A bobbin rolls without slipping on a horizontal surface so that the velocity \(v\) of the end of the thread (point \(A\)) is directed along the horizontal. A board hinged at point \(B\) leans against the bobbin making an angle \(2x\) with the horizontal. The inner and outer radii of the bobbin are \(r\) and \(R\) respectively.

If the angular velocity \(w\) of the board can be expressed in terms of \(x\) as \(\dfrac{kv(\sin x)^{2}}{r+R}\), find the magnitude of \(k\).

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