A uniform shelf, $AB$, of length 4 m and mass $2M$ kg, is at rest with $A$ against a rough vertical wall. The shelf is held in a horizontal position by a steel cable attached to $B$ and to the vertical wall at the point $C$. A heavy vase of mass $4M$ is placed on the shelf at $P$ where $AP=3$ m. The angle between the cable and the shelf is $\alpha$, where $\tan \alpha =\frac{4}{3}$. The shelf is modelled as a uniform rod, the vase is modelled as a particle and the cable is modelled as a light inextensible string.