Confusion regarding direction of buoyant force

Question

In this question, we are given the density of oil, water and the block. We are given that the block is floating, and thus we need to find what fraction of the block's volume lies in the oil. I have a very simple question; which equation is correct:

Weight of block + weight of displaced oil = weight of displaced water (implying that oil exerts buoyant force downwards)

OR

weight of block = weight of displaced oil + weight of displaced water (implying that buoyant force due to oil is upwards, which doesn't make sense to me whatsoever).

This question involves the exact same doubt, but clearly there was some confusion between the writer of the accepted answer and the asker (see comments). In the end however, the answerer said that oil exerts force downwards. Is this buoyant force? (if it was I could say that the downwards buoyant force due to oil is equal to weight of volume of oil displaced, which would be equal to the fraction of volume of block in the oil) Or is it the force due to pressure of the oil above the block (in which case I could calculate it using the formula $\rho_0 + \rho gh$. If I do use this formula would I need to find the net force due to difference of pressures at the bottom and top? How exactly do I do that when the density changes at some unknown depth? Are they equivalent?

Reading through this thread honestly just confused me even more, some people involved said the force due to oil is downwards, but I'm not sure if it's right to call that the buoyant force. I also read somewhere that I should just find the total amount of oil and water displaced (fraction of block's volume in each liquid times the respective density times $g$) and their sum would be equal to the weight of the block. This would imply that despite the oil being entirely above the block (none below it) it somehow contributes to the buoyant force.

Clearly, I have some major confusion about what causes buoyant force, and the difference between buoyancy and pressure. I would really appreciate it if someone could systematically answer my questions and clear my confusion :)

Answer 1

If you replace the block with a layer of oil and a layer of water (such that the heights all match up), you'd agree that everything is stable.

Therefore the weight of the block must be equal to the water and oil (your second equation).

Yes, the net buoyant force is always upward. Some of the oil does push downward on the block, but some of the oil pushes downward on the water.

Imagine removing all the oil that is above the top of the block. Now none of the oil is pushing downward on the block. Instead it's pushing downward on the water where the block is not present. This force from the oil pushes the block upward in the water.

Any oil you add above the top of the block doesn't change the position of the block in the water. It just adds to the total pressure. It has no more effect than it would if you put the tank in a pressure chamber so that the air pressure on top of the oil instead of 1bar were raised to 2bar.

If the net effect of the oil's upward and downward force cancels out, then removing all the oil should not change anything, right? Why would I factor in the oil at all then?

No, I said the oil above the block has no effect (because the pressure from it manifests as a force that pushes down both on the block and the water supporting the block).

But the oil beside the block (above the water, but below the top of the block) does have an effect. It's not pushing down on the block, but is is pushing down on the water. If you remove the oil, the pressure in the water is reduced, but the gravitational force on the block is not. The block will sit lower in the water.

Answer 2

Ask yourself what would happen to the block if you were to remove oil such that the level of the oil is exactly level with the top of the block? The answer is nothing! The block will continue to float completely submerged in exactly the same location, and yet now there is no oil above the block exerting a downward force on the block.

Now you may ask how that can be? If there is less force acting down on the block you might expect the block to move up. The reason it doesn't is the force the oil exerts downward on the water, which pushes the block upward, is also now reduced.

So your second equation is correct and is (where $b$, $o$, and $w$ denote the block, oil and water)

$$\rho_{b}gV_{b}=\rho_{o}gV_{o}+\rho_{w}gV_{w}\tag{1}$$

The only thing that matters is the volume of the oil and water displaced by the block. That can be determined by realizing if the block is floating completely submerged, the total volume of the oil and water displaced must equal the volume of the block or

$$V_{b}=V_{o}+V_{w}\tag{2}$$

Equations (1) and (2) can be solved for the two unknowns, $V_w$ and $V_o$.

Hope this helps.