1. Nov 30, 2014

### Graeme M

I recently posted a question about solids that led to a very productive discussion about solids, weight, gravity and pressure. I learned that many intuitive ideas I had about these concepts were wrong and I now have a better understanding as a result. However, in the course of my reading I came across the buoyant force.

Now of course at a general level I know what buoyancy is, but after reading a number of sources it became apparent that it’s much more complicated than I thought.

I now think I have a fair understanding of this force, but there are a few things I can’t get straight in my head, even after reading a lot.

Here’s my understanding, followed by my question. These points are at a very simplified level – that is, a homogeneous liquid, in this case water, with all other forces removed. I am talking about a static situation so I am disregarding flow, drags, and other complicating factors.

1. A submerged object displaces its volume in water.
2. The weight of the water displaced is equivalent to the upward buoyant force on the object.
3. The buoyant force arises from the difference in pressure between the top of the submerged object and the bottom of the submerged object.
4. The pressure in a column of water increases with depth and hence the buoyant force will increase with depth, all other factors remaining equal.
5. The apparent weight of a submerged object is reduced by the magnitude of the buoyant force. Because weight is a force, the net force applied to the object is the difference between its weight and the buoyant force.
6. If an object’s apparent weight is greater than the buoyant force, the object will sink. And vice versa – if less than the buoyant force, the object will rise.

The total apparent weight of a container of water and an object outside of the container of water is the sum of the two weights. If the object is placed in the container of water, the total weight of the container and water is increased by the weight of the object. However the apparent weight of the object, inside the container, is reduced by the buoyant force.

Assuming that the object’s mass and shape is such that the buoyant force is exactly equal to its weight and the object does not sink, the object’s apparent weight in the container will be zero. However, the total system weight of container, water and object is increased by the weight of the object.

What is happening here? The upward force exactly balances the downward force, yet there is still a nett downward force.

2. Nov 30, 2014

### Bystander

What is that "net downward force.?" Reread your excellent essay, and think carefully about all you've said.

3. Nov 30, 2014

### Staff: Mentor

Items 2 and 4 are inconsistent. Item 2 is correct, and item 4 is incorrect.

Regarding your question, if you added water to the container, the total weight of material in the container would increase. If instead you add an object to the container and the object is neutrally buoyant, this would be the same as adding water to the container.

Chet

4. Nov 30, 2014

### Staff: Mentor

We seem to get questions like this a lot. I wonder if schools are doing a poor job of teaching what "net force" is?

1. All forces always come in pairs, if you look hard enough. Typically when people talk of "net force" they are talking about the overall external forces on an object. And when they say the sum is non-zero, that just means they aren't counting the force from inertia and acceleration. So that has nothing to do with the situation here...

2. The internal forces on the floating object don't really need to be considered to determine the the external forces between the container and what is supporting it. You can sum them if you want, to find how the forces are transmitted through the container, but you really don't need to.

3. So if you do want to sum the forces, the upwards force on the object matches the downwards force on the water. Right. But that isn't a "net force" of zero because the two forces are acting on different things -- one acts on the water, the other on the object. Net force is all forces on one "object" summed together. So the water and object in it have other forces on them....

5. Nov 30, 2014

### Graeme M

Chestermiller, I get that in terms of weight for the whole container, adding water or a solid of same weight has the same effect.

However adding water results in no buoyant force whereas adding a solid object does. In the former case there is no other force involved, while in the latter there is but it doesn't matter whether it is nett up, down or neutral. The buoyant force affects apparent weight of the object, but not the whole system.

In other words, the buoyant force can be of any magnitude yet the total system weight is not changed. I can't get my head around that.

6. Nov 30, 2014

### Graeme M

Also, are you saying that my number 4 is wrong because the pressure differential between upper and lower surfaces remains the same regardless of depth and pressure? I did read on some science site somewhere that an object can sink until a point at which the buoyant force is sufficient to offset the weight, so that surely would arise from the change in pressure?

