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 the 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 = \[\frac{F}{A}\]
F = PA
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),
The mass density of the fluid ρ
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.
FAQs on Pascal Law - Formula, Application & Derivation
1.Can Pascal’s Law be applied on solids and gases?
Pascal’s Law is mainly applicable on incompressible fluids. Although it could be applied on gas, it would not be as fruitful as the liquid. It is not possible for solids since fluids help to determine the pressure through resistance to flow. This is the reason why all the hydraulic systems like hydraulic brakes, hydraulic jacks, hydraulic press, etc. are used for such purposes. One typical form is found in the automotive repair stores that have a lift in which the air from the air pressure is applied to the top of the oil container. This further leads to the oil to apply pressure on the pistol which lifts the car. But in case a solid object is dropped into the fluid within an enclosed container, the solid object will also feel the pressure when force is exerted on it.
2.What are the applications of Pascal’s Law?
Pascal’s Law is mainly used in hydraulics systems. Some of them are:
Hydraulic Lift- It is used for lifting heavy objects. The force applied creates ‘lift’ and ‘work’. It is used in industries, construction, transport sector, etc. The mathematical representation of Pascal's law suggests that the pressure determined on the fluid in the piston provides enough force to lift and move the object.
Hydraulic Jack- It is based on a closed container and used to lift cars from the ground for repair and maintenance. It contains a large and a small cylinder which are connected. Once the handle is pressed, the valve closes which causes the small piston to exert force on another liquid to the large cylinder which further exerts pressure to lift the object through continuous up and down movement of the handle.
Hydraulic Brakes- These are used to halt the movement of cars.
Hydraulic Pump- Used in the automobile industry, these are used in discharge of fluid.
Aircraft Hydraulic System- It is used to slow down the movement of aircrafts on the runway as well as control gears.
3.Is NCERT enough to study Pascal’s Law?
NCERTs are known to be the best books for understanding the basics of different topics due to their conceptualised and concise explanations on difficult chapters. Also, students must keep an eye on the examples and the question banks after every chapter are very useful for the students since such questions are often asked in several competitive exams like JEE and NEET. In order for students to clear all doubts, they can first study the topic in the NCERT and then move forward with other reference books if needed for numericals.