# Can the formula of buoyancy be used in this arrangement?

The buoyant force exerted by an object in water is given by $$\rho.g.v_{submerged}$$

Which only depends on the volume submerged and nothing else(I think)

But can it be applied to partially submerged bodies like in the diagram below?

I feel like it can be applied to the part which is submerged, but my teacher says otherwise.

An intuitive explaination on why it can/cannot be would be much appreciated.

• Draw a submerged cylinder oriented vertically. Hydrostatic forces always act normal to the surface, so all forces on the side of the cylinder "cancel out". The buoyancy force is due to the difference between the force on the top of the cylinder and the force on the bottom of the cylinder (pressure increases with depth). Now analyze your particular problem with the same method. Is your teacher correct? Nov 9, 2020 at 19:46

I think it's useful to consider what causes Archimedes' principle to be true to see why it does not apply as you are suggesting here.

The fluid on the bottom is actually necessary for the principle to apply. We know from hydrostatics that the pressure of a fluid increases with depth. This means that for any object submerged in a way where there is fluid below it, the pressure below should be higher than the pressure above, where the magnitude of that difference depends on height difference, fluid density, and gravity.

We also know that the force due to pressure depends on the area that the pressure acts on. So the object will have a greater force acting below it than above it, and you can see how the larger the area of the object in the horizontal direction, the greater the difference in vertical force should be due to the vertical pressure difference. The area in the vertical direction(s) should make no difference, because the forces on each side will cancel out for an object totally surrounded by water.

So then when you consider that the pressure on the bottom varies based on the height of the object in each location, and that the force due to that pressure will depend on how much area the bottom has at each location, you might be able to get an intuitive grasp for why Archimedes' principle says the buoyant force depends on the submerged volume. It is really due to these pressure differences and the forces that show up due to them, and how when all combined the force scales directly with the volume.

But the important thing to remember is that this only applies when fluid is below it. In this picture you have, the fluid is only above, so there is pressure pushing down on the object due to the water, but no water below the object to push back up, so Archimedes' principle no longer applies. The mistake of assuming it works when there is no fluid below the object is not uncommon though, because when Archimedes' principle is taught, it's rarely explained how the principle is derived, and so what constitutes a "submerged" object is often seen in the way you thought here. But yes your teacher is correct.

I apologize for the really long winded answer, but this is how I got an intuitive understanding for this exact topic, and if you do manage to follow what I'm saying I think you will feel a lot more comfortable with buoyancy and Archimedes' principle too.

• No sir not long at all, brilliant explanation. Nov 10, 2020 at 1:50
• I find all of the above reasonable and agreeable. And yet I'm having trouble getting past a related scenario: an object (say, just like the cylinder in this example), with a bottom face that is perfectly smooth, resting on the bottom of a tank that is similarly perfectly smooth. I.e. there is no water between the face and the tank's bottom surface. My intuition says that the object will still have buoyancy. Yet, according to the explanation, it won't. Is this simply because it's impossible to literally have perfectly smooth surfaces, or is something else going on? Nov 10, 2020 at 3:38
• (I think the previous comment is within the scope of your answer, seeking clarification of that answer. But if you think it's better as a separate question, just say so...thanks!) Nov 10, 2020 at 3:39
• @PeterDuniho I just clarified that with my teacher, He said if nothing is mentioned in the question we are to assume irregularities between the surfaces exist which provide buoyant force, Thanks for mentioning that. Nov 10, 2020 at 5:39
• @PeterDuniho physics.stackexchange.com/questions/59866/… That has actually been asked here before. The consensus is indeed that it shouldn't experience buoyancy if no water can get under it. Typically you at least get a layer of water. Also worth noting that an obvious example is a suction cup. Stick a suction cup to a container, and adding water should actually push it down more, not raise it due to buoyancy; because what makes suction cups stick in the first place is the extra pressure above them.
– JMac
Nov 10, 2020 at 14:04