Anaya Nolan
Blaise Pascal's 1653 publication, 'Treatise on the Equilibrium of Liquids', laid the foundation for the development of hydraulics, a significant advancement in modern mechanical technology. Pascal's Law, or Pascal's Principle, states that when pressure is applied to a confined static fluid, it is transmitted undiminished in all directions throughout the fluid and to the container's walls. This principle is applied in hydraulic systems, which are used in automotive brakes, hydraulic jacks, and other mechanical systems. A hydraulic lift, for instance, consists of two pistons of different cross-sectional areas connected by a cylinder filled with an incompressible fluid. When force is applied to the smaller piston, the pressure is transmitted equally to the larger piston, resulting in a greater force that can be used to lift heavy objects.
| Characteristics | Values |
|---|---|
| Pascal's Law | "The external static pressure applied on a confined liquid is distributed or transmitted evenly throughout the liquid in all directions" |
| The static pressure acts at right angles to any surface in contact with the fluid | |
| The pressure at a point for a static fluid would be the same across all planes passing through that point in that fluid | |
| Pascal's law is also known as Pascal's principle or principle of transmission of fluid-pressure | |
| A change in pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid and to the walls of its container | |
| The total pressure in a fluid is the sum of the pressures from different sources | |
| Pascal's law applies to gases | |
| Hydraulic Lift | A narrow cylinder is connected to a wider cylinder |
| They are fitted with airtight pistons on either end | |
| The inside of the cylinders is filled with fluid that cannot be compressed | |
| Pressure applied at piston A is transmitted equally to piston B without diminishing the use of the fluid that cannot be compressed |
What You'll Learn
Pascal's Law and hydraulic lifts
Pascals Law and Hydraulic Lifts
Pascal's Law, also known as Pascal's Principle, was discovered by French mathematician Blaise Pascal in 1653. The law states that:
> The external static pressure applied on a confined liquid is distributed or transmitted evenly throughout the liquid in all directions.
In other words, when pressure is applied to an enclosed fluid, it is transmitted undiminished to all portions of the fluid and the walls of its container. This means that the pressure at any point in the fluid is equal in all directions.
Hydraulic Lifts
Hydraulic lifts are a practical application of Pascal's Law. A hydraulic lift consists of two pistons separated by a space filled with liquid. The pistons are fitted at either end of two cylinders, one narrow and the other wider, and the cylinders are filled with an incompressible fluid.
When pressure is applied to the smaller piston, it is transmitted equally to the larger piston, which then serves as a platform to lift heavy objects. The force applied to the smaller piston is amplified by the ratio of the larger piston's cross-sectional area to the smaller piston's cross-sectional area.
For example, if the larger piston has an area five times greater than the smaller piston, a 100-N force applied to the smaller piston will result in a 500-N force on the larger piston. This mechanical advantage is the basis for how hydraulic lifts are able to lift very heavy weights.
Limitations and Approximations
It is important to note that Pascal's Law assumes the fluid is at rest and in static equilibrium. In reality, when using a hydraulic lift, the fluid is flowing and not truly in static equilibrium. Therefore, strictly speaking, Pascal's Law is not applicable in this dynamic situation.
However, if the lift is rising slowly, the deviations from Pascal's Law are small and can be considered a reasonable approximation. This approximation is useful for simplifying calculations and gaining a basic understanding of the underlying principles.
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Pascal's Law formula
Pascal's law, also known as Pascal's principle, states that:
> The external static pressure applied on a confined liquid is distributed or transmitted evenly throughout the liquid in all directions.
Pascal's law was stated by French mathematician Blaise Pascal in 1653. The law applies to both liquids and gases.
The formula for Pascal's law is:
> F = PA
Where:
- F is the force applied
- P is the pressure transmitted
- A is the cross-sectional area
This means that the force applied is equal to the pressure multiplied by the cross-sectional area.
Pascal's law can be applied to hydraulic lifts. A hydraulic lift consists of two pistons separated by a space filled with liquid. The pressure exerted in the column is transmitted throughout the liquid, resulting in pressure being applied to the other piston. The force felt by the second piston is given by:
> F2 = F1 * (A2 / A1)
Where:
- F1 is the force applied to the first piston
- F2 is the force felt by the second piston
- A1 is the cross-sectional area of the first piston
- A2 is the cross-sectional area of the second piston
This equation shows that the applied force is increased by the factor of A2/A1. This property is useful in hydraulic systems for lifting heavy weights.
