How Is Buoyancy Calculated for a Cubical Block Floating in a Denser Liquid?

[hvthvt]

Homework Statement

A cubical block of density ρ_b with sides lengths L floats in a liquid of greater density ρL. (A) what fraction of the blocks volume is above the surface of the liquid?
(b) the liquid is denser than water (density ρw) and does not mix with it. If water is poured on the surface of the liquid, how deep must the water layer be so that the water surface just rises to the top of the block? Express your answer in terms of L, ρ_b, ρL and ρw. (c) Find the depth of the water layer in (b) if the liquid is mercury, the block is made of iron, and its side length is 10 cm.

Homework Equations

Fbuoyancy=ρVg
F=mg

The Attempt at a Solution

I really need to know the answers to these questions, because the idea is pretty simple, but I'm simply stuck! For (a), I thought of the following: ρ_b*L³=ρ_L*L²*(L-x) where x is the part which is above water. How can I find a fraction??
For (b), my mind is really freaking out. I understand that there is a surface of water ON top of the liquid ρL. If the water layer rises on top of the block, the height of it will be (L-x) while the height of the liquid with density ρL will be x itself. Can anybody please help me out?? I would really appreciate your help!

 

[haruspex]

For a), what (in terms of L and x) is the volume of the block above the surface? What fraction is that of the block's total volume?
(For part (a), you don't really need to assume the block floats level - it could float at any angle. You can just use volumes rather than lengths.)

For (b), I understand that there is a surface of water ON top of the liquid ρL. If the water layer rises on top of the block, the height of it will be (L-x) while the height of the liquid with density ρL will be x itself.

Yes, but a different x now. What weight of the denser liquid is displaced by the block now? What weight of water is displaced by the block? What equation can you write connecting those with the weight of the block?

 

[hvthvt]

Hmm. I do not understand how I can give the fraction for the volumes. Can anybody show this idea to me? Please?

 

[haruspex]

Hmm. I do not understand how I can give the fraction for the volumes. Can anybody show this idea to me? Please?

Fr part (a), you are taking the block to have side L, and a length x of that is above the water.
What volume of the block is above the water?
To get that as a fraction of the whole volume, you divide that by the whole volume, L³.
You have an equation which gives you x as a function of L and the densities.

 

[HalfLostAndSearching]

I would approach these questions by first understanding the concept of buoyancy and the factors that affect it. Buoyancy is the upward force exerted on an object immersed in a fluid, and it depends on the density of the fluid, the volume of the object, and the gravitational acceleration.

For (a), we can use the formula for buoyancy (Fbuoyancy=ρVg) and the weight of the block (F=mg) to find the fraction of the block's volume above the surface of the liquid. Since the block is floating, the buoyant force (Fbuoyancy) must be equal to the weight of the block (F). So we can set these two equations equal to each other and solve for the fraction of the block's volume above the surface (x):

ρbL^3 = ρL(L^2)(L-x)

x = ρbL / ρL

Therefore, the fraction of the block's volume above the surface is equal to the ratio of the block's density to the liquid's density.

For (b), we can use the same concept of balancing forces to find the depth of the water layer needed to just reach the top of the block. Again, the buoyant force must be equal to the weight of the block. So we can set these two equations equal to each other and solve for the depth of the water layer (h):

ρw(Vw)g = ρbL^3g

h = ρbL^3 / (ρwVw)

We know that the volume of the water layer (Vw) is equal to the area of the block (L^2) multiplied by the depth of the water layer (h). So we can substitute this into the equation:

h = ρbL^3 / (ρwL^2h)

Solving for h, we get:

h = ρb / ρw

Therefore, the depth of the water layer needed to just reach the top of the block is equal to the ratio of the block's density to the water's density.

For (c), we can use the same method as in (b), but with the given values for the block's material and size. Substituting in the values for iron (ρb = 7.87 g/cm^3) and mercury (ρw = 13.6 g/cm^3