Weight of a Box between two fluids of different densities

[viciousp]

Homework Statement

A box with a height L, and a cross sectional Area A is floating between two fluids of densities p1 and p2. Determine the mass of the box using ho, h1, h2, p1, p2, and A.

Homework Equations

F= pVg

The Attempt at a Solution

In the file attached I drew my free body diagram and the image of the box. Solving my FBD I get:

mg+p1(A)(ho)g=p2(A)(h2)g

Solving for m:

p2(A)(h2)-p1(A)(ho)

but with Archimedes principal the answer should be p2(A)(h2)+p1(A)(h1), so what did I do wrong/forget?

Thanks

 

[gneill]

The bottom liquid is under the top liquid (obviously), but this has consequences for the pressures that are felt by the box in the bottom liquid.

 

[viciousp]

So the only thing I can think of is the pressure of the top liquid acting on the bottom liquid, as well as the buoyancy of the bottom liquid with respect to the top liquid.

Im assuming these two forces are equal to each other Fb= (h1+h0)A*p1. If I include the buoyancy of liquid 2 on liquid 1 on the box diagram it does give me the answer I want, but I don't conceptually understand why.

 

[gneill]

Write the expressions for the pressures at the bottom of the box and the top of the box (at depths h0 and h0+h1+h2).

 

[viciousp]

Ahh, I see thanks now it makes sense.