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Pascal’s law was given in 1647 by Blaise Pascal, a well-known French mathematician, physicist, and philosopher. Pascal’s law states that pressure exerted in a resting liquid is the same in all directions. When external pressure is applied to any section of a fluid enclosed in a vessel, it is transmitted in all directions undiminished and evenly. This law governs the operation of hydraulic power machines. Pascal’s law has many applications in daily life. Several devices, such as hydraulic lifts and hydraulic brakes, are based on Pascal's law.
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Key Terms: Application of Pascal Law, Pascal Law Definition, Pascal Law Formula, Pascal's Principle, Hydraulic Lift Pascal's Law
Pascal Law Definition
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Pascal Law describes how applying pressure to a fluid in a closed container has an impact. It states that pressure exerted on a contained fluid is distributed evenly across the container.
Pascal Law
Pressure is created when a liquid is confined and a force is applied. When fluid pressure is applied to a moving output piston, the output force is generated. Pascal created the first hydraulic equipment, which consisted of two sealed containers connected by a tube.
The pistons inside the cylinders seal against the cylinder walls, preventing liquid from seeping out and air from entering the cylinder. The pressure rises throughout the system when the piston in the first cylinder applies a force. The force is transferred to the second cylinder via the connecting tube.
The second cylinder's pressurized fluid exerts a force on the bottom of the second piston, propelling it upward and carrying a load over it. Pascal discovered that he could increase the force available to work by utilizing this tool, just as he did with a lever or gear.
Pascal discovered that when a force is given to a liquid, it creates pressure or the transfer of force through the liquid. When a liquid is defined and put under pressure, these investigations revealed two fundamental elements.
- The pressure that is applied to it is distributed evenly in all directions
- operates with equal force throughout the container.
Pascal Law Formula
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The Pascal law formula is as follows:
F = PA
- F is the applied force
- P is the transmitted pressure
- A is the cross-sectional area
From the above equation, we can also define Pascal's Law as Pressure is equal to the force divided by the area on which the force acts.
Pascal Law and its Applications
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Pascal Law and its Applications in Hydraulic Lifts are elaborated below:
Hydraulic Lift: The given image depicts a hydraulic lift in a basic line diagram. The functioning principle of a hydraulic lift is as follows. The idea of equal pressure transmission throughout a fluid (Pascal's Law) governs its operation.
Pascals Law in Hydraulic Lift
A narrow cylinder (in this example, A) is connected to a larger cylinder (in this case B) (in this case B). On both ends, airtight pistons are installed. The insides of the cylinders are filled with a non-compressible fluid.
Pressure applied to piston A is equally transmitted to piston B, allowing the fluid that cannot be compressed to be used. As a result, piston B can be used to raise huge things such as large machinery or automobiles. A hydraulic jack and hydraulic press are two further examples, and most cars employ forced amplification in their braking systems.
Derivation of Pascal’s Law
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Consider an arbitrary right-angled prismatic triangle in a liquid having density rho. Due to the small size of the prismatic element, each point is regarded to be at the same depth as the liquid surface. As a result, T is the same at all of these locations.
Derivation of Pascal’s Law
Now, ad, bd, and cd are the areas of the faces ABFE, ABDC, and CDFE, respectively.
Let P1, P2, and P3 be the pressure on the faces ABFE, ABDC, and CDFE.
The force exerted by pressure 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, sinθ=ba cosθ=ca
Because the prism is in equilibrium, the net force acting on it will be zero.
F1 sin θ = F2
F1 cos θ = F3
P1 ad ba = P2 bd (equ 1)
P1 ad ca = P3 cd (equ 2)
From 1 and 2
P1 = P2 and P1 = P3
∴ P1= P2= P3
Things to Remember
- Pascal’s law states that pressure is distributed evenly across an area.
- The formula for Pascal's law is F=PA.
- Pascal’s law is applied to hydraulic machines.
Previous Year Questions
Sample Questions
Ques: Explain why, although pressure is force divided by area, hydrostatic pressure is a scalar quantity? (1 mark)
Ans: When a force is applied to a fluid, the pressure generated is uniformly transmitted in all directions, resulting in a scalar quantity with no particular direction. Only while considering the magnitudes, do we talk about force divided by area. The actual vector form of the relation is 
Ques: A 50 kilograms girl balances on a single heel while wearing high heels. With a diameter of 1.0 cm, the heel is circular. How much pressure does the heel put on the horizontal floor? (2 marks)
Ans: Mass of the girl m = 50 kg.
Gravitational acceleration g = 9.8 m s-2
Weight of the girl (W) , mg = 490 N
Ques: Torricelli's barometer was made of mercury. Pascal replicated it with French wine with a density of 984 kg m-3. Determine the height of the wine column at normal atmospheric pressure. (2 marks)
Ans:
Atmospheric pressure is 
The density of French wine 
The height of the wine column hw would be
Ques: A hydraulic car lift can raise vehicles weighing up to 3000 kg. The piston carrying the load has a cross-sectional area of 425 cm2. What would be the maximum pressure that the smaller piston would have to handle? (2 marks)
Ans: Maximum Pressure which the piston would have to bear is
Ques: A hydraulic automobile lift is designed to lift cars with a maximum mass of 3000 kg. The area of the cross-section of the piston carrying the load is 425 cm2. What maximum pressure would the smaller piston have to bear? (3 Marks)
Ans: Pressure on the piston P=F/A
Force F=m×a
=3000×9.8
=29400 N
Area of cross-section A=425×10−4sqm
Therefore the pressure P=3000×9.8/425×10−4=6.92×105Pa.
Ques: The rise or fall of a liquid against gravitational force through fine tubes is known as capillarity. [MAY-2011] (5 Marks)
a) Give an example of capillarity from practical life.
b) Derive an expression for the capillary rise through a capillary tube.
Ans: a) 1) A blotting paper absorbs ink by capillary action.
2) It is by capillary action that water reaches every branch of a plant from the stem.
3) We use a towel to dry our bodies after a bath.
4) Due to capillary action, the water rises to the surface of the fields and gets evaporated. To avoid this loss of water, the fields are plowed.
5) A sponge retains water,
b) Consider a capillary tube of radius ‘a’ dipped in a liquid of density p and surface tension S. If the liquid has a concave meniscus it will rise in the capillary tube. Let h be the rise of the liquid in the tube. Let p, be the pressure on the concave side of the meniscus and p0, that on the other side.
The excess pressure on the concave side of the meniscus can be written as \(p_i - p_0 = \frac{2S}{R}\)… (1)
Where R is the radius of the concave meniscus. The tangent to the meniscus at point A makes an angle θ with the wall of the tube. In the right-angled triangle ACO
Considering two points M and N in the same hori-zontal level of a liquid at rest, pressure at N = pressure at M But pressure at M = ρ1, the pressure over the concave meniscus and pressure at N = p0 +hρ
Ques: Hydraulic brakes are based on? (2 Marks)
Ans: Hydraulic brakes work on the principle of Pascal’s law. According to this law whenever pressure is applied to the fluid it travels uniformly in all directions. Therefore, when we apply force on a small piston, the pressure gets created which is transmitted through the fluid to a larger piston.
As a result of this larger force, uniform braking is applied on all four wheels. As braking force is generated due to hydraulic pressure, they are known as hydraulic brakes. Liquids are used instead of gas as liquids are incompressible.
Ques: The small piston of a hydraulic lift has an area of 0.20m2. A car weighing 1.20×104N sits on a rack mounted on a large piston. The large piston has an area of 0.90m2. How large a force must be applied to the small piston to support the car? (3 Marks)
Ans: Hydraulic lift works on the principle of pascal's law
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