": {"templateType": "anything", "description": "", "definition": "massa+massb", "group": "Ungrouped variables", "name": "mass"}, "massb": {"templateType": "randrange", "description": "", "definition": "random(1.6..3#0.1)", "group": "Ungrouped variables", "name": "massb"}, "g": {"templateType": "anything", "description": "", "definition": "9.8", "group": "Ungrouped variables", "name": "g"}, "aexact": {"templateType": "anything", "description": "", "definition": "(T-massa*g-massb*g)/mass", "group": "Ungrouped variables", "name": "aexact"}, "massa": {"templateType": "randrange", "description": "", "definition": "random(0.25..1.5#0.25)", "group": "Ungrouped variables", "name": "massa"}, "T": {"templateType": "randrange", "description": "tension in string
", "definition": "random(45..80#0.05)", "group": "Ungrouped variables", "name": "T"}}, "advice": "a) We can draw a diagram to show the forces acting on the system. As all parts are moving in the same straight line (vertically) we can resolve for the whole system.
\n![](\"resources/question-resources/Connected_particles2.png\")
We have that
\n\\begin{align} T - m_1g - m_2g & = (m_1 + m_2)a, \\\\
\\var{T} - \\var{massa}g - \\var{massb}g & = \\var{mass}a, \\\\
a & =\\frac{\\var{T} - \\var{massa}g - \\var{massb}g}{\\var{mass}}, \\\\
& = \\var{a}\\mathrm{ms^{-2}}.\\end{align}
The acceleration of the system is $\\var{a}\\mathrm{ms^{-2}}$.
\nb) To find the force exerted on mass $B$ by mass $A$ we can find the force exerted on $A$ by $B$ (the normal reaction, $R$) and use Newton's 3rd Law to say that the force exerted on $B$ by $A$ will have the same magnitude.
\n![](\"resources/question-resources/Connected_particles3.png\")
We resolve the forces to find $R$, using the 3d.p. value for $a$ from part a).
\n\\begin{align} R - m_1g & = m_1a, \\\\
R & = m_1g + m_1a, \\\\
& = \\var{massa} \\left(9.8 + \\var{precround(aexact,4)}\\right), \\\\
& = \\var{precround(massa*(9.8+aexact),3)} \\mathrm{N}.\\end{align}
So the force exerted on $B$ by $A$ is $\\var{precround(massa*(9.8+aexact),3)} \\mathrm{N}$.
\nc) Let X be the force exerted on mass $B$ by the box.
\nResolving forces acting on mass $B$, with upwards being the positive direction, and using our answer from (b), we obtain:
\n\\begin{align} X - \\var{precround(massa*(9.8+aexact),4)} - m_2g & = m_2a, \\\\
X & = \\var{precround(massa*(9.8+aexact),4)}+\\var{massb}g+\\var{massb}\\times \\var{precround(aexact,4)}, \\\\
X & = \\var{precround(massa*(9.8+aexact)+massb*g+massb*aexact,3)} \\mathrm{N} \\end{align}