A$.2$ A vertical rectangular sluice door, $AB$ of $80cm$ wide and $120cm$ tall is rested frictionlessly at the exit of a water tank and hinged at $A$ as shown in Fig. A$\mathrm{.}2.$ The door is kept close by a weight $W,$ hanged at the end of a horizontal cantilever beam, which is fixed to the centre of the door. The cantilever beam is $30cm$ long. Determine the minimum weight $W$ required to keep the sluice door close by neglecting the weight of the door and the cantilever beam.

$(10$ marks$)$

Fig. A$.2$