Explaining the Hydrostatic Pressure Paradox: What's Wrong with the Flask?

In summary, the hydrostatic pressure of water is the same at the bottom of each container, but the force acting on the balance is reduced because of the perpendicular force acting on the side of the flask.
  • #1
dobry_den
115
0
Hi! I came over this paradox: an Erlenmeyer flask and a beaker are filled with water to the same height - it is assumed that they have the same base area and the same mass. Then the hydrostatic pressure of water is the same at the bottom of each container (since the depths are the same) and when put on a balance, they should exert the same force on it and therefore weigh the same. The question is - what's wrong?

I think it might be that one should take into account all the forces acting on the containers. In the case of the beaker, there's the force that's acts on the base plus forces that act on the sides of the beaker. The forces acting on the sides should cancel themselves - when we look at every vertical cut of the beaker that goes through its axis, the forces on opposite sides are equal in magnitude (due to the Pascal's Law - "the fluid pressure at all points in a connected body of an incompressible fluid at rest, which are at the same absolute height, are the same") and opposite in directions.

In the case of the flask, these forces are perpendicular to its sides. Horizontal components cancel out. But for the paradox stated above, their vertical components are important, since they are responsible for reducing the force acting on the balance - they reduce the effect of the hydrostatic force acting on the base of the flask.

I also attached a drawing. Is this a good answer to the initial question "What's wrong (in the case of the flask)?"
 

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  • #2
Yes, your reasoning is correct. This is kind of an inverse application of Archimedes' law...
 
  • #3
Thanks very much for making me sure... I have one more question - how the force acting on the side could be computed? I think it would require integration, wouldn't it? Like taking an infinitesimaly narrow stripe around the flask whose all points are in the same depth, calculating the hydrostatic pressure and then using the formula p = F/A, where A is the area of the stripe?
 

What is the Hydrostatic Pressure paradox?

The Hydrostatic Pressure paradox is a phenomenon that occurs when a fluid is contained within a vessel and subjected to different pressures at different points. According to Pascal's principle, the pressure at any point in a static fluid is equal in all directions. However, in the Hydrostatic Pressure paradox, the pressure at the bottom of the vessel is greater than the pressure at the top, even though the points are at the same depth.

What causes the Hydrostatic Pressure paradox?

The Hydrostatic Pressure paradox is caused by the weight of the fluid in the vessel. As the depth increases, the weight of the fluid above also increases, resulting in a higher pressure at the bottom of the vessel.

Is the Hydrostatic Pressure paradox a real paradox?

No, the Hydrostatic Pressure paradox is not a true paradox. It is a result of our common understanding of pressure, which is based on our everyday experiences with solid objects. In reality, fluids behave differently than solids and can exert pressure in different directions.

How is the Hydrostatic Pressure paradox relevant in science?

The Hydrostatic Pressure paradox has important implications in many fields of science, such as fluid mechanics, hydrology, and geology. Understanding this paradox is crucial in designing and maintaining structures that hold fluids, such as dams, pipelines, and water tanks.

Can the Hydrostatic Pressure paradox be explained by any other theories?

Yes, the Hydrostatic Pressure paradox can also be explained by Bernoulli's principle, which states that as the speed of a fluid increases, its pressure decreases. In the case of the Hydrostatic Pressure paradox, the fluid at the bottom of the vessel is moving faster due to the weight of the fluid above, resulting in a lower pressure at the bottom.

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