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

[bigplanet401]

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

$$p_\text{gauge} = \rho g h$$
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...

 

[Bystander]

and that will be rho g a.

only height that seems to matter

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?

 

[bigplanet401]

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?

 

[pbuk]

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?

 

[pbuk]

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?