Pascals Law and its Application - Definition, Formula, Applications, FAQs

Pascal's Law is one of the basic principles of fluid mechanics. This principle was coined by Blaise Pascal and is the basis of many hydraulic systems and devices. The principle is such that it can be used in diesel engines, hydraulic lifts, brakes, presses, etc.; that is, even in your everyday life, it will find application in engineering efficiencies such as force transmission and multiplication. This is where the strength of fluid mechanics may be showcased in terms of real-world problems, solved with a lot of precision and indeed much efficiency.

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Pascal Law Definition

Pascal's Law states that "The pressure externally applied on a confined liquid is transmitted or distributed uniformly through the entirety of liquid in all directions".

Static pressure acts perpendicular to any surface in contact with the fluid. Pascal also found that at a point, the pressure would be the same across any arbitrary planes passing through that point within that fluid for the case of a static fluid. Pascal's law is also referred to as Pascal's principle or principle of transmission of fluid pressure. This law was formulated in 1653 by French mathematician Blaise Pascal.

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Pascal Law Formula

Pascal's law is expressed as follows:

$$
F=P A
$$
Where,
F is the applied force, while P is the transmitted pressure.
A represents the cross-sectional area.

Derivation of Pascal Law

Consider a right-angled triangle in a liquid of density. Due to the small size of the element, each point is presumed to be at the same depth as the liquid surface. Gravity has the same effect at all of these locations.

Let $a b$, bd, and cd represent the cross-sectional areas of the ABFE, ABDC, and CDFE faces, respectively.

Let P₁, P₂, and P₃ represent the pressures communicated on the faces $A B F E, A B D C$, and CDFE, respectively.

The pressure exerts a force parallel to the surface. Allow P₁ to apply force F₁ to the surface ABFE, P₂ to apply force F2 to the surface ABDC, and P₃ to apply force F₃ to the surface CDFE.

Here,
$F_1=P_1 \times$ area of $A B F E=P_1 \mathrm{ab}$
$F_2=P_2 \times$ area of $A B D C=P_2$ bd
And, $F_3=P_3 \times$ area of $C D F E=P_3 c d$
We also know that,

$$
\begin{aligned}
& \sin \theta=\frac{b}{a} \\
& \cos \theta=\frac{c}{a}
\end{aligned}
$$
The prism's net force will be zero because the prism is in equilibrium.

$$
\begin{aligned}
& F_1 \sin \theta=F_2 \\
& F_1 \cos \theta=F_3
\end{aligned}
$$

Pressure is expressed as,

$$
\begin{aligned}
& P_1 a d b / a=P_2 b d(\text { equation-i) } \\
& P_1 a d c / a=P_3 c d \text { (equation-ii) }
\end{aligned}
$$
From (i) and (ii),

$$
\begin{gathered}
\mathbf{P}_1=\mathrm{P}_2 \text { and } \mathrm{P}_1=\mathbf{P}_3 \\
\therefore \mathrm{P}_1=\mathrm{P}_2=\mathrm{P}_3
\end{gathered}
$$

Applications of Pascal's law

1. Fluid Lift

It has many applications in everyday life. Many devices, such as hydraulic lifting and pressure brakes, are based on Pascal's law. The liquid is used to transfer pressure to all these devices. In a hydraulic lift, the two pistons are separated by a space filled with liquid. The cross-section piston at the small cross A is used to apply force F directly to that liquid. The pressure P = F / A is transferred across the liquid to a larger cylinder fitted with a large area piston B, resulting in a higher force of (P × B. ). Platform B can be moved up or down. Therefore, the force used is increased by the B / A factor

2. Brake Fluid

In automobiles, pressure brakes also serve the same purpose. When using less force on the foot, the master piston moves inside the master cylinder, and the resulting pressure is transferred by the oil brakes to form a larger area piston. A high force was then applied to the piston and it was pulled down, stretching the braces on the brake line. As a result, the small force at the base produces excessive force returning to the wheel. The main advantage of the system is that the pressure, which is stopped by pressing the pedal, is transmitted evenly across all cylinders, attached to four wheels to make the braking effort equal to all the wheels.