Hydrostatic pressure — the formula p = ρgh and the hydrostatic paradox
What we will do: We will measure the pressure at different depths in vessels of different shapes and check what it really depends on.
What you see on screen: Connected vessels of different shapes with liquid at a shared level, a pressure sensor at an adjustable depth, and sliders for depth and the liquid's density. The sensor shows the same pressure at the same depth regardless of the vessel. Readouts: depth, density, pressure and liquid level.
I will give you a target pressure — tune the depth and the liquid's density to reach it. p = ρ·g·h.
What just happened?
In both vessels — the wide one and the narrow one — the pressure at the same depth turned out to be identical. The shape and amount of liquid made no difference.
Hydrostatic pressure:
is the push of a column of liquid on a given spot. It depends only on depth and the density of the liquid:
ρ — the density of the liquid, h — the depth below the surface, g — gravitational acceleration.
Hydrostatic paradox:
pressure does NOT depend on the shape of the vessel or the amount of liquid — only on how DEEP you are and how dense the liquid is. A narrow tube and a wide barrel give the same pressure at the same depth.
Connected vessels:
because pressure depends only on depth, the liquid settles by itself at the SAME level in all connected arms — regardless of their shape.
Why you need this on the exam:
p = ρgh and the principle of connected vessels are the basis of hydrostatics problems (a diver, a dam, a hydraulic press, a water level). The trick: depth is what counts, not the amount of liquid.
Remember: hydrostatic pressure depends only on depth and the density of the liquid (p = ρgh), not on the shape of the vessel. That is why in connected vessels the liquid stands at a single level.
Theory & formulas
Hydrostatic pressure
The pressure of a column of liquid depends only on depth and the density of the liquid: It does not depend on the shape of the vessel or on the amount of liquid.
Hydrostatic paradox
At the same depth the pressure is identical regardless of whether the vessel is wide, narrow or slanted — only the height of the column counts.
Connected vessels
Because the pressures at the shared bottom must be equal, the same liquid settles at the same level in every arm.
Notation: — hydrostatic pressure, — the density of the liquid, — gravitational acceleration, — depth.
Frequently asked questions
›What does hydrostatic pressure depend on?
Only on depth and the density of the liquid: p = ρgh — not on the shape of the vessel or the amount of liquid.
›What is the hydrostatic paradox?
It is the fact that at the same depth the pressure is the same regardless of the shape and volume of the vessel.
›Why does liquid stand at a single level in connected vessels?
Because pressure depends only on depth — the equilibrium of the system forces an equal level in all arms.
Teaching? Embed this experiment on your site.
Free widget — paste it on your school site, blog or notes. It links to the full version.