6. \begin{figure}[h] \end{figure} A \(\operatorname { rod } A B\) has mass \(M\) and length \(2 a\).
The rod has its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall. The rod makes an angle \(\theta\) with the ground, as shown in Figure 3.
The rod is at rest in limiting equilibrium.

1. State the direction (left or right on Figure 3 above) of the frictional force acting on the \(\operatorname { rod }\) at \(A\). Give a reason for your answer. The magnitude of the normal reaction of the wall on the rod at \(B\) is \(S\).
   In an initial model, the rod is modelled as being uniform.
   Use this initial model to answer parts (b), (c) and (d).
2. By taking moments about \(A\), show that $$S = \frac { 1 } { 2 } M g \cot \theta$$ The coefficient of friction between the rod and the ground is \(\mu\)
   Given that \(\tan \theta = \frac { 3 } { 4 }\)
3. find the value of \(\mu\)
4. find, in terms of \(M\) and \(g\), the magnitude of the resultant force acting on the rod at \(A\). In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to \(B\) than it is to \(A\). A new value for \(S\) is calculated using this new model, with \(\tan \theta = \frac { 3 } { 4 }\)
5. State whether this new value for \(S\) is larger, smaller or equal to the value that \(S\) would take using the initial model. Give a reason for your answer.