PASCAL'S PRINCIPLE: HISTORY, APPLICATIONS, EXAMPLES - PHYSICAL - 2025

The principle of Pascal, Pascal or law states that a change in pressure of a fluid confined in any point is transmitted unchanged to all other points within the fluid.

This principle was discovered by the French scientist Blaise Pascal (1623 - 1662). Due to the importance of the contributions made by Pascal to science, the pressure unit in the International System has been named in his honor.

Figure 1. A backhoe makes use of Pascal's principle to lift heavy weights. Source: Source: publicdomainpictures.net

Since pressure is defined as the ratio of the force perpendicular to a surface and its area, 1 Pascal (Pa) is equal to 1 newton / m ².

History

To test his principle, Pascal devised a fairly powerful proof. He took a hollow sphere and drilled in several places, put plugs in all the holes except one, through which he filled it with water. In this he placed a syringe fitted with a plunger.

By sufficiently increasing the pressure in the plunger, the plugs are released at the same time, because the pressure is transmitted equally to all points of the fluid and in all directions, thus demonstrating Pascal's law.

Figure 2. Pascal's syringe. source: Wikimedia Commons.

Blaise Pascal had a short life, marked by illness. The incredible scope of his mind led him to inquire into various aspects of nature and philosophy. His contributions were not limited to studying the behavior of fluids, Pascal was also a pioneer in computing.

And it is that at age 19, Pascal created a mechanical calculator for his father to use in his work in the French tax system: the pascalin.

Also, together with his friend and colleague the great mathematician Pierre de Fermat, they gave shape to the theory of probabilities, indispensable in Physics and Statistics. Pascal passed away in Paris, at the age of 39.

Explanation of Pascal's principle

The following experiment is quite simple: a U-tube is filled with water and plugs are placed on each end that can slide smoothly and easily, like pistons. Pressure is made against the left piston, sinking it a little and it is observed that the one on the right rises, pushed by the fluid (figure 3).

Figure 3. Application of Pascal's principle. Source: self made.

This happens because the pressure is transmitted without any decrease to all points of the fluid, including those that are in contact with the piston on the right.

Liquids such as water or oil are incompressible, but at the same time the molecules have sufficient freedom of movement, which makes it possible for the pressure to be distributed over the right piston.

Thanks to this, the right piston receives a force that is exactly the same in magnitude and direction as that applied to the left, but in the opposite direction.

The pressure in a static fluid is independent of the shape of the container. It will be shown shortly that pressure varies linearly with depth, and Pascal's principle follows from this.

A change in pressure at any point causes the pressure at another point to change by the same amount. Otherwise there would be an extra pressure that would make the liquid flow.

The relationship between pressure and depth

A fluid at rest exerts a force on the walls of the container that contains it and also on the surface of any object immersed in it. In Pascal's syringe experiment it is seen that the streams of water come out perpendicular to the sphere.

The fluids distribute the force perpendicularly on the surface on which it acts, so it is convenient to introduce the concept of mean pressure P _m as the perpendicular force exerted F _⊥ by area A, whose SI unit is the pascal:

Pressure increases with depth. It can be seen by isolating a small portion of fluid in static equilibrium and applying Newton's second law:

Figure 4. Free-body diagram of a small portion of fluid in static equilibrium in the shape of a cube. Source: E-xuao

The horizontal forces cancel out in pairs, but in the vertical direction the forces are grouped like this:

Expressing mass in terms of density ρ = mass / volume:

The volume of the fluid portion is the product A xh:

Applications

Pascal's principle has been used to build numerous devices that multiply force and facilitate tasks such as lifting weights, stamping on metal, or pressing objects. Among them are:

-Hydraulic press

-The brake system of automobiles

-Mechanical shovels and mechanical arms

-Hydraulic jack

-Cranes and elevators

Next, let's see how Pascal's Principle turns small forces into large forces to do all these jobs. The hydraulic press is the most characteristic example and will be analyzed below.

The hydraulic press

To build a hydraulic press, the same device as in figure 3 is taken, that is, a U-shaped container, of which we already know that the same force is transmitted from one piston to the other. The difference will be the size of the pistons and this is what makes the device work.

The following figure shows Pascal's principle in action. The pressure is the same at all points in the fluid, both in the small and large piston:

Figure 5. Diagram of the hydraulic press. Source: Wikimedia Commons.

p = F ₁ / S ₁ = F ₂ / S ₂

The magnitude of the force that is transmitted to the large piston is:

F ₂ = (S ₂ / S ₁). F ₁

Since S ₂ > S ₁, it results in F ₂ > F ₁, therefore the output force has been multiplied by the factor given by the quotient between the areas.

Examples

This section presents application examples.

Hydraulic brakes

Car brakes make use of Pascal's principle through a hydraulic fluid that fills tubes connected to the wheels. When he needs to stop, the driver applies force by depressing the brake pedal and creating fluid pressure.

At the other extreme, pressure pushes the brake pads against the drum or brake discs that rotate in conjunction with the wheels (not the tires). The resulting friction causes the disc to slow down, also slowing down the wheels.

Figure 6. Hydraulic brake system. Source: F. Zapata

Mechanical Advantage of Hydraulic Press

In the hydraulic press of figure 5, the input work must equal the output work as long as friction is not taken into account.

The input force F ₁ causes the piston to travel a distance d ₁ while descending, while the output force F ₂ allows a travel d ₂ of the rising piston. If the mechanical work done by both forces is the same:

The mechanical advantage M is the quotient between the magnitudes of the input force and the output force:

And as demonstrated in the previous section, it can also be expressed as the quotient between the areas:

It seems that work can be done for free, but in truth energy is not being created with this device, since the mechanical advantage is obtained at the expense of the displacement of the small piston d ₁.

Thus, to optimize performance, a valve system is added to the device in such a way that the outlet piston rises thanks to short pulses on the inlet piston.

In this way, the operator of a hydraulic garage jack pumps several times to gradually lift a vehicle.

Exercise resolved

In the hydraulic press of Figure 5, the piston areas are 0.5 square inches (small piston) and 25 square inches (large piston). Find:

a) The mechanical advantage of this press.

b) The force necessary to lift a 1-ton load.

c) The distance that the input force must act to lift said load by 1 inch.

Express all results in units of the British system and the SI International System.

Solution

a) The mechanical advantage is:

M = F ₂ / F ₁ = S ₂ / S ₁ = 25 in ² / 0.5 in ² = 50

b) 1 ton equals 2000 lb-force. The necessary force is F ₁:

F ₁ = F ₂ / M = 2000 lb-force / 50 = 40 lb-force

To express the result in the International System, the following conversion factor is required:

1 lb-force = 4.448 N

Therefore the magnitude of F1 is 177.92 N.

c) M = d ₁ / d _{2 →} d ₁ = Md ₂ = 50 x 1 in = 50 in

The required conversion factor is: 1 in = 2.54 cm

References

1. Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill. 417-450.
2. College Physics. Pascal's begin. Recovered from: opentextbc.ca.
3. Figueroa, D. (2005). Series: Physics for Science and Engineering. Volume 4. Fluids and Thermodynamics. Edited by Douglas Figueroa (USB). 4 - 12.
4. Rex, A. 2011. Fundamentals of Physics. Pearson. 246-255.
5. Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. McGraw Hill. 301-320.