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. `2h//3`
B. `5h//3`
C. `5h//3`
D. `5h//2`

Answer

C

Explanation

Consider the eqilibrium of wooden block. Forces acting in the downward direction are as follows

a. Weight of wooden cylinder
`=pi(4r)^(2)xxhxxrho/3xxg=pixx16r^(2)(hrho)/3g`
b. Force due to pressure `(P_(1))` created by liquid of height `h_(1)` above the wooden block is
`=P_(1)xxpi(4r^(2))=[P_(0)+h_(1)rhog]xxpi(4r^(2))`
`=[rho_(0)+h_(1)rhog]pixx16r^(2)`
Force acting on the upward direction due to pressure `p_(0)` exerted from below wooden block and atmospheric pressure is
`=p_(2)xxpi[(4r)^(2)-(2r)^(2)]+P_(0)xx(2r)^(2)`
`[rho_(0)+h_(1)rhog]pixx16r^(2)`
Force acting on the upward directioin due to pressure `P_(0)` exerted from below the wooden block and atmpsheric pressure is
`=p_(2)xxpi[(4r)^(2)-(2r)^(2)]+P_(0)xx(2r)^(2)`
`=[P_(0)+(h_(1)+h)rhog]xxpixx12r^(2)+4r^(2)P_(0)`
At the verge of rising
`[P_(0)+(h_(1)+h)rhog]pixx12r^(2)+4r^(2)P_(0)`
`pixx10r^(2)hxxrho/3g+[P_(0)+h_(1)rhog)xxpixx16r^(2)`
`12h_(1)+12h=(16h)/3+16h`
`12h_(1)-(16h)/3=4h_(1)implies(5h)/(3h_(1))`