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A solid cube floats in a liquid of densi...

A solid cube floats in a liquid of density `rho` with half of its volume submerged as shown. The side length of the cube is a, which is very small compared to the size of the container. Now a liquid of density `(rho)/(3)` (immiscible with the liquid already in the container) is poured slowly into the container. The column of the liquid of density `(rho)/(3)` that must be poured so that the cube is fully submerged (with its top surface coincident with the surface of the poured liquid) is :

A

`(a)/(3)`

B

`(a)/(2)`

C

`(2a)/(3)`

D

`(3a)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
D
Let the density of the cube be `rho_(o)`
Then, because cube is in equilibrium `rho_(o)a^(3)g=rho((a^(3))/(2))g`
`impliesrho_(o)=(rho)/(2)`
Let the final equilibrium position of the cube be as shown
Then, `rho_(o)a^(3)g=((rho)/(3))(a^(2)x)g+rho(a-x)(a^(2))g`
`impliesx=(3)/(4)a`
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