Archimedes principle problem? *1 fluid, 1 solid*

[jrrodri7]

I'm having trouble arriving at a finalized answer dealing with the sum of the forces with this problem dealing with Archimedes principle (buoyancy is weight of the displacement).

Here's the question:

A cube of wood whose edge is 12 mm is floating in a liquid in a glass with one of its faces parallel to the liquid surface. The density of wood is 762 kg/m^3, that of liquid is 1296 kg/m^3. How far (h) below the liquid surface is the bottom face of the cube?


Relevant equations

$$\Sigma$$ F = ma = 0 (it's floating)

$$\rho$$ = F / A

Here's what I tried doing.

$$\Sigma$$ F_y = B - F_l - F_w = 0 , with B as Buoyancy, F_l is the liquids force, and the F_w is the forceo f the wood . I ended up using Volume and density to plug in for Forces obtaining this...

$$\rho$$ * V_l * g - $$\rho$$ * V_w = $$\rho$$ * V_w * H

I'm not sure if this is near the correct approach.

 

[Doc Al]

The buoyant force is the force of the liquid on the wood. Only two forces act on the wood: its weight and the buoyant force.

Hint: What volume of liquid must be displaced to support the wood? How does that compare to the volume of the wood?

 

[jrrodri7]

Oh, so it's similar to the case of the fully sugmerged object, only with different densities given and g is negligible.