# Pascal's law and variation of pressure with depth

1. Sep 18, 2011

### logearav

1. The problem statement, all variables and given/known data

Pascal's law states that " The pressure in a fluid at rest is the same at all points if they are at the same height"
Also we know " Pressure increases with depth"
I get confused. When pressure increases with distance, how pressure is same at all points.

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 18, 2011

### high noon

3. Sep 18, 2011

### logearav

Thanks for the reply Mr. High noon. But i dont understand from that link. Can u provide some examples to explain this.

4. Sep 21, 2011

### logearav

Mr. High noon, let me explain the scenario. I have a beaker of four feet height and i fill it with water full to the brim, and I mark three points A, B and C at the top, middle and bottom of the beaker. I presume that the pressure will be same at these points, since pascal's law states that " Pressure is same everywhere if the fluid is at rest"
Also, i have another statement, " Pressure increases with height". This is what confuses me, since according to this statement, pressure at the points A, B and C will be different, since they are located at different heights from the top of the beaker. Please clarify, sir

5. Sep 21, 2011

### SteamKing

Staff Emeritus
I think the key point you are missing in Pascal's Law ("the pressure is the same everywhere if the fluid is at rest") is this situation applies only in those cases where the fluid is confined.

For example, in your beaker scenario, I am presuming the beaker is open at the top. In that situation, after the beaker is filled, the pressure in the fluid at points A, B, and C will vary with depth from the free surface of the fluid. The pressure will not be the same unless A, B, and C are at the same depth relative to the surface of the fluid.

Now, instead of a beaker with an open top, suppose I had a beaker which was closed by a tight fitting piston, which was able to slide along the sides of the beaker without allowing any leakage. If I filled the space between the piston and the bottom of the beaker with a fluid, and then applied a force to the piston, the pressure at all points in the enclosed volume contained by the sides and bottom of the beaker and the piston would be equal. This is the mechanical principle behind hydraulic mechanisms.

6. Sep 21, 2011

### olivermsun

It works out because same height = same depth. One is just measured from the bottom up, and the other is measured from the top down.

7. Sep 22, 2011

### logearav

Mr. SteamKing,
In my attachment, Pressure at points C and D will be same and Point B will have more pressure than Point A. Points C and D will have more pressure than A and B. Am I right?
Instead of an open beaker, I fit a piston and compress the liquid. What will happen to pressure at these points?
I am interpreting in this way. Due to compression the point A will experience an increase in pressure, say 2 pascal, and at the same time point B will also experience an increase in pressure by 2 pascal. Points C and D will have the same pressure.
Correct me if i am wrong.

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8. Sep 24, 2011

### logearav

Consider my attachment.
Is point E in the top-sealed tube, above or below atmospheric pressure?
I think it is below atmospheric pressure, since it lies above A. Am i right?

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9. Oct 10, 2011

### high noon

"Mr. High noon, let me explain the scenario. I have a beaker of four feet height and i fill it with water full to the brim, and I mark three points A, B and C at the top, middle and bottom of the beaker. I presume that the pressure will be same at these points, since pascal's law states that " Pressure is same everywhere if the fluid is at rest"
Also, i have another statement, " Pressure increases with height". This is what confuses me, since according to this statement, pressure at the points A, B and C will be different, since they are located at different heights from the top of the beaker. Please clarify, sir"

what is pressure actually? Its the force experienced by the wall of the beaker from the colliding molecules of the fluid, is not it? then if you think about the fluid in a space where there is no gravity, the whole beaker will see same pressure at all the points on it.
now when you put some liquid in cylinder, think about the whole liq as some disks of liq with finite width, and a disk which is the upper most feels only the atmospheric pressure, and one which is below some disks feels atm + the weight of some liq disks thats why the pressure changes with height, only beacuse of gravity (constant force in a direction!).

and see this-
Pascal's Principle of Hydrostatics. Pascal actually has three separate principles of hydrostatics. When a textbook refers to Pascal's Principle it should specify which is meant.

Pascal 1: The pressure at any point in a liquid exerts force equally in all directions. This means that an infinitessimal surface area placed at that point will experience the same force due to pressure no matter what its orientation.

Pascal 2: When pressure is changed (increased or decreased) at any point in a homogenous, incompressible fluid, all other points experience the same change of pressure.

Except for minor edits and insertion of the words 'homogenous' and 'incompressible', this is the statement of the principle given in John A. Eldridge's textbook College Physics (McGraw-Hill, 1937). Yet over half of the textbooks I've checked, including recent ones, omit the important word 'changed'. Some textbooks add the qualification 'enclosed fluid'. This gives the false impression that the fluid must be in a closed container, which isn't a necessary condition of Pascal's principle at all.

Some of these textbooks do indicate that Pascal's principle applies only to changes in pressure, but do so in the surrounding text, not in the bold, highlighted, and boxed statement of the principle. Students, of course, read the emphasized statement of the principle and not the surrounding text. Few books give any examples of the principle applied to anything other than enclosed liquids. The usual example is the hydraulic press. Too few show that Pascal's principle is derivable in one step from Bernoulli's equation. Therefore students have the false impression that these are independent laws.

Pascal 3. The hydraulic lever. The hydraulic jack is a problem in fluid equilibrium, just as a pulley system is a problem in mechanical equilibrium (no accelerations involved). It's the static situation in which a small force on a small piston balances a large force on a large piston. No change of pressure need be involved here. A constant force on one piston slowly lifts a different piston with a constant force on it. At all times during this process the fluid is in near-equilibrium. This 'principle' is no more than an application of the definition of pressure as F/A, the quotient of net force to the area over which the force acts. However, it also uses the principle that pressure in a fluid is uniform throughout the fluid at all points of the same height.

This hydraulic jack lifitng process is done at constant speed. If the two pistons are at different levels, as they usually are in real jacks used for lifting, there's a pressure difference between the two pistons due to height difference (rho)gh. In textbook examples this is generally considered small enough to neglect and may not even be mentioned.

Pascal's own discussion of the principle is not concisely stated and can be misleading if hastily read. See his On the Equilibrium of Liquids, 1663. He inroduces the principle with the example of a piston as part of an enclosed vessel and considers what happens if a force is applied to that piston. He concludes that each portion of the vessel is pressed in proportion to its area. He does mention parenthetically that he is 'excluding the weight of the water..., for I am speaking only of the piston's effect.'