A uniform cylinder with a rough surface and of radius \(a\) is fixed with its axis horizontal. Two identical uniform rods \(A B\) and \(B C\), each of weight \(W\) and length \(2 a\), are rigidly joined at \(B\) with \(A B\) perpendicular to \(B C\). The rods rest on the cylinder in a vertical plane perpendicular to the axis of the cylinder with \(A B\) at an angle \(\theta\) to the horizontal. \(D\) and \(E\) are the midpoints of \(A B\) and \(B C\) respectively and also the points of contact of the rods with the cylinder (see diagram). The rods are about to slip in a clockwise direction. The coefficient of friction between each rod and the cylinder is \(\mu\).

The normal reaction between \(A B\) and the cylinder is \(R\) and the normal reaction between \(B C\) and the cylinder is \(N\).
(a) Find the ratio \(R: N\) in terms of \(\mu\).

(b) Given that \(\mu=\frac{1}{3}\), find the value of \(\tan \theta\).

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