U-tube: a denser liquid and its level

The left arm of a U-tube holds a 24 cm water column, and the right arm holds another liquid of density 1.2 g/cm³ (water density 1.0 g/cm³). How high is the column in the right arm at rest, and what is the height difference between the two surfaces?

Step-by-step

1. ρ_water·h_water = ρ_oil·h_oil (equal pressure at the same level)
   In connected vessels the same liquid has equal pressure at the same level
2. h_oil = ρ_water·h_water/ρ_oil = 1×24/1.2 = 20
   The lower the density, the taller the column needed to balance
3. Δh = |h_oil − h_water| = 4 cm
   Height difference between the two liquid levels

Why it works

The two columns balance the same pressure at the bottom, so ρ_water × h_water = ρ_liquid × h_liquid. The denser liquid stands lower, and the difference is the gap between the two heights.

Common mistakes

• Multiplying by 1.2 to get 28.8 cm instead of dividing.
• Forgetting the difference is the gap between the two column heights.

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