Pascal's Principle Homework: Pressure Variation w/ Height & Diameter

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In a system where a small tube is connected to a larger tube filled with water, the pressure at the bottom of the larger tube increases with the height of the water column in the small tube, as described by Pascal's principle. When the height of the small tube is varied, the pressure at the bottom of the larger tube increases by an amount proportional to the height, represented as ρg a. The diameter of the upper tube does not affect the pressure at the bottom, as pressure is dependent solely on the height of the liquid column above. This leads to confusion regarding why a small column of liquid can exert the same pressure as a larger one if both are of equal height. Ultimately, the discussion emphasizes that height is the critical factor in determining pressure, regardless of the diameter of the tube.
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Homework Statement


A small tube is connected to the top of a larger one and the whole thing is filled with water. The small tube has height a and the larger tube has height b.

What happens to the pressure at the bottom of the larger tube as (1) a is varied, and (2) a is held constant but the diameter of the upper tube is increased?

Homework Equations


<br /> p_\text{gauge} = \rho g h<br />
Pascal's principle.

The Attempt at a Solution



(1) According to Pascal's principle, the larger tube will see a pressure increase of rho g a. This will increase the downward force at the bottom of the larger barrel, and that will be rho g a.

(2) I don't think the diameter matters, but intuitively I can't see why! If b is the diameter of a straw (a few millimeters), the smaller tube will increase the pressure on the larger one just as much as a big tube on top. It is only height that seems to matter, then, and if I took a really tall straw and put it over a large vat of water, I would see a huge increase in force at the bottom of the vat. Confused...

 
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bigplanet401 said:
and that will be rho g a.

bigplanet401 said:
only height that seems to matter

bigplanet401 said:
really tall straw and put it over a large vat of water, I would see a huge increase in force at the bottom of the vat. Confused...
... and, your question is --- what?
 
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Is height really the only thing that matters here? Intuitively, this just doesn't make sense to me. How can a small straw of liquid (say 10 cm high) exert the same pressure at the surface of the barrel as, say, a huge vat that is just as high?
 
Dive 2 m down to the bottom of a swimming pool. Dive the same distance below the surface of the ocean: do you feel more pressure?
 
MrAnchovy said:
Dive 2 m down to the bottom of a swimming pool. Dive the same distance below the surface of the ocean: do you feel more pressure?
Well you do of course because ocean water is denser due to dissolved salts, but do you feel thousands of times more pressure?
 

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