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Two identical cylindrical vessel with th...

Two identical cylindrical vessel with their bases at the same level each contain a liquid of density `rho`. The height of the liquid in one vessel is `h_(1)` and in the other is `h_(2)` the area of either base is A. What is the work done by gravity is equalising the levels when the two vessels are connected?

Text Solution

Verified by Experts

The correct Answer is:
`(gArho)/(4)(h_(t) - h_(2))^(2)`
Let `h` be level in equilibrium. Equating the volumes, we have
`Ah_(1) + Ah_(2) = 2Ah`
`:. H = ((h_(1) + h_(2))/(2))`
Work done by gravity `= U_(i) - U_(f)`
Work `W = (m_(1)gh_(1)/(2) + m_(2)gh_(2)/(2)) - (m_(1) + m_(2))g h/(2)`
`= (Ah_(1)rhogh_(1))/(2) + (Ah_(2)rhogh_(2))/(2) - [Ah_(1) - Ah_(2)rho + Ah_(2)rholg((h_(1) + h_(2))/(4))`
Simplifying this, we get
`W = (rhoAg)/(4)(h_(1) - h_(2))^(2)`
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