A uniform shelf, ABABAB, of length a a\,a m and mass m m\,m kg, is at rest with A A\,A against a rough vertical wall. The shelf is held in a horizontal position by a wire attached to B B\,B and to the vertical wall at CCC. The tension in the wire is 6mg. A decorative ornament of mass 4m is placed on the shelf at P P\,P where AP=xAP = xAP=x. The angle between the wire and the shelf is 30∘30^\circ30∘. The ornament is modelled as a particle and the wire is modelled as a light inextensible string.
Find x x\,x in terms of aaa.
The reaction force at A A\,A makes an angle θ \theta\,θ with the shelf. Find the value of tanθ\tan \thetatanθ.