Trying to understand hydrostatic pressure with different vessel widths

[abrek]

TL;DR Summary

hydraulic pressure at minimum pipe diameter

will the hydrostatic pressure be the same on the vessels shown with a different diameter of 1 meter and 1 centimeter? and will it be the same in both vessels if the first pipe has a diameter even less than 1 millimeter, 1 thousandth of a millimeter, 1 atom?

 

[DaveE]

Yes. Same pressure. PSI = pounds per square inch. Regardless of the pipe diameter, you can think of the weight of a thin vertical column of liquid. If you double the cross sectional area of that column, you also double the weight, so the change cancels out.

 

[DaveE]

When you get down to atomic distance scales these rules are likely to break down. But the answer will depend on a lot of specific things, like which sort of atoms, etc. There's no simple answer in the nanoscale cases.

 

[berkeman]

will the hydrostatic pressure be the same on the vessels shown with a different diameter of 1 meter and 1 centimeter?

As already covered by @DaveE the answer is yes. The weights of the two water pipes will be different (more water total means a heavier pipe+water combination), but the pressure distributions will be the same.