[Physics] the basic reason behind buoyancy

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Yesterday my friends and I were solving this easy-looking question:

A metal cube is placed in an empty vessel. When water is filled in the vessel so that the cube is completely immersed in the water, the force on the bottom of the vessel in contact with the cube:

(A) Will Increase

(B) Will Decrease

(C) Will remain the same

(D) Will become zero

At first we thought the answer to be (B) considering the buoyant force acting opposite to the weight of the cube.
But then, a question popped up in my mind: "What causes this buoyant force even if no fluid was actually displaced?"

So I referred to the Archimedes' Statement:

Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.
— Archimedes of Syracuse

But according to the question, no such fluid was displaced by the block !

So, here are my questions:

  1. What is the basic reason behind buoyancy? Is it the pressure difference in fluids or the 'liquid-displaced' theory?

  2. What must be the answer to the question (That I was solving) ?

  3. Considering the case that you may say- "The reason behind buoyancy is the pressure difference in the fluids"; Will there be no buoyant force applied to the iron cube as there is no fluid layer present below the cube i.e. no pressure difference?

(Considering the iron cube is VERY dense and thus not allowing any fluid layer to push its way below it)

  1. Or you might add another statement to the above question that-"Yes, buoyant force is applied as there is always a fluid layer present below it…" .Please do explain yourself.

  2. If your answer is according to the Principle given by Archimedes of Syracuse , will there be no buoyant force on the cube as the cube didn't actually displace any fluid? Just like the case where you may consider it to be a part of the vessel itself.

Best Answer

  1. The reason behind buoyancy is the pressure difference in fluids. More specifically, the difference in hydrostatic pressure on different levels, since the hydrostatic pressure of water increases with depth ($p=\rho gh$ wher $h$ is the distance below the surface). The part of the body which is subject to higher hydrostatic pressure will be pushed more upwards than the part of the body subject to lower hydrostatic pressure will be pushed downwards. Also, in space there would be no buoyancy because there would be no gravity, hence no difference in hydrostatic pressure.
  2. The answer is actually A. Since, as you said, no water gets below the cube, there will be no force directed upwards due to pressure. The liquid will actually push the cube downwards, to the floor, so the force on the bottom of the vessel in contact with the cube will be the cube's weight + the weight of the liquid above.
  3. Yes, since the reason behind buoyancy is the pressure difference, if there's no water below the cube, there will actually be no buoyant force. The liquid will push the cube downwards.
  4. I would argue that the cube does actually displace water (if you removed it, the water level would decline), but in this case you're right when you treat it as a part of the vessel, since there's no water below it. Every body immersed in a fluid will displace some part of it (because the fluid was there in the first place, when you immersed the body it had to go somewhere), but the Archimedes principle saying the force on the body will be equal to the weight of the displaced fluid is only right when the body is completely immersed in it. Actually, the derivation of this law is fairly simple:

Let's consider a volume of water $V$, let it be in the shape of, for instance, a potato. The water is subject to two forces: its weight and the buoyant force. Since it doesn't move, we conclude that the buoyant force is equal the the weight of $V$ of water. Now, let's say we displace the water with a potato of the same shape and volume $V$ - it occupies exactly the same space as the water discussed. Since the buoyant force is the result of the difference in pressure and does not depend on any characteristics of the body, it will be exactly the same as it was with the water volume. That's because the pressure around the body doesn't change. That's why we conclude the bouyant force is equal to the weight of the displaced fluid.

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