5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-12_424_1118_221_420}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
A uniform rod \(A B\) has length \(8 a\) and weight \(W\).
The end \(A\) of the rod is freely hinged to horizontal ground.
The rod rests in equilibrium against a block which is also fixed to the ground.
The block is modelled as a smooth solid hemisphere with radius \(2 a\) and centre \(D\).
The point of contact between the rod and the block is \(C\), where \(A C = 5 a\)
The rod is at an angle \(\theta\) to the ground, as shown in Figure 1.
Points \(A , B , C\) and \(D\) all lie in the same vertical plane.
- Show that \(A D = \sqrt { 29 } a\)
- Show that the magnitude of the normal reaction at \(C\) between the rod and the block is \(\frac { 4 } { \sqrt { 29 } } W\)
The resultant force acting on the rod at \(A\) has magnitude \(k W\) and acts at an angle \(\alpha\) to the ground.
- Find (i) the exact value of \(k\)
(ii) the exact value of \(\tan \alpha\)
\includegraphics[max width=\textwidth, alt={}, center]{1732eb73-8c16-4a45-8d3b-a88e659e47ea-15_72_1819_2709_114}