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## Homework Statement

In a cylindrical container (with radius R) there are 2 fluids (separated like water and oil, fluid 1 lies under fluid 2) with given volumes V_i, given densities ρ_i.

You let them rotate with respective angular frequencies ω_i.

There is no friction.

Find the functions of form [itex]y_i=a_i x^2+b_i[/itex] (cylindrical coordinates) that describe the surface (y_i) of each of the 2 fluids.

[Ignore cases of the surfaces touching each other or the bottom of the container]

**2. The attempt at a solution**

I was thinking about finding the functions y_i that would minimize the overall energy for the given values.

I wrote down the following:

[itex]E=E_{pot1}+E_{pot2}+E_{rot1}+E_{rot2}[/itex]

This is a function of ρ, ω, V, a_1, b_1, a_2, b_2

But because Volume is conserved, b_i depends on a_i so we can leave out b_i as an argument of the function.

Now we want to minimize the energy by solving the following two equations for the 2 unknown a.

[itex]\frac{∂E}{a_1}=0[/itex]

[itex]\frac{∂E}{a_2}=0[/itex]

By solving this we obtain a_i --> b_i --> y_i(r)

Is "minimizing the energy" a legitimate way of finding a solution for this problem?

I'm being careful, because I tried this on the same problem but exclusively for 1 fluid and it didn't work, because ∂E/∂a was independent of a.