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Pascal's Law derivation
The French mathematician Blaise Pascal stated Pascal's Law in 1653 (or 1647, according to another source). Pascal's Law states that:
> The external static pressure applied on a confined liquid is distributed or transmitted evenly throughout the liquid in all directions.
The static pressure acts at right angles to any surface in contact with the fluid. This means that, in a fluid at rest, the pressure is the same at all points, provided they are at the same height.
The formula for Pascal's Law is:
> P = F/A
Where:
- P = Pressure
- F = Force
- A = Area of contact
Pascal's Law can be derived as follows:
Consider a right-angled triangle prism submerged in a liquid of density rho. The size of the submerged element is negligible compared to the volume of the liquid, and all points on the element experience the same gravitational force. Now, the area of the faces PQRS, PSUT, and QRUT of the prism is ps, qs, and rs respectively. Assume the pressure applied by the liquid on these faces is P1, P2, and P3 respectively.
The exerted force by this pressure to the faces in the perpendicular inward direction is F1, F2, and F3. Thus, F1 = P1 × Area of PQRS = P1 × ps; F2 = P2 × Area of PSUT = P2 × qs; and F3 = P3 × Area of QRUT = P3 × rs.
Now, in triangle PQT, sin θ = p/r and cos θ = q/r. The net force on the prism will be zero since the prism is in equilibrium. F3 sin θ = F1 and F3 cos θ = F2 (putting values of F1, F2, and F3 from the above values)
⇒ P3 × rs × p/r = P1 × ps and P3 × rs × q/r = P2 × qs
⇒ P3 = P1 and P3 = P2
Therefore, the pressure throughout the liquid remains the same.
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Applications of Pascal's Law
Pascal's Law, or Pascal's Principle, states that "the external static pressure applied on a confined liquid is distributed or transmitted evenly throughout the liquid in all directions". This principle was derived by French mathematician Blaise Pascal in 1653.
Pascal's Law is applied in the working of a hydraulic lift. A hydraulic lift is constructed with a narrow cylinder (cylinder A) connected to a wider cylinder (cylinder B). Both cylinders are fitted with airtight pistons at either end, and the cylinders are filled with an incompressible fluid. When pressure is applied to piston A, it is transmitted equally to piston B, allowing piston B to serve as a platform to lift heavy objects.
Other applications of Pascal's Law include the hydraulic jack and hydraulic press, as well as forced amplification in the braking system of most cars.
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Pascal's Law FAQs
Pascals Law FAQs
Pascal's Law, also known as Pascal's Principle, states that when a change in pressure is applied to an enclosed fluid, it is transmitted undiminished in all directions and to all portions of the fluid and the walls of its container.
Pascal's Law is applied in hydraulic systems, which are used in automotive brakes, hydraulic jacks, and other mechanical systems. Hydraulic lifts, for example, use this principle to lift heavy objects.
Yes, Pascal's Law is applicable to gases. It states that in a closed container, pressure applied to one part of the container is transmitted equally to all parts.
Who stated Pascal's Law?
Pascal's Law was stated by French mathematician and philosopher Blaise Pascal in 1653, in his work "Treatise on the Equilibrium of Liquids".
Pascal's Law says that pressure applied to an enclosed fluid will be transmitted without a change in magnitude to all points of the fluid and the container walls. The pressure at any point in the fluid is equal in all directions.
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Frequently asked questions
Pascal's Law states that the external static pressure applied to a confined liquid is distributed or transmitted evenly throughout the liquid in all directions.
Pascal's Law is the principle behind how hydraulic lifts work. A narrow cylinder (A) is connected to a wider cylinder (B), both fitted with airtight pistons and filled with an incompressible fluid. When pressure is applied to piston A, it is transmitted equally to piston B, allowing it to lift heavy objects.
The formula for Pascal's Law is F = PA, where F is the force applied, P is the pressure transmitted, and A is the cross-sectional area.
Pascal's Law has various applications, including hydraulic jacks, hydraulic presses, and hydraulic braking systems in cars.
