## Pascal Law

**Pascal Law:** The question that pop-up in our mind is how does a hydraulic crane lift heavy objects such as large containers and equipment? A hydraulic crane is one of the applications of Pascal Law. The crane contains a hydraulic pump that is electrified with the help of the crane's engine that applies pressure to the oil or fluid contained in the system. As we know, Oil is incompressible, the pressure which is exerted on one part of the system is distributed uniformly throughout the other part of the hydraulic crane. This allows the device to lift up heavy equipment and containers that are relatively more difficult to carry than other lifting devices.

## What is Pascal Law?

The Pascal Law (also referred to as the Pascal Principle or Principle of Transmission of Fluid-Pressure) is a principle given by French mathematician **Blaise Pascal **in the year 1653 and finally formulated in the year 1663. **According to Pascal's law, “Pressure exerted at any point in a confined incompressible fluid is distributed uniformly throughout the fluid such that the same change occurs everywhere.”**

## Pascal Law Formula

Pascal Law signifies the relationship between pressure, force applied and area of contact, that is, the following is the formula of Pascal's law:

**P = F/A, **

**or F = PA **

Where,

P is Pressure,

F is Force

A is the Area of contact

**Based on Formula one the example is-**

**Example: A pressure of 4000 Pa is transmitted throughout the** **hydraulic pump due to a force being applied on a piston. If the piston has an area of 0.4 m², what force is applied?**

**Solution:** According to the question, P = 4000 Pa = N/m²

A = 0.1 m²

As we know, F = PA

By putting the value, we get, F = 400 N

So the force applied upon the piston is 400N.

## Pascal Law Derivation

Let us suppose that the ad, bd and cd are the areas of the faces ADFC, ADEB and BEFC.

And the force that acts upon the faces of the prism is Fa, Fb and Fc.

Similarly letting the pressure act on the three faces of the prism P1, P2 and P3 respectively.

As we know, Mathematically, Pressure is equal to the force divided by the area.

P=F/A

So, we can write force as, F=P×A

Therefore, Force Fa, Fb and Fc can be written as-

Fa=P1× area of BEFC =P1×cd

Fb=P2× area of ADFC =P2×ad

Fc=P3× area of ADEB =P3×bd

Let the angle BAC= θ,

So now in the ΔBAC,

sinθ=b/a and cosθ=c/a

Since the above triangular prism is at equilibrium then the net force on the prism will be zero.

So, by balancing the forces for the prism to be in equilibrium,

Fa=Fbcosθ ----------- (1)

and Fbsinθ=Fc ------------ (2)

On substituting the values of sinθ and cosθ value of the forces in (1) and (2), we get,

For equation (1),

P1×cd=P2×ad×c/a

P1×cd=P2×cd

P1=P2 ------------- (3)

Similar to (2)

P2×ad×ba=P3×bd

P2×bd=P3×bd

P2=P3 --------- (4)

So now from (3) and (4), we can infer that P1=P2 and P2=P3,

Therefore, this implies that all the values of the pressure are mutually equal to each other.

P1=P2=P3 (Hence Proved)

## Application of Pascal Law

The Principle application of Pascal's Law is seen in different instruments such as-

- Hydraulic Lift
- Hydraulic Jack
- Hydraulic Brakes
- Hydraulic Pumps
- Aircraft Hydraulic System