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`

Options

A. `Arhog[((h_1-h_2))/(2)]`
B. `Arhog[((h_1-h_2))/(2)]^2`
C. `Arho[((h_1-h_2))/(4)]`
D. `Arhog[((h_1-h_2))/(4)]^2`

Answer

B

Explanation

Since the volume remains same.
`Ah_1+Ah_2=Ah+ahimpliesh=(h_1+h_2)/(2)`
`U_1=([(rhoAh_1)gh_1+(rhoAh_2)gh_2])/(2)`
`U_2=((rhoAh_1)gh)/(2)xx2`
`W_(gr)=-(U_2-U_1)=U_1-U_2`
`=(rhoAg)/(2)[(h_1^2+h_2^2)-2h^2]`
`=(rhoAg)/(2)[(h_1^2+h_2^2)-2((h_1+h_2)/(2))^2]`
`=(rhoAg)/(2)[((h_1+h_2)/(2))^2]=rhoAg[(h_1-h_2)/(2)]^2`