A liquid of density `rho` is filled in two indentical cylindrical vessels with base area A and height of the liquid in one vessel is `h_(1)` and that in the other vessel is `h_(2)`. If both the vesels are connected with their base at the same level then calculate the work done by gravity in equalizing the levels.

Options

A. `(h_(1)-h_(2))grho`
B. `(h_(1)-h_(2))gArho`
C. `1/2(h_(1)-h_(2))^(2)gArho`
D. `1/4(h_(1)-h_(2))^(2)gArho`

Answer

D

Explanation

Here h is the average of `h_(1)` and `h_(2)`
So `h=(h_(1)+h_(2))/2`
And
`Deltah=h_(1)-h`
`=h_(1)-((h_(1)+h_(2))/2)`
`=((h_(1)-h_(2))/2)`
Mass of liquid dispalced is
`rho=m/V`
`m=rhoV`
`m=rhoxxAxxDeltah`
`m=rhoxxAxx((h_(1)-h_(2))/2)`
Where `rho=` density of liquid
`A=` area of vessel
V=volume of liquid displaced
`W=-DeltaU`
`W=mgDeltah`
`W=rhogA=((h_(1)-h_(2))/2)xx((h_(1)-h_(2))/2)`
`W=(rhogA)/4(h_(1)-h_(2))^(2)`