A cylindrical tank has a hole of diameter 2r in its bottom. The hole is covered wooden cylindrical block of diameter 4r, height h and density `rho//3`.

Situation I: Initially, the tank is filled with water of density `rho` to a height such that the height of water above the top of the block is `h_1` (measured from the top of the block).
Situation II: The water is removed from the tank to a height `h_2` (measured from the bottom of the block), as shown in the figure. The height `h_2` is smaller than h (height of the block) and thus the block is exposed to the atmosphere.
Find the minimum value of height `h_1` (in situation 1), for which the block just starts to move up?

Options

A. (a) `(2h)/(3)`
B. (b) `(5h)/(4)`
C. (c) `(5h)/(3)`
D. (d) `(5h)/(2)`

Answer

C

Explanation

Consider the equilibrium of wooden block. Forces acting in the downward direction are
(a) Weight of wooden cylinder
`=pi(2r)^2xxhxxrho/3xxg`
`=pixx4r^2(hrho)/(3)g`

(b) Force due to pressure (`P_1`) created by liquid of height `h_1` above the wooden block is
`=P_1xxpi(2r)^2=[P_0+h_1rhog]xxpi(2r)^2`
`=[P_0+(h_1+h)rhog]xxpixx3r^2+P_0pir^2`
At the verge of rising
`[P_0+(h_1+h)rhog]xx(pixx3r^2)+pir^2P_0`
`=[P_0+h_1rhog]xx4pir^2+(pixx4r^2hrhog)/(3)` or `h_1=(5h)/(3)`