Pascal's Law

Consider the image given below. The column is filled with water. The ends of each column A and B have been blocked by a piston. If piston A is pressed, what do you think will happen to piston B? Piston B is going to rise up. We have just applied Pascal’s law to our fluid pressure. So what is Pascal’s law?

Hydraulic Lift

 

What is Pascal’s Law?

According to Pascal’s law,

The external static pressure applied on a confined liquid is distributed or transmitted evenly throughout the liquid in all directions.

The static pressure acts at right angles to any surface in contact with the fluid. Pascal also found that the pressure at a point for a static fluid would be same across all planes passing through that point in that fluid. Pascal’s law is also known as Pascal’s principle.

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Pascal Law Formula

Following is the formula of Pascal law:

F = PA

Where,

  • F is the force applied
  • P is the pressure transmitted
  • A is the cross sectional area

Example of Pascal’s Law

Let us understand the working principle of Pascal’s law with an example.

A pressure of 2000 Pa is transmitted throughout a liquid column due a force being applied on a piston. If the piston has an area of 0.1 m2, what is the force applied?

This can be calculated using Pascal’s Law formula.

F = PA

Here,

P = 2000 Pa = N/m2

A = 0.1 m2

Substituting values, we arrive at F = 200 N

Applications of Pascal’s Law

  • Hydraulic Lift: The image you saw at the beginning of this article is a simple line diagram of a hydraulic lift. This is the principle of working of hydraulic lift. It works based on the principle of equal pressure transmission throughout a fluid (Pascal’s Law).
  • The construction is such that a narrow cylinder (in this case A) is connected to a wider cylinder (in this case B). They are fitted with airtight pistons on either end. The inside of the cylinders are filled with an incompressible fluid.
  • Pressure applied at piston A is transmitted equally to piston B without diminishing, on use of an incompressible fluid. Piston B effectively serves as a platform to lift heavy objects like big machines or vehicles. Few more applications include a hydraulic jack and hydraulic press and forced amplification is used in the braking system of most cars.

Pascal’s Law Derivation

Consider an arbitrary right angled prismatic triangle in the liquid of density rho. Since the prismatic element is very small, every point is considered to be at the same depth from the liquid surface. The effect of gravity is also same at all these points.

Pascal's law derivation

Let ad, bd, and cd be the area of the faces ABFE, ABDC, and CDFE respectively.

Let P1, P2, and P3 be the pressure on the faces ABFE, ABDC, and CDFE.

Pressure exerts force which is normal to the surface. Let P1 exert force F1 on the surface ABFE, P2 exert force F2 on the surface ABDC, and P3 exert force F3 on the surface CDFE.

Therefore, Force F1, F2, and F3 is given as:
F1 = P1 × area of ABFE = P1 ad
F2 = P2 × area of ABDC = P2 bd
F3 = P3 × area of CDFE = P3 cd

Also, \(sin\theta =\frac{b}{a}\) \(sin\theta =\frac{c}{a}\)

The net force on the prism will be zero since the prism is in equilibrium.

F1 sin θ = F2
F1 cos θ = F3

P1 ad \(\frac{b}{a}\) = P2 bd (equ 1)
P1 ad \(\frac{c}{a}\) = P3 cd (equ 2)

From 1 and 2
P1 = P2 and P1 = P3

∴ P1= P2= P3

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