Bouyancy, Archimedes principle

[lavankohsa]

Homework Statement

I have confusion. Suppose i have metal cube placed in empty vessel. I have seal tight the base of cube so that the water or any fluid can't seep below the cube. Now i pour water into the vessel. Will there be bouyancy on the metal cube ?? If i measure the weight of cube, will it be lesser than its actual weight ?

Homework Equations

The Attempt at a Solution

I know that bouyancy force is equal to weight of the displaced fluid. The cube will displace the fluid. But since its base is seal tight, there will be no fluid under the base. So there is no fluid under the cube to exert the bouyancy force .

 

[ehild]

How do you measure the weight of the cube separately from the vessel, if you stick it to the vessel?

 

[haruspex]

The set-up you describe is essentially the same as pressing a rubber suction cup onto a smooth surface. Since no air can get between the rubber and the surface, the atmospheric pressure holds it to the surface.
Not only is there no buoyancy, the weight of water pressing down on top will make the cube's weight seem even greater.

 

[AlephNumbers]

Wait a minute. Wouldn't there still be an upwards "buoyant force" because the pressure exerted on the cube at the top of the cube would be less than that of the pressure exerted by the water on the bottom of the cube?

 

[haruspex]

Wait a minute. Wouldn't there still be an upwards "buoyant force" because the pressure exerted on the cube at the top of the cube would be less than that of the pressure exerted by the water on the bottom of the cube?

lavankohsa specifiies that there is no water between the underside of the cube and the container.

 

[AlephNumbers]

Yeah I got a little mixed up there. Inevitably some water would get in, and that's why a suction cup won't hold forever. I was misattributing that to some sort of upward buoyant force. Yay suction cups!

 

[lavankohsa]

So there will be no bouyancy force on the cube because there is no fluid below it. Suppose i remove the seal than water will come below the cube then there should be bouyancy force on the cube, right ??

 

[AlephNumbers]

Yes. But as long as it is sealed, there is actually a downward force from the water exerted on the cube

 

[lavankohsa]

But even when it is sealed , it is displacing the fluid and according to archimedes principle, bouyancy force= weight of displaced fluid.

 

[Delta2]

But even when it is sealed , it is displacing the fluid and according to archimedes principle, bouyancy force= weight of displaced fluid.

Well archimede's principle assumes that there will be water or fluid under the cube, otherwise the principle doesn't hold. I know this assumption isn't being mentioned or empasized in most textbooks but it is a requirement for the principle to hold.

Archimedes principle also needs the force of gravity (which creates the hydrostatic pressure in the fluid) in order to hold but this is also another assumption that isn't emphasized.

 

[haruspex]

So there will be no bouyancy force on the cube because there is no fluid below it. Suppose i remove the seal than water will come below the cube then there should be bouyancy force on the cube, right ??

Yes. The pressure will be greater at the bottom of the cube than at the top, hence a net upward force.

 

[Vibhor]

Hello ,

Suppose the cube mentioned by the OP rests on the bottom of the container such that the surfaces are not in complete contact ( some water has crept in) . I mean some parts of the bottom surface of cube touches the container (direct contact) whereas some parts has water between cube and bottom of container .

My doubts are

1) Does buoyant force acts on the cube ? I think yes , since there is some fluid beneath the cube .

2) Does Archimedes principle apply ? I don't think since the cube is not completely surrounded by fluid ( as there are some parts touching the base of container ) .

3) Is the buoyant force still given by V_cube ρ_liquid g ? I am not sure since the cube is not completely surrounded by fluid ( as there are some parts touching the base of container ) .
If not , then , Is the buoyant force = Weight - Normal reaction from the base of the container ??

Many Thanks.

 

[haruspex]

Hello ,

Suppose the cube mentioned by the OP rests on the bottom of the container such that the surfaces are not in complete contact ( some water has crept in) . I mean some parts of the bottom surface of cube touches the container (direct contact) whereas some parts has water between cube and bottom of container .

My doubts are

1) Does buoyant force acts on the cube ? I think yes , since there is some fluid beneath the cube .

2) Does Archimedes principle apply ? I don't think since the cube is not completely surrounded by fluid ( as there are some parts touching the base of container ) .

3) Is the buoyant force still given by V_cube ρ_liquid g ? I am not sure since the cube is not completely surrounded by fluid ( as there are some parts touching the base of container ) .
If not , then , Is the buoyant force = Weight - Normal reaction from the base of the container ??

Many Thanks.

You are right that if there is a significant area of the base which is not wetted then you cannot apply Archimedes' principle. A familiar example is a rubber suction cup. But it's pretty hard to get the same result with 'flat' surfaces. They're never completely flat, nor completely rigid, so the water pressure in a wet region will tend to prise apart the surfaces along the boundary and seep in.

 

[Vibhor]

Thanks . Just to reassure that I have understood it correctly .

So, if even a very small amount of water is under the cube then buoyant force starts acting on the cube , but it would not be equal to V_cube ρ_liquid g . Right ??

Buoyant force equal to V_cube ρ_liquid g would act only when each and every point on cube (V_cube) is in contact with the liquid ??

Edit : V_cube is volume of cube submerged in the liquid

 

[haruspex]

Thanks . Just to reassure that I have understood it correctly .

So, if even a very small amount of water is under the cube then buoyant force starts acting on the cube , but it would not be equal to V_cube ρ_liquid g . Right ??

Buoyant force equal to V_cube ρ_liquid g would act only when each and every point on cube (V_cube) is in contact with the liquid ??

Edit : V_cube is volume of cube submerged in the liquid

Yes.

 

[Vibhor]

Thanks a lot

You are right that if there is a significant area of the base which is not wetted then you cannot apply Archimedes' principle.

Suppose there is a metal sphere instead of a cube (completely submerged in the liquid , resting at the bottom) . In this case there would be a single point in contact with the bottom of container . I think in this case Archimedes principle would be applicable and the upward buoyant force would be given by V_sphere ρ_liquid g .

Right ??

 

[haruspex]

Thanks a lot
Suppose there is a metal sphere instead of a cube (completely submerged in the liquid , resting at the bottom) . In this case there would be a single point in contact with the bottom of container . I think in this case Archimedes principle would be applicable and the upward buoyant force would be given by V_sphere ρ_liquid g .

Right ??

Yes, unless the floor has the same curvature.

 

[Vibhor]

Please see the question :

Ans given is C) .

I believe all three A), B),C) can be correct under different conditions .

I think C) can only be correct under the assumption that no water enters between the bottom of cube and vessel AND water is filled just upto the height of the cube . If former condition is not fulfilled then answer would be B) . But if former is fulfilled and latter is not ( water is filled to a height greater than the height of the cube) then A) is correct .

Do you agree ??

 

[Chestermiller]

What does your free body diagram on the cube tell you?

 

[Vibhor]

I have written my analysis on the basis of FBD of cube .Which part of my post#18 do you not agree with ?

 

[Chestermiller]

I have written my analysis on the basis of FBD of cube .Which part of my post#18 do you not agree with ?

I totally agree with your analysis.

 

[Vibhor]

Ok . Thank you very much . Would you like to add anything to the discussion between posts#12 and #17 ?

 

[Chestermiller]

Ok . Thank you very much . Would you like to add anything to the discussion between posts#12 and #17 ?

No. I think that you and haruspex did a fine job of analyzing the situation.