Communicating vessels — liquid level

What just happened?

After pouring the same liquid, the levels in both arms evened out, even though the tubes differ in width. The cross-section does not matter — what counts is the height of the liquid column.

Law of communicating vessels:

— at the base, where the liquids meet, the pressures must be equal (otherwise the liquid would flow). Hence the column heights are inversely proportional to density.

What does it mean in practice?

For a homogeneous liquid ($\rho_1=\rho_2$) the columns are equal — $h_1 = h_2$ — regardless of the shape and width of the arms. For two immiscible liquids:

The lighter liquid stands higher, the heavier one lower — so that the product $\rho\, h$ is the same on both sides.

Where will you meet this?

A water level on a construction site, the trap under a sink, a water tower, a U-tube as a manometer for comparing liquid densities — all of these work thanks to the equality of pressures at the base.

Why you need this for the exam:

The condition $\rho_1 h_1 = \rho_2 h_2$ solves typical U-tube problems: find an unknown density or a level difference. Trick: equate the pressures at the base and cancel $g$.

Remember: in communicating vessels a homogeneous liquid stands at the same level, because the pressures at the base must be equal: ρ·g·h decides, not the shape of the vessel.