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. What is Pascal’s Law and how does it apply to fluids at rest?
Pascal’s Law states that when an external pressure is applied to a confined fluid, it is transmitted undiminished and equally in all directions throughout the fluid. This principle is applicable to all incompressible fluids at rest, ensuring uniform pressure distribution regardless of direction.
2. State the mathematical formula for Pascal’s Law and explain each term.
The formula is P = F / A, where:
- P stands for pressure (in pascals, Pa),
- F is the force applied (in newtons, N),
- A is the area over which the force is applied (in square meters, m2).
3. Describe three important real-life applications of Pascal’s Law.
Key applications include:
- Hydraulic lifts: Used to lift heavy loads like vehicles using fluid pressure transmission.
- Hydraulic brakes: Widely utilized in automobiles to apply braking force equally to all wheels.
- Hydraulic jacks and pumps: Employed in machines to increase force for lifting or moving objects efficiently.
4. How does Pascal’s Law provide a mechanical advantage in hydraulic systems?
Hydraulic systems based on Pascal’s Law use a small applied force on a small-area piston, which is transmitted as a much larger force on a large-area piston. The mechanical advantage is the ratio of the areas (output area/input area), allowing small efforts to lift heavy loads efficiently in accordance with the principle.
5. Can Pascal’s Law be extended to gases and solids? Why or why not?
Pascal’s Law applies mainly to incompressible fluids like liquids. In gases, pressure can be transmitted, but since gases are compressible, the law is less effective. Solids do not transmit pressure in the same way due to their rigid structure, so the law does not apply to solids.
6. How does the pressure in a fluid vary with depth according to Pascal’s Law?
Pressure in a fluid increases with depth as described by the formula P = Pa + ρgh, where:
- Pa is atmospheric pressure,
- ρ is the fluid density,
- g is acceleration due to gravity,
- h is the depth below the surface.
7. Derive Pascal’s Law for an element of fluid at rest within a container.
Consider a small prismatic element of fluid at depth. Fluid equilibrium requires equal pressure on all faces at the same depth. By balancing forces on each face and applying geometry, it is found that pressure is equal in every direction at a point in a resting fluid. This leads to the mathematical statement of Pascal’s Law.
8. Why are hydraulic brakes more efficient and safer than mechanical brakes in automobiles?
Hydraulic brakes transmit pressure equally to all wheels using fluid, ensuring uniform force and better control. This equal distribution reduces the risk of wheel locking and enhances overall braking efficiency and safety compared to mechanical systems that may not distribute force evenly.
9. What might happen if air bubbles are present in a hydraulic system based on Pascal's Law?
If air bubbles are present, the compressibility of air causes ineffective pressure transmission. This can lead to delayed or reduced movement in the hydraulic system, as part of the applied force compresses the air rather than being transmitted, violating the ideal conditions required by Pascal’s Law.
10. How does understanding Pascal’s Law benefit students preparing for CBSE board exams and competitive exams like JEE or NEET?
Mastering Pascal’s Law addresses both theoretical and numerical questions, which are frequent in CBSE exams and form the foundation for problem-solving in competitive tests. Grasping its applications, derivation, and limitations is essential for stronger conceptual understanding and exam success as outlined in the current syllabus.