7. Nov 30, 2014

### Graeme M

russ_watters, you say
"So if you do want to sum the forces, the upwards force on the object matches the downwards force on the water. Right. But that isn't a "net force" of zero because the two forces are acting on different things -- one acts on the water, the other on the object. Net force is all forces on one "object" summed together. So the water and object in it have other forces on them...."

But the downwards force is on the object (gravity), and the upwards force is on the object (buoyancy). The upwards force isn't matching a downwards force on the water?

8. Nov 30, 2014

### Staff: Mentor

Incorrect. What do you think keeps a parcel of water at the surface from sinking?
Right.
That would be rare, but it would be because water isn't quite incompressible. So its density increases a little with depth. More common, though, is the opposite: as objects sink, they get compressed and become less buoyant. That's a problem for scuba divers: they have to continuously adjust their buoyancy to avoid sinking to the bottom or shooting to the surface.
Well sure it is: what force moves the water out of the way for the object to replace it?

9. Dec 1, 2014

### Graeme M

What keeps a parcel of water from sinking?

Hmmm... I wouldn't have thought of it like that! My thinking would say this. Water is a single thing much like a solid. The solid retains its form and shape from the bonds that hold its molecules/atoms in place. I understand that water is similar but that its bonds are not so resilient - that is, water molecules constantly create and break their bonds. So in effect water has the same form as a solid but its shape is flexible. What keeps the water at the top above the water at the bottom is the attractive/repelling forces. In fact I'd have thought that in the absence of currents etc, ie an entirely still column of water, there'd be no specific buoyany force as such, just the same forces that keep a solid together.

So... Are you saying that the buoyant force is simply the same force that keeps molecules/atoms apart? That is, repelling forces at the molecular level? So buoyancy is no more than the effect of two objects with defined form interacting.

That would explain how lowering an object into a container of water reduces the apparent weight of the object while increasing the apparent weight of the container of water. In effect, it would be similar to sliding say a brick across the upper surface of two other bricks - if a a scale were under each of the two lower bricks, the gaining brick will increase in weight proportionally to the rate at which the other brick loses weight.

10. Dec 1, 2014

### Staff: Mentor

No, that has nothing to do with buoyant force. The "parcel of water" thought experiment is simply for this: instead of pouring more water into the container, put the additional water in a plastic bag and put that bag into the container. What happens and why?

11. Dec 1, 2014

### Graeme M

OK... not sure I see the relevance. You have introduced a surface. A plastic bag filled with water is effectively no different to a solid object of equivalent mass. The water in the container cannot know that behind the surface lies more water. The bag filled with water will displace an equivalent volume of water in the container and the buoyant force will equal the weight of that volume of water. Providing your bag has thin walls, the total weight of the bag and water should be very similar to the buoyant force and the bag should float just under the surface. If though your bag has very thick walls, its overall weight may overcome the buoyant force and it will sink (here I am assuming something about the relative densities of plastic and water. And the shape of the bag.).

I don't see that tells me anything about the behaviour of a body of water.

12. Dec 1, 2014

### Sanky123

Good example

13. Dec 1, 2014

### Staff: Mentor

The point of the example is that the parcel of water behaves the same whether the bag is there or not. The bag is just an arbitrary boundary around an arbitrary volume of water, which has no impact on how you conceive the effects of buoyancy.

That idea becomes more useful when the parcel of water becomes warmer or cooler than its surrounding water and its net force changes. Or better yet, air in the atmosphere. Changes in buoyancy/density drive convection, which in the atmosphere drives the weather.

Last edited: Dec 1, 2014
14. Dec 1, 2014

### sophiecentaur

If you are having a problem with the concepts involved then perhaps you could work out a specific example (say a disc of wood of density 0.8) and a 1l beaker of water. (Choose some disc and beaker dimensions.) Put the two, side by side, on the platform of some scales. Now put the disc in the water.and, bearing in mind the total measured mass much be the same, work out the forces on the disc (once it has stopped moving) and the pressures in the water, at the new level. It's all got to work out so just find what you need to add to what to get it to balance. Actually, Archimedes' Principle is a half way house in the argument.
You have either discovered an Earth-shattering fact, hitherto unknown to mankind OR you have got your reasoning wrong, somewhere. Assume the latter. :)

15. Dec 1, 2014

### Staff: Mentor

Not that Russ needs my help, but I fully support what he is saying. The water surrounding the bag doesn't know whether there is a bag full of water there, a solid object of the same shape there, or water with no bag around it filling the same space. In all three situations, the hydrostatic pressure forces exerted by the surrounding water cause the same upward force on whatever is filling that space.

