 # Pascal Law - Formula, Application & Derivation

What is Pascal’s Law

This law was given by a well known French mathematician, physicist, and philosopher Blaise Pascal in the year 1647.

This law states that pressure exerted in some liquid which is at rest is same in all the directions.

OR

Whenever an external pressure is applied on any part of a fluid contained in a vessel, it is transmitted undiminished and equally in all directions.

Hydraulic Power machines work on the basis of this law.

Pascal’s Law Formula

Pascal's Law formula shows the relationship between pressure, force applied and area of contact i.e,

P = $\frac{F}{A}$

Where, P= Pressure, F=Force and A=Area of contact

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

A Pressure of 2000 Pa is Transmitted Throughout a Liquid Column by Applying a Force on a Piston. If the Piston has an Area of 0.1 m2, What is the Force Applied?

We can calculate the value of force using Pascal’s Law formula.

F = PA

Here,

P = 2000 Pa = N/m2

A = 0.1 m2

After substituting the values, we arrive at Force = 20N or F = 200 N

Applications of Pascal’s Law

1. Hydraulic Lift

It has many applications in daily life. Several devices, such as hydraulic lift and hydraulic brakes, are based on Pascal's law. Fluids are used for transmitting pressure in all these devices. In a hydraulic lift, as shown in the figure above, two pistons are separated by the space filled with a liquid. A piston of small cross-section A is used to exert a force F directly on the liquid. The pressure P =F/A is transmitted throughout the liquid to the larger cylinder attached with a larger piston of area B, which results in an upward force of  P × B. Therefore, the piston is capable of supporting a large force (large weight of, say a car or a truck placed on the platform). By changing the force at A, the platform can be moved up or down. Thus, the applied force has been increased by a factor of B/A and this factor is the mechanical advantage of the device.

2. Hydraulic Brake

In automobiles, the hydraulic brakes also work on the same principle. When we apply a little force on the pedal with our foot, the master piston moves inside the master cylinder, and the pressure caused is transmitted through the brake oil for acting on a piston of the larger surface area. A large force then acts on the piston and is pushed down, which expands the brake shoes against brake lining. Consequently, a small force on the pedal produces an extremely retarding force on the wheel. A significant advantage of the system is that the pressure, which is set up by pressing pedal is transmitted equally to all cylinders, which are attached to the four wheels to make the braking effort equal on all wheels.

3. Variation of Pressure with Depth

Consider a fluid at rest in a container. In the figure above point 1 is at height h from a point 2. P1 and P2 denote the pressure at points 1 and 2 respectively. Consider a cylindrical element of fluid having an area of base A and height h. Since the fluid is at rest, the resultant horizontal forces should be zero along with the resultant vertical forces balancing the weight of the element. The forces, which are acting in the vertical direction, are due to the fluid pressure at the top (P1A) acting downward and at the bottom (P2A) acting upward. If mg is the weight of the fluid in the cylinder then we can say that,

(P2 −P1 ) A = mg

Now, if ρ is the mass density of the fluid then the mass of fluid will be

m = ρV= ρhA

so that  (P2 −P1) = ρgh

Pressure difference depends on

The vertical distance h between the points (1 and 2),

1. The mass density of the fluid ρ

2. Acceleration due to gravity g.

If the point 1 under discussion is shifted to the top of the fluid (say, water), which is open to the atmosphere, P1 may be replaced by atmospheric pressure (Pa ) and we replace P2 by P. Then the above equation gives,

P = Pa + ρgh.

Derivation of Pascal’s Law

Blaise Pascal, a French scientist observed that the pressure in a fluid at rest is the same at all points provided they are at the same height. This fact may be demonstrated directly. The figure above shows an element in the interior of a fluid at rest. This element AEC-BDF is in the form of a right-angled prism. In this principle, the prismatic element is extremely small, due to which, every part of it can be considered at the same depth from the liquid surface and hence, at all these points, the effect of the gravity is the same. The forces on this element are the ones exerted by the rest of the fluid and they must be normal or perpendicular to the surfaces of the element. Thus, the fluid exerts pressures Pa, Pb, and Pc on this element of an area corresponding to the normal forces Fa, Fb and Fc as shown in the figure above on the faces ABFE, ABDC and CDFE denoted by Aa, Ab and Ac respectively.

Then

Fa sinθ = Fb , Fa cosθ = Fc (by equilibrium)

Aa sinθ = Ab , Aa cosθ = Ac (by geometry)

$\frac{F_{a}}{A_{a}}$ = $\frac{F_{b}}{A_{b}}$ = $\frac{F_{c}}{A_{c}}$

Therefore, the pressure exerted is the same in all directions in the fluid, which is at rest. We can say that like other types of stress, pressure is not a vector quantity. No direction can be assigned to it. The force against any area within (or bounding) a fluid at rest and under pressure is normal to the area, regardless of the orientation of the area.