6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-20_755_579_267_703}
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\caption{Figure 5}
\end{figure}
A uniform rod, \(A B\), of mass \(8 m\) and length \(2 a\), has its end \(A\) resting against a rough vertical wall. One end of a light inextensible string is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\), which is vertically above \(A\). The angle between the rod and the string is \(30 ^ { \circ }\). A particle of mass \(k m\) is fixed to the rod at \(C\), where \(A C = 0.5 a\). The rod is in equilibrium in a vertical plane perpendicular to the wall, and is at an angle of \(60 ^ { \circ }\) to the wall, as shown in Figure 5. The tension in the string is \(T\).
- Show that \(T = \frac { \sqrt { 3 } } { 4 } ( 16 + k ) m g\)
The coefficient of friction between the wall and the rod is \(\frac { 2 } { 3 } \sqrt { 3 }\).
Given that the rod is in limiting equilibrium, - find the value of \(k\).
\includegraphics[max width=\textwidth, alt={}, center]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-23_67_65_2656_1886}