(i) A non uniform cube of side length a is kept inside a container as shown in the figure. The average density of the material of the cube is 2 `rho` where `rho` is the density of water. Water is gradually fille din the container. It is observed that the cube begins to topple, about its edge into the plane of the figure passing through pointA, when the height of the water in the container becomes `(a)/(2)`. Find the distance of the centre of mass of the cube from the face AB of the cube. Assume that water seeps under the cube.

(ii) A rectangular concrete block (specific gravity =2.5) is used as a retatining wall in a reservoir of water. The height and width of the block are x and y respectively the height of water in the reservoir is `z=(3)/(4)x`. the concrete block cannot slide on the horizontal base but can rotate about an axis perpendicular to the plane of the figure and passing through point A

(a) Calculate the minimum value of the ratio `(y)/(x)` for which the block will not begin to overturn about A.
(b) Redo the above problem for the case when there is a seepage and a thin film of water is present under the block. Assume that a seal at A prevents the water from flowing out underneath the block.

Answer

(i) `x=(9)/(8)`
(ii) (a) `(3)/(4sqrt(10))`
(b) `(3)/(4sqrt(7))`