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Wize University Physics Textbook (Master) > Fluid Mechanics

Pascal's Principle

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Pascal's Principle


Liquids are nearly incompressible, so the change in the pressure at one point appears at all points in the fluid, exactly with the same value. This is known as Pascal's principle.


Wize Concept
Pascal's Principle tells us that:
  • Liquids rise to the same height in all connected open branches of a container.
  • The pressure is the same at all points on a horizontal line through a connected liquid.

Watch Out!
Pascal's principle is valid for fluids in hydrostatic equilibrium when the fluid is not flowing around and moving.

Exam Tip
To use Pascal's principle for static fluid, draw a horizontal line passing through the fluid and set the pressure of different points on this line equal to each other.



Example: Pouring Oil in a U-Tube Filled with Water


In a U-tube filled with some water, we slowly pour vegetable oil in one arm.
a) Find the density of the oil if h=1h=1 cm (the density of water is 10001000 kg/m3).
b) What is the difference in the water levels in the right arm, before and after pouring the oil ?



Part a)


Since we have a static fluid and points A and B are on a horizontal line (at the same level), they have equal pressures by Pascal's principle.

PA=PBP_A=P_B

ρwg(0.04)=ρoilg(0.05)\rho_wg\left(0.04\right)=\rho_{oil}g\left(0.05\right)

ρoil=ρw(0.040.05)=800\rho_{oil}=\rho_w\left(\dfrac{0.04}{0.05}\right)=800



Part b)


The difference in the water level in the right arm, before and after pouring the oil, is the same as the average between the water levels now: that is, the current difference between the water levels in the left and right arms divided by 22.

difference=0.042=0.02difference=\dfrac{0.04}{2}=0.02 m

Practice: U-Tube Filled with Multiple Fluids


A U-tube of uniform cross-sectional area and open to the atmosphere is partially filled with mercury (ρmercury=13.6 ρwater\rho_{mercury}=13.6 \ \rho_{water}). Water is then poured into both arms. The equilibrium configuration of the tube is shown in the figure.

a) Find the value of hh.

b) How would the value of hh change if, instead of water we had a liquid with density ρ\rho, and instead of mercury we had a liquid with density 15 ρ15 \ \rho?