Questions About Fluids & Buoyancy

[Red_CCF]

I have some questions regarding fluids.

1. This may seem like a dumb question, but what's the explanation (preferrably with calculations) on why a denser object sinks when placed in a less dense (presumably liquid) object in terms of buoyancy?

2. When I place a solid in a liquid such that it floats, how do I predict beforehand where the water level will be relative to the side of the solid.

3. I'm really shaky with water pressure and its relation to buoyancy. What's the role of water pressure in keeping something afloat? I know buoyant force keeps things afloat so what does water pressure do? Also, when something is sinking, how do we involve water pressure and buoyant force to calculate how fast it sinks (ignoring drag force).

Any help is appreciated. Thanks.

 

[Doc Al]

I have some questions regarding fluids.

1. This may seem like a dumb question, but what's the explanation (preferrably with calculations) on why a denser object sinks when placed in a less dense (presumably liquid) object in terms of buoyancy?

Consider the forces acting on the body. The buoyant force equal to the weight of the displaced fluid acting upward; the weight of the body acting downward. If the body is more dense than the fluid, there will be a net downward force.

2. When I place a solid in a liquid such that it floats, how do I predict beforehand where the water level will be relative to the side of the solid.

Compare the density of the solid to the density of the liquid. That will give you the fraction of the body that will be submerged.

3. I'm really shaky with water pressure and its relation to buoyancy. What's the role of water pressure in keeping something afloat? I know buoyant force keeps things afloat so what does water pressure do? Also, when something is sinking, how do we involve water pressure and buoyant force to calculate how fast it sinks (ignoring drag force).

The net force due to water pressure on an object is the buoyant force.

 

[Red_CCF]

Thanks for the response.

Compare the density of the solid to the density of the liquid. That will give you the fraction of the body that will be submerged.

I don't really understand what you mean. Can you do a numeric example?

The net force due to water pressure on an object is the buoyant force.

So does this mean that if I have a cube, the pressure pushing the bottom of the cube up subtracting the pressure pushing the top of the cube down and multiplied by the area of its face should equal the buoyant force acting on it?

 

[Doc Al]

I don't really understand what you mean. Can you do a numeric example?

I'll give you a hint so you can figure it out on your own. You want to compare the volume of liquid displaced to the volume of the body itself. That will tell you what fraction of the body is submerged. (Now set up the equation I described in my last post. You'll need the densities of liquid and body.)

So does this mean that if I have a cube, the pressure pushing the bottom of the cube up subtracting the pressure pushing the top of the cube down and multiplied by the area of its face should equal the buoyant force acting on it?

Yes, assuming the cube is oriented so that its sides are vertical.

 

[Red_CCF]

I'll give you a hint so you can figure it out on your own. You want to compare the volume of liquid displaced to the volume of the body itself. That will tell you what fraction of the body is submerged. (Now set up the equation I described in my last post. You'll need the densities of liquid and body.)

I did some searching and found that in this statement

For a floating object, only the submerged volume displaces water.

here: http://en.wikipedia.org/wiki/Buoyancy

Does this mean that the volume of water (or any fluid) displaced is equal to the volume of the object underwater provided that it is floating?

Thanks in advance

 

[Doc Al]

Does this mean that the volume of water (or any fluid) displaced is equal to the volume of the object underwater provided that it is floating?

Yes. That's true even if it's not floating.

 

[Red_CCF]

Yes. That's true even if it's not floating.

Is it possible to prove this mathematically or can it only be done experimentally?

 

[Doc Al]

Is it possible to prove this mathematically or can it only be done experimentally?

I'm unclear what you're looking to have proven. That the volume of fluid displaced is equal to the volume of the object underwater? That's true by definition.

If you looking to prove what fraction of a floating object is submerged, then set up the force equation. For a floating object, the upward buoyant force must equal the weight of the object.

