KAHIBARO
Table of Contents
Pressure Transmission in a Confined Fluid
Pascal's principle describes what happens when the pressure of a confined fluid is changed. If an external pressure is applied to a fluid at rest in a closed container, that pressure change is transmitted undiminished to every part of the fluid and to the walls of the container.
This idea is very important because it explains how fluids can transmit force from one place to another. A small force applied over a small area can create the same pressure increase everywhere in the fluid. If that same pressure acts on a larger area elsewhere, the resulting force can be much larger.
Pascal's principle states that a pressure change applied to an enclosed fluid is transmitted equally and undiminished throughout the fluid.
Pressure and Force
Pressure is defined as force per unit area,
$$
P = \frac{F}{A}
$$
If a force $F_1$ is applied to a piston of area $A_1$, the pressure added to the fluid is
$$
P = \frac{F_1}{A_1}
$$
Because of Pascal's principle, this same pressure acts at another piston of area $A_2$. The force at the second piston is then
$$
F_2 = P A_2
$$
Substituting for $P$ gives
$$
F_2 = \frac{F_1}{A_1} A_2
$$
or
$$
\frac{F_1}{A_1} = \frac{F_2}{A_2}
$$
This is the basic equation used in hydraulic systems.
For an ideal hydraulic system,
$$
\frac{F_1}{A_1} = \frac{F_2}{A_2}
$$
A larger output area produces a larger output force.
Hydraulic Multiplication of Force
Pascal's principle does not create energy from nothing. It multiplies force by using a larger area, but this comes with a tradeoff. The piston with the smaller area must move a greater distance than the piston with the larger area.
If the small piston moves down by a distance $d_1$, and the large piston rises by a distance $d_2$, then the volumes displaced must match:
$$
A_1 d_1 = A_2 d_2
$$
So if $A_2$ is much larger than $A_1$, then $d_2$ is much smaller than $d_1$.
This means a hydraulic machine can increase force, but it decreases the distance moved at the output.
Hydraulic systems multiply force, but they do not multiply work for free.
In an ideal system,
$$
A_1 d_1 = A_2 d_2
$$
and
$$
F_1 d_1 = F_2 d_2
$$
Hydraulic Press
A hydraulic press is the classic application of Pascal's principle. It uses two pistons connected by a confined fluid.
If the input piston has a small area and the output piston has a large area, then a modest input force can produce a large output force.
Suppose
$$
A_1 = 2 \,\text{cm}^2, \qquad A_2 = 50 \,\text{cm}^2, \qquad F_1 = 40 \,\text{N}
$$
Then
$$
F_2 = F_1 \frac{A_2}{A_1} = 40 \times \frac{50}{2} = 1000 \,\text{N}
$$
So a 40 newton input force produces a 1000 newton output force.
Common Devices Based on Pascal's Principle
Many useful machines rely on this principle. In each case, pressure applied in one place is transmitted through the fluid to produce force somewhere else.
| Device | Use of Pascal's principle |
|---|---|
| Hydraulic press | Multiplies force to compress or lift heavy loads |
| Hydraulic jack | Lifts cars or other heavy objects |
| Car brake system | Transmits force from brake pedal to brake pads |
| Hydraulic lift | Raises platforms or elevators |
In hydraulic brakes, pressing the brake pedal increases pressure in brake fluid. That pressure is transmitted through the fluid to the brake cylinders at the wheels, where forces are applied to slow the car.
Conditions for Pascal's Principle
Pascal's principle applies when the fluid is enclosed and at rest, or at least when the pressure is transmitted through the confined fluid without significant loss. Liquids are especially useful because they are nearly incompressible.
In real systems, some effects reduce ideal performance. These include friction, leakage, trapped air bubbles, and deformation of parts. Air bubbles are especially undesirable in hydraulic systems because gases compress much more easily than liquids.
Pascal's principle works best in enclosed fluids, especially nearly incompressible liquids.
Trapped gas reduces efficient pressure transmission.
Visual Model of a Hydraulic System
The left piston applies pressure to the fluid. That pressure is transmitted through the fluid to the right piston. If $A_2 > A_1$, then the output force is greater than the input force.
Key Idea
Pascal's principle is the rule that makes hydraulic machines possible. The important point is not just that force is transmitted, but that pressure is transmitted equally throughout the enclosed fluid. Because force equals pressure times area, changing the area changes the force.
The central idea is:
same pressure, different area, different force.
$$
F = PA
$$
Understanding this principle provides the foundation for hydraulic tools, lifts, and braking systems.