Object in equilibrium between 2 fluids

[Epiclightning]

Homework Statement

if an object is in equilibrium at the interface of 2 fluids in an open tank, why do both provide upthrust?

Homework Equations

P=P' +dgh

The Attempt at a Solution

The upper fluid is not in contact with the lower part of the object, so why would the fluid provide an upward buoyant force? There is no area at which the fluid could exert a force.
Yet almost all the questions I have seen that deal with such a principle equate mg with the buoyant force due to the lower and upper liquids.

 

[CrazyNinja]

Find the pressure at the lower end of the block and you will understand where you are going wrong.

 

[Epiclightning]

Find the pressure at the lower end of the block and you will understand where you are going wrong.

That would be P₀ + h₁d₁g + h₂d₂g + (h₁+h₂)(d_objectg), right?

h1, h2 = heights of object in each liquid
d1, d2 = their densities
but how would this simplification help further?

 

[Chestermiller]

That would be P₀ + h₁d₁g + h₂d₂g + (h₁+h₂)(d_objectg), right?

h1, h2 = heights of object in each liquid
d1, d2 = their densities
but how would this simplification help further?

The pressure on the top of the block is P₀.

The pressure on the bottom of the block is P₀ + h₁d₁g + h₂d₂g

The difference in pressure between the bottom of the block and the top of the block is h₁d₁g + h₂d₂g.
So the force is A(h₁d₁g + h₂d₂g)=V₁d₁g+V₂d₂g. But this is just equal to the sum of the weights of the displaced fluids.

 

[Epiclightning]

So we don't include the weight of the block when we write the pressure?

 

[CrazyNinja]

So we don't include the weight of the block when we write the pressure?

No you don't.