 

[jack action]

Whenever an object is put in a container filled with a liquid, it raises the liquid level, like this:

When the container is an ocean and the object is a pebble (or even a boat), the level difference is not noticeable but it is still there.

How much liquid is raised? All we know is that the sum of the volumes for both the liquid and the object must be constant, whether there are in separate container or not.

If there is a volume of liquid that is raised, then you need a force to lift it. That force is the weight of the object. It is like a scale such as this one:

http://francais.istockphoto.com/file_thumbview_approve/4361116/2/istockphoto_4361116-balance-scale.jpg

where you put the object on one side and the water displaced on the other. One is the counterweight of the other.

If the volume of liquid displaced is not large enough to compensate for the weight of the object (object denser than liquid), then the "scale will tip over", i.e. the object will sink to the bottom. If you drop slowly the object in the liquid, and the liquid is denser than the object, then the object will stop sinking (hence it will float) when the weight of the liquid displaced will be equal to the weight of the object.

 

[Red_CCF]

I'm unclear what you're looking to have proven. That the volume of fluid displaced is equal to the volume of the object underwater? That's true by definition.

If you looking to prove what fraction of a floating object is submerged, then set up the force equation. For a floating object, the upward buoyant force must equal the weight of the object.

I'm confused as to how the two are different. If I know volume of the object underwater, couldn't I just divide that by the total volume that that should give how much the object is submerged?

 

[Red_CCF]

Whenever an object is put in a container filled with a liquid, it raises the liquid level, like this:

When the container is an ocean and the object is a pebble (or even a boat), the level difference is not noticeable but it is still there.

How much liquid is raised? All we know is that the sum of the volumes for both the liquid and the object must be constant, whether there are in separate container or not.

If there is a volume of liquid that is raised, then you need a force to lift it. That force is the weight of the object. It is like a scale such as this one:

http://francais.istockphoto.com/file_thumbview_approve/4361116/2/istockphoto_4361116-balance-scale.jpg

where you put the object on one side and the water displaced on the other. One is the counterweight of the other.

If the volume of liquid displaced is not large enough to compensate for the weight of the object (object denser than liquid), then the "scale will tip over", i.e. the object will sink to the bottom. If you drop slowly the object in the liquid, and the liquid is denser than the object, then the object will stop sinking (hence it will float) when the weight of the liquid displaced will be equal to the weight of the object.

Thanks for such a detailed illustration. So is this basically the explanation behind Archimedes' principle?

 

[Doc Al]

I'm confused as to how the two are different. If I know volume of the object underwater, couldn't I just divide that by the total volume that that should give how much the object is submerged?

Sure. But how can you predict how much of an object will be under water?

 

[Red_CCF]

Sure. But how can you predict how much of an object will be under water?

Oh I see now; so basically if an object's less dense than the fluid I just take the gravitational force of the object and divide it by the density of the fluid and g to get volume of fluid displaced (or volume of object under water) and then divide that by the object's total volume to get what fraction of the object is submerged?

 

[Doc Al]

Oh I see now; so basically if an object's less dense than the fluid I just take the gravitational force of the object and divide it by the density of the fluid and g to get volume of fluid displaced (or volume of object under water) and then divide that by the object's total volume to get what fraction of the object is submerged?

And when you do all that, you end up with a simple result:

Archimedes' Principle tells us:
ρ_obj*g*V_obj = ρ_fl*g*V_fl

So, the fraction of the floating object that is submerged is:
V_fl/V_obj = ρ_obj/ρ_fl

So, if the object has 1/3 the density of water, 1/3 of it will be submerged when floating in water.

 

[jack action]

Thanks for such a detailed illustration. So is this basically the explanation behind Archimedes' principle?

