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A uniform rod $A B$, of weight $W$ and length $2 a$, rests with the end $A$ on a rough horizontal plane. A light inextensible string $B C$ is attached to the rod at $B$ and passes over a small smooth fixed peg $P$, which is at a distance $h$ vertically above $A$. A particle is attached at $C$ and hangs vertically. The points $A , B$ and $C$ are all in the same vertical plane. In equilibrium the rod is inclined at an angle $\theta$ to the horizontal (see diagram). The coefficient of friction between the rod and the plane is $\mu$. Show that

$$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$

Given that the particle attached at $C$ has weight $k W$, angle $A B P = 90 ^ { \circ }$ and $h = 3 a$, find\\
(i) the value of $k$,\\
(ii) the horizontal component of the force on $P$, in terms of $W$.

\hfill \mbox{\textit{CAIE FP2 2011 Q10 EITHER}}