Chet

16. Dec 1, 2014

### Graeme M

Haha SophieCentaur, if ever I worked out an earth shattering fact no-one else had, it'd be the greatest miracle in history. I'm flat out figuring out how to get out the door each morning!

Now, to my mind there's a couple of things here, which may be just too hard for me to put into words. Chestermiller I disagree with what you say. Not because I am right and you are wrong, but because what you say makes no sense to me. Of course the problem is I just don't have the breadth of education, but I'm game to take a stab at this.

There is no 'parcel' of water. The buoyant force surely depends on there being a physical surface. If you just have water, there can be no surface - that is, if you examine any section of a still body of water it should look exactly the same as any other.

So the forces involved must be only those forces that inhabit any body. On the other thread, I learned that bonding forces - attractive/repelling - are what keep a solid, solid. A liquid has exactly the same internal forces at the molecular level, except (as I understand it) that the bonds are more short-lived. So in a way, a liquid is a very flexible solid.

What keeps water 'up' is the same thing that keeps the top of a solid 'up', surely? Ignoring currents and temperature and so on, a still cylindrical container of liquid must be broadly the same thing as a solid cylinder. Differences in density of the liquid just give rise to a different profile as the matter responds to gravity.

On the other thread I learned that the atmosphere has a pressure gradient as a response to gravity. So if we had two liquids mixed, one denser than the other, then the more dense will obviously sink to the bottom and the lighter one will be on top. But that is just a response to gravity is it not? It's not the buoyant force at work.

The imaginary parcel is just that. I can see how it's a useful model, but I think it's misleading. I can see how a solid object in a liquid can have a 'buoyant force' operating on it, but I cannot see how an arbitrary chunk of water in a body of water can have anything other than the normal molecular level forces acting on it.

17. Dec 1, 2014

### Khashishi

Regarding your original question, the mass of the full system clearly increases when you add more objects. The mass of the system is simply the sum of all the masses in the system. The apparent weight only matters when measuring the weight submerged in some fluid. If you put your scale outside of the fluid and measure the weight of the whole container, there is no buoyant force so you feel the full weight.

18. Dec 1, 2014

### nasu

It seems that at least part of your original problem relies on the assumption that the zero net force on the body is not compatible with the expected total weight.
In order to see that this is not realy a problem, you can take a step back and consider the following configuartion, which have nothing to do with buoyant forces.

1. What weight is measured by the spring scale?
2. What is the net force on the block on the right? (or on the left)

19. Dec 1, 2014

### A.T.

The total buoyant force on the solid object is just the sum of all molecular level forces exerted on it by the water.

20. Dec 1, 2014

### Staff: Mentor

OK GraemeM, I hear you. Let me first say that I still stand by what Russ and I have been saying and am confident in it, even regarding masses of liquid immersed within a larger body of the same liquid. However, I obviously have not done a good job of explaining the fundamentals of buoyancy to you. So, if it is all right with you, I would like to take a step backwards and, for now, confine attention exclusively to solid bodies of arbitrary shape immersed within a surrounding liquid. We will be focusing on determining the net force exerted by the surrounding liquid on the solid body. We are going to be doing some modeling to quantify the force. (Later we'll come back to fluid parcels).

I assume you are familiar with hydrostatics, and in particular, the equation for the fluid pressure p at a depth z below the air interface: $$p=p_a+\rho g z$$
where pa is the air pressure, ρ is the fluid density, and g is the acceleration of gravity.

Are you also familiar with Pascal's Law indicating that the pressure p acts equally in all directions at a given location in a fluid? This means that if there is an object immersed in a fluid, the pressure acts normal to the surface of the object at all locations on the surface.

I will continue after getting confirmation that you are comfortable with both these concepts.

Chet