#### fluidistic

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**1. Homework Statement**

Consider a liquid whose density [tex]\rho[/tex] varies such that [tex]\rho (x)=Kx[/tex] where [tex]K[/tex] is a constant and [tex]x[/tex] is measured from the surface of the liquid. The liquid is contained into a cylinder, and the liquid height is [tex]L=\frac{2m}{AeK}[/tex].

We introduce a parallelepiped horizontally where its upper and bottom surfaces are worth [tex]A[/tex], its mass is [tex]m[/tex] and its height is [tex]2e[/tex].

Depreciate the viscosity of the liquid.

1)Determine the equilibrium position of the mass.

2)Determine the motion equation of it, if we apart it a [tex]\Delta x[/tex] from its equilibrium position.

**2. Homework Equations**

None given.

**3. The Attempt at a Solution**

I don't think I can apply Archimedes' principle because the fluid hasn't a constant density.

What I did was to calculate the pressure difference between the upper and bottom surfaces of the mass.

if X is the distance between the surface of the liquid and the center of mass of the parallelepiped, I get that the force acting on the upper surface is [tex]AK\int _0^{X-e}xdx=-\frac{AK}{2}(X-e)^2[/tex].

Similarly I get the force acting on the bottom surface :[tex]\frac{AK}{2}(X+e)^2[/tex].

I sum them up to get the total force due to the pressure's difference : [tex]2AKeX[/tex].

This force acts upward. However to calculate the net force on the mass, I have to add the last force : its weight : [tex]mg[/tex].

When the mass is at equilibrium, [tex]mg=2AKeX \Rightarrow X=\frac{mg}{2AKe}[/tex].

Am I right? If so, I'll try to continue alone.