A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre $$\(O\)$$ and radius $$\(5 a\)$$. A uniform $$\(\operatorname{rod} A B\)$$, of length $$\(16 a\)$$ and weight $$\(W\)$$, rests in equilibrium on the hemisphere with end $$\(A\)$$ on the ground. The rod rests on the hemisphere at the point $$\(C\)$$, where $$\(A C=12 a\)$$ and angle $$\(C A O=\alpha\)$$, as shown in Figure 1. Points $$\(A, C, B\)$$ and $$\(O\)$$ all lie in the same vertical plane. (a) Explain why $$\(A O=13 a\)$$ (1) The normal reaction on the rod at $$\(C\)$$ has magnitude $$\(k W\)$$ (b) Show that $$\(k=\frac{8}{13}\)$$ (3) The resultant force acting on the $$\(\operatorname{rod}\)$$ at $$\(A\)$$ has magnitude $$\(R\)$$ and acts upwards at $$\(\theta^{\circ}\)$$ to the horizontal. (c) Find (i) an expression for $$\(R\)$$ in terms of $$\(W\)$$ (ii) the value of $$\(\theta\)$$ (8)

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