Yes. Here's the story of how Archimedes found that principle:

http://twistedphysics.typepad.com/cocktail_party_physics/how_cool_is_that/page/2/]Archimedes[/PLAIN] was a Greek mathematician, famous for all kinds of things, but among the most oft-repeated tales is how he came to the aid of his friend, Hiero, king of the Greek city of Syracuse. Hiero suspected that a goldsmith charged with making him a royal crown -- one assumes he needed a spare -- had kept some of the gold provided for himself, and mixed in silver to ensure the weight of the final crown matched that of the original lump of gold provided. He didn't want to melt the crown down to discover the truth, but the thought just nagged at him, and he asked Archimedes to help. Inspiration hit one day as Archimedes lowered himself into one of the public baths in the city and noticed displaced water flowing over the sides of the tub. Legend has it that he was so excited with his insight, he leapt out of the tub and ran (naked?) through the streets of Syracuse yelling, "Eureka! Eureka!" ("I found it! I found it!")

A theoretical insight must be backed up by experiment, so Archimedes took a lump of gold and of silver, each weighing the same as the king's crown, although the lump of silver was much larger because silver is lighter than gold. He put each lump in a vessel filled to the rim with water, and noted that the larger amount of silver caused more water to overflow than the lump of gold, because there was more material, even though both weighed the same. He concluded that a solid material will push away an amount of water equal to its own bulkiness (volume). So if the king's crown were indeed made of pure gold, it would have to displace the same amount of water as the lump of pure gold that weighed the same. Unfortunately for the dishonest goldsmith, the crown made more water overflow than the pure lump of gold, proving that the goldsmith had added silver to the crown to make it bulkier. The goldsmith's fate was probably not a happy one.

 

[Red_CCF]

Yes. Here's the story of how Archimedes found that principle:

lol that's pretty funny. How did he discover buoyant force and that it equals the weight of the water displaced?

 

[jack action]

lol that's pretty funny. How did he discover buoyant force and that it equals the weight of the water displaced?

That's Archimedes Principle, that's what he found:

http://twistedphysics.typepad.com/cocktail_party_physics/how_cool_is_that/page/2/]He[/PLAIN] concluded that a solid material will push away an amount of water equal to its own bulkiness (volume).

If something pushes something else, it implies there is a force involved. Today we call this force buoyancy.

 

[Red_CCF]

That's Archimedes Principle, that's what he found:



If something pushes something else, it implies there is a force involved. Today we call this force buoyancy.

I was wondering how he figured out that the buoyant force was actually equal to the weight of the water displaced?

He concluded that a solid material will push away an amount of water equal to its own bulkiness (volume).

Have a question about this, does this only apply if the object sinks because if an object is floating it shouldn't be possible to push away water of equal volume to itself?

 

[jack action]

I was wondering how he figured out that the buoyant force was actually equal to the weight of the water displaced?

Like I said earlier, we know that the sum of the volumes for both the liquid and the object must be constant, whether there are in separate container or not. But you can easily measure the volume of water displaced and compare it with the known volume of the object if you want to prove it.

Have a question about this, does this only apply if the object sinks because if an object is floating it shouldn't be possible to push away water of equal volume to itself?

Only the volume of the object that is submerged will displace an equal volume of water.

 

[Red_CCF]

Like I said earlier, we know that the sum of the volumes for both the liquid and the object must be constant, whether there are in separate container or not. But you can easily measure the volume of water displaced and compare it with the known volume of the object if you want to prove it.

I think we have a misunderstanding here. I was talking about the weight of the displaced water and how to prove that this weight is pushing the object upwards

 

[Doc Al]

I was talking about the weight of the displaced water and how to prove that this weight is pushing the object upwards

If you want to understand Archimedes' Principle, study the simple argument on this page: http://hyperphysics.phy-astr.gsu.edu/hbase/pbuoy.html"

 

[Red_CCF]

If you want to understand Archimedes' Principle, study the simple argument on this page: http://hyperphysics.phy-astr.gsu.edu/hbase/pbuoy.html"

Thanks I think this answers my questions. So it's the fact that pressure changes (due to the displacement of the fluid by the object) that creates this buoyant force, which happens to equal the weight of the displaced fluid.