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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 $B$.

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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 $g=9.8 \\, \\mathrm{ms^{-2}}$.

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Give your answers to the following questions to 3 decimal places.

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acceleration 3d.p.

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tension in string

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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.

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We have that 

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\\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}

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The acceleration of the system is $\\var{a}\\mathrm{ms^{-2}}$.

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b) 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.

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We resolve the forces to find $R$, using the 3d.p. value for $a$ from part a).

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\\begin{align} R - m_1g & = m_1a, \\\\
                                    R & = m_1g + m_1a, \\\\
                                       & = \\var{massa} \\left(9.8 + \\var{a}\\right), \\\\
                                        & = \\var{precround(massa*(9.8+a),3)} \\mathrm{N}.\\end{align}

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So the force exerted on $B$ by $A$ is $\\var{precround(massa*(9.8+a),3)} \\mathrm{N}$.

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c) To find the force exerted on mass $B$ by the box we find the force exerted on the box by $B$ and then use Newton's 3rd Law to say that the force exerted on $B$ by the box has the same magnitude but is in the opposite direction.

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Looking only at the box we have

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\\begin{align} T - F & = 0 \\times a, \\\\
                       T & = F, \\\\
                         \\var{T} \\mathrm{N} & = F. \\end{align}

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Where here $F$ is the force exerted on the box by $B$ and the mass of the box is $0$ as it is modelled as being light.

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So the force exerted on $B$ by the box is $\\var{T} \\mathrm{N}$ upwards.

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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.

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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?

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Using the acceleration in $\\mathrm{ms^{-2}}$ found in part a), find the force in Newtons ($\\mathrm{N}$) exerted on mass $B$ by mass $A$.

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Find the force in Newtons ($\\mathrm{N}$) exerted on mass $B$ by the box.

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