Krotov problem: how to write Energy conservation for this fluid?

[ermia]

Homework Statement

I uploaded the problem. when I want to write the potential energy of the fluid in the initial and final state I don't find the right answer. I know the rest of the problem.

Relevant Equations

$$ U=\rho g h A h_{cm} $$

I wrote some potentials but they were wrong. I used the cm of all fluid parts and I used the radius which is $$ \sqrt S/ \pi $$ .

 

[ermia]

That equation for radius has a "type" error.
$$r =\sqrt \frac{S}{\pi} $$

 

[haruspex]

Per forum rules, please post your working. Otherwise we have no way to know where you are going wrong.

 

[Lnewqban]

Could you explain how is that radius, which is not shown as data in the problem, relevant?

 

[haruspex]

Could you explain how is that radius, which is not shown as data in the problem, relevant?

S is the cross sectional area, so $S=\pi r^2$. But if you are asking why bother calculating the radius (you don't even have to assume it is a circular cross section) then I agree.