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.
What is the difference in height between the top surface of the glycerin and the top surface of the alcohol?
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
To find the difference in height between the top surface of the glycerin and the top surface of the alcohol, we need to consider the pressure differences between the two arms of the U-shaped tube.
When the liquids are at the same level, the pressure at the same height should be equal. Since the two liquids do not mix, we can treat each side of the U-shaped tube as a separate fluid column.
Let's assume the density of glycerin is ρg and the density of ethyl alcohol is ρa.
We know that pressure is given by the formula: P = ρgh, where P is pressure, ρ is density, g is the acceleration due to gravity, and h is the height of the fluid column.
In this case, both sides of the U-shaped tube have the same height, 20 cm. Hence, the pressures on both sides are equal:
Pglycerin = ρg * 9.8 * 20
Palcohol = ρa * 9.8 * 20
Since the pressures on both sides of the tube are equal, we can write:
Pglycerin = Palcohol
Thus, ρg * 9.8 * 20 = ρa * 9.8 * 20
Now, divide both sides of the equation by 9.8 to cancel out the gravitational acceleration:
ρg * 20 = ρa * 20
Divide both sides of the equation by 20:
ρg = ρa
Since the densities of glycerin and ethyl alcohol are equal, the heights of the liquids on both sides of the U-shaped tube will also be equal.
Therefore, the difference in height between the top surface of the glycerin and the top surface of the alcohol is 0 cm.
To find the difference in height between the top surface of the glycerin and the top surface of the alcohol, we need to consider the pressure difference between the two fluids. The pressure at any given point in a fluid is determined by the height of the fluid column above that point.
In this scenario, since both arms of the U-shaped tube contain the same height of glycerin (20 cm), the pressure at the bottom of each arm will be the same.
When the ethyl alcohol is poured into one arm, it creates an additional fluid column that exerts its own pressure. The pressure at the bottom of this column is determined by the height of the alcohol column (20 cm).
Since the two liquids do not mix, the pressure at the bottom of the alcohol column should be equal to the pressure at the bottom of the glycerin column. This means that the difference in height between the top surface of the glycerin and the top surface of the alcohol is equal to the height of the alcohol column, which is 20 cm.