Standard +0.8 This is a multi-part mechanics problem requiring equilibrium of moments and forces with a non-standard geometry (tension at angle 2θ to the rod), followed by friction analysis. The setup requires careful resolution of forces and taking moments about a point, with some algebraic manipulation to reach the given result. The friction coefficient application adds complexity. This is above average difficulty but uses standard A-level mechanics techniques without requiring exceptional insight.
The diagram below shows a uniform rod \(A B\) of weight \(W N\) and length \(2 l\), with its lower end \(A\) resting on a rough horizontal floor. A light cable is attached to the other end \(B\). The rod is in equilibrium when it is inclined at an angle of \(\theta\) to the floor, where \(0 ^ { \circ } < \theta \leqslant 45 ^ { \circ }\). The tension in the cable is \(T \mathrm {~N}\) acting at an angle of \(2 \theta\) to the rod, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-18_508_1105_559_479}
(i) Show that \(T = \frac { W } { 4 } \operatorname { cosec } \theta\).
(ii) Hence determine the normal reaction of the floor on the rod at \(A\), giving your answer in terms of \(W\).
Given that the coefficient of friction between the floor and the rod is \(\frac { \sqrt { 3 } } { 3 }\), calculate the minimum possible value for \(\theta\).
The region \(R\), shown in the diagram below, is bounded by the coordinate axes and the curve
$$y = \frac { a } { b } \sqrt { b ^ { 2 } - x ^ { 2 } }$$
where \(a , b\) are constants.
\includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-21_451_1116_644_468}
The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a uniform solid \(S\). The volume of \(S\) is \(\frac { 2 } { 3 } \pi a ^ { 2 } b\).
Use integration to show that the distance of the centre of mass of \(S\) from the \(y\)-axis is \(\frac { 3 b } { 8 }\).
The diagram below shows a small tree growing in a pot. The uniform solid \(S\) described on the previous page may be used to model the part of the tree above the pot. This part of the tree has height \(h \mathrm {~cm}\) and base radius \(\frac { h } { 4 } \mathrm {~cm}\). The pot, including its contents, may be modelled as a solid cylinder of height 50 cm and radius 25 cm .
\includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-22_846_839_1596_612}
You may assume that the density of the pot, including its contents, is equal to 20 times the density of the part of the tree above the pot.
A gardener suggests that a tree is said to have outgrown its pot if the centre of mass, of both the tree and its pot, lies above the height of the pot. Determine the maximum value of \(h\) before the tree outgrows its pot.
Identify one possible limitation of the model used that could affect your answer to part (b).
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\begin{enumerate}
\item The diagram below shows a uniform rod $A B$ of weight $W N$ and length $2 l$, with its lower end $A$ resting on a rough horizontal floor. A light cable is attached to the other end $B$. The rod is in equilibrium when it is inclined at an angle of $\theta$ to the floor, where $0 ^ { \circ } < \theta \leqslant 45 ^ { \circ }$. The tension in the cable is $T \mathrm {~N}$ acting at an angle of $2 \theta$ to the rod, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-18_508_1105_559_479}\\
(a) (i) Show that $T = \frac { W } { 4 } \operatorname { cosec } \theta$.\\
(ii) Hence determine the normal reaction of the floor on the rod at $A$, giving your answer in terms of $W$.\\
(b) Given that the coefficient of friction between the floor and the rod is $\frac { \sqrt { 3 } } { 3 }$, calculate the minimum possible value for $\theta$.\\
\item The region $R$, shown in the diagram below, is bounded by the coordinate axes and the curve
\end{enumerate}
$$y = \frac { a } { b } \sqrt { b ^ { 2 } - x ^ { 2 } }$$
where $a , b$ are constants.\\
\includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-21_451_1116_644_468}
The region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis to form a uniform solid $S$. The volume of $S$ is $\frac { 2 } { 3 } \pi a ^ { 2 } b$.\\
(a) Use integration to show that the distance of the centre of mass of $S$ from the $y$-axis is $\frac { 3 b } { 8 }$.\\
The diagram below shows a small tree growing in a pot. The uniform solid $S$ described on the previous page may be used to model the part of the tree above the pot. This part of the tree has height $h \mathrm {~cm}$ and base radius $\frac { h } { 4 } \mathrm {~cm}$. The pot, including its contents, may be modelled as a solid cylinder of height 50 cm and radius 25 cm .\\
\includegraphics[max width=\textwidth, alt={}, center]{36112cfa-20c4-4ba8-b972-6b7b44e5182f-22_846_839_1596_612}
You may assume that the density of the pot, including its contents, is equal to 20 times the density of the part of the tree above the pot.\\
(b) A gardener suggests that a tree is said to have outgrown its pot if the centre of mass, of both the tree and its pot, lies above the height of the pot. Determine the maximum value of $h$ before the tree outgrows its pot.\\
(c) Identify one possible limitation of the model used that could affect your answer to part (b).
\section*{END OF PAPER}
Additional page, if required. Write the question number(s) in the left-hand margin.
\section*{PLEASE DO NOT WRITE ON THIS PAGE}
\section*{PLEASE DO NOT WRITE ON THIS PAGE}
\hfill \mbox{\textit{WJEC Further Unit 6 2024 Q5}}