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Difficulty: MediumHydrostatic Pressure

An open-ended U-shaped tube contains two immiscible static liquids, Liquid A and Liquid B, which meet at a distinct separation line. Above this horizontal separation line, the column of Liquid A on the left side is exactly 20 cm20\text{ cm} tall. The column of Liquid B on the right side is exactly 30 cm30\text{ cm} tall. If the density of Liquid A is 1.5 g/cm31.5\text{ g/cm}^3, what must be the density of Liquid B to maintain this equilibrium?

  1. 1.0 g/cm31.0\text{ g/cm}^3Answer
  2. B
    2.25 g/cm32.25\text{ g/cm}^3
  3. C
    0.67 g/cm30.67\text{ g/cm}^3
  4. D
    1.5 g/cm31.5\text{ g/cm}^3

Answer

The density of Liquid B is 1.0 g/cm31.0\text{ g/cm}^3.
In an open U-tube manometer at equilibrium, the hydrostatic pressure exerted by both fluid columns at their meeting point must be equal. Since pressure equals density times gravity times height (P=ρghP = \rho g h), we can drop the constant gravity and set ρA×hA=ρB×hB\rho_A \times h_A = \rho_B \times h_B. Multiplying Liquid A's density (1.51.5) by its height (2020) gives 3030. To balance this, Liquid B must have a density that, when multiplied by its height (3030), also equals 3030. Therefore, Liquid B's density is 1.0 g/cm31.0\text{ g/cm}^3.

Step-by-Step Solution

1
Identify the relevant physics principle for a static fluid system.
The hydrostatic pressure at the horizontal interface must be equal on both sides (PA=PBP_A = P_B).
If the pressures were not equal at the interface, the fluid would move until equilibrium is reached.
2
Set up the pressure equivalence equation.
ρA×g×hA=ρB×g×hB\rho_A \times g \times h_A = \rho_B \times g \times h_B
Hydrostatic pressure is the product of fluid density, gravitational acceleration, and depth.
3
Cancel out gravity (gg) and substitute the given values into the equation.
1.5×20=ρB×301.5 \times 20 = \rho_B \times 30
Gravity is constant on both sides, and the values provided are density (1.5 g/cm31.5\text{ g/cm}^3) and heights (20 cm20\text{ cm} and 30 cm30\text{ cm}).
4
Solve for the unknown density (ρB\rho_B).
30=30×ρB⇒ρB=1.0 g/cm330 = 30 \times \rho_B \Rightarrow \rho_B = 1.0\text{ g/cm}^3
Isolating the variable gives the exact density required to balance the pressure.

Key Concept

Hydrostatic pressure depends on the product of a fluid's density and its vertical depth.
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Topics

Hydrostatic Pressure

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