Glycerin is poured into an open U-shaped tube until the height in both sides is 20 {\rm cm}. Ethyl alcohol is then poured into one arm until the height of the alcohol column is 20 {\rm cm}. The two liquids do not mix.

Question

What is the difference in height between the top surface of the glycerin and the top surface of the alcohol?

Answer 1

The alcohol in the first arm floats on top of the denser glycerin and pressures it down distance h from the initial level. As a result, the glycerin rises by the distance h in the second arm. Due to the equilibrium condition, the point of the two liquids contact in the 1st arm and symmetrical point in the 2nd arm are at the equaled pressures
p₁=p₂
p₀ +ρ₁gh₁=p₀ +ρ₂gh₂
The level of glycerin in the 1st arm came down by h and the level of glycerin in the 2nd arm came up by h => h₂=2h, Since h₁=0.2 m,
ρ₁gh₁= ρ₂g2h
h= ρ₁h₁/2 ρ₂=790•0.2/2•1260 =0.0627 cm
Then
Δh=0.2-2•0.0627 = 0.0746 m