Buoyant force and volume submerged

[swell9]

Hello,

This is bugging me so I'd appreciate any help.
I have two objects made from the same material with the same mass.
The base of Object 2 is 3 times that of Object 1. If I place the objects base first in a liquid, why is the buoyant force on them equal?

What here is what I'm thinking:
V$_{ submerged}$= (Density$_{ object}$ X Volume$_{ object}$) / (Density$_{ liquid}$)

Since the volume of the two objects has to be different if they are of the same material and of the same mass, the volume submerged of each will also be different.
Doesn't this give that the Buoyant force will also be different?

Thanks for your time.

 

[Travis_King]

if material (therefore density) and mass are the same, then volume is also the same.

 

[Nugatory]

I have two objects made from the same material with the same mass.
The base of Object 2 is 3 times that of Object 1. If I place the objects base first in a liquid, why is the buoyant force on them equal?

If the objects are made of the same material and they have the same mass, then they necessarily have the same volume although their shapes are different.

If the objects are denser than water, they'll sink, the entire volume will end up underwater, and it won't matter if that one of them is wider in the other.

If they are not denser than water, they will float. Both will displace a volume of water equal to their weight; the one with the narrower base will sink deeper into the water to displace the same volume.

 

[swell9]

if material (therefore density) and mass are the same, then volume is also the same.

Oh wow I'm officially an idiot now. Thanks though :P

 

[swell9]

If they are not denser than water, they will float. Both will displace a volume of water equal to their weight; the one with the narrower base will sink deeper into the water to displace the same volume.

Is this scenario referring to objects that do not have the same volume?

Basically what I'm saying: if they have the same volume then they necessarily displace the same liquid volume. Is this statement correct?

 

[Nugatory]

Is this scenario referring to objects that do not have the same volume?

Basically what I'm saying: if they have the same volume then they necessarily displace the same liquid volume. Is this statement correct?

They necessarily displace the same volume. But if their shapes are different, as suggested by "the base of Object 2 is 3 times that of Object 1" in your first post, then the depth below the water and the height above the water will be different. Suppose that one of the objects is 10 cm by 30 cm by 1 cm and the other is 10 cm by 10 cm by 3 cm, and they both have a density of .5 grams/cm³. They'll both have a volume of 300 cm[sup3[/sup], a mass of 150 grams, and will displace 150 cm³ of water. However, the first object will sink .5 cm into the water and the second will sink 1.5 cm into the water.

 

[swell9]

They necessarily displace the same volume. But if their shapes are different, as suggested by "the base of Object 2 is 3 times that of Object 1" in your first post, then the depth below the water and the height above the water will be different. Suppose that one of the objects is 10 cm by 30 cm by 1 cm and the other is 10 cm by 10 cm by 3 cm, and they both have a density of .5 grams/cm³. They'll both have a volume of 300 cm[sup3[/sup], a mass of 150 grams, and will displace 150 cm³ of water. However, the first object will sink .5 cm into the water and the second will sink 1.5 cm into the water.

Thanks so much. It is clear now.

 

[jfizzix]

I used to have a similar question a few years ago in undergrad about buoyancy. If I had two objects of the same mass, but one with a much larger surface area (comparing say, a sphere and a koosh ball of the same material) it seemed to me that the pressure times a larger surface area would lead to a larger buoyant force.

Once I figured out how to do surface integrals, I figured out that no matter the shape, the buoyant force is always equal to the weight of the displaced fluid. It doesn't matter if the fluid is compressible, or if we assume local gravity, universal gravity, or even with some GR thrown in; the result is the same.

 

[A.T.]

I have two objects made from the same material with the same mass.
The base of Object 2 is 3 times that of Object 1. If I place the objects base first in a liquid, why is the buoyant force on them equal?

The buoyant force can be different here, if the material's density is greater than water's, and the objects are shaped such that one sinks while the other one floats on the surface.