Two cyllindrical vessels of equal cross sectional ara A contain water upto heights `h_1` and `h_2`. The vessels are interconnected so that the levels in them become equal. Calculate the work done by the force of gravity during the process. The density of water is `rho`

Explanation

According to the question, we draw the following diagram.
Since the mass of water remains same
`rhoAh_(1)+rhoAh_(2)=rhoAh+rhoAhimplies h =(h_(1)+h_(2))/(2)`
Potential energy of first vessel
`U_(1)=(rhoAh_(1))"g"(h_(1))/(2)+(rhoAh_(2))"g"(h_(2))/(2)=(rhoAg)/(2)(h_(1)^(2)+h_(2)^(2))` Potential energy of second vessel
`U_(2)=(rhoAh)"g"(h)/(2)xx2=(rhoAg(h_(1)+h_(2))^(2))/(4)`
Since the gravitational force is a conservative force, then
`W_(gr)=U_(1)-U_(2)=(rhoAg)/(4)(h_(1)-h_(2))^(2)`