# Pascal’s Principle: history, applications, examples

The **principle of Pascal** , Pascal or law states that a change in pressure of a fluid confined at 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 Pascal’s contributions to science, the pressure unit in the International System was named after him.

Figure 1. A backhoe uses the Pascal principle to lift large weights. Source: Source: publicdomainpictures.net

As pressure is defined as the ratio of the force perpendicular to a surface over its area, 1 Pascal (Pa) is equal to 1 newton / m ^{2} .

**Story**

To prove his principle, Pascal planned a very powerful demonstration. He took a hollow sphere and pierced it in several places, plugging it into every hole except one, through which he filled it with water. Into this he placed a syringe provided with a plunger.

By sufficiently increasing the pressure on the piston, the plugs are actuated at the same time, because the pressure is transmitted equally to all points of the fluid and in all directions, demonstrating Pascal’s law.

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

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

And at the age of 19, Pascal created a mechanical calculator for his father to use in his work on the French tax system: the *pascalina* .

Furthermore, together with his friend and colleague, the great mathematician Pierre de Fermat, they shaped the theory of probability, indispensable in physics and statistics. Pascal died in Paris at 39 years old.

__Explanation of Pascal’s Principle__

__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 a piston. Pressure is pressed against the left piston, sinking it a little and it is observed that the right piston rises, pushed by the fluid (figure 3).

This is because pressure is transmitted without any decrease at any point in the fluid, including those in contact with the right piston.

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

Thanks to this, the right piston receives a force exactly equal in magnitude and direction to which it was applied on the left, but in the opposite direction.

The pressure in a static fluid is independent of the shape of the container. It will be immediately demonstrated that pressure varies linearly with depth and Pascal’s principle is a consequence of this.

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

**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 submerged in it. In the Pascal syringe experiment, it is observed that the water streams come out *perpendicular* to the sphere.

Force fluids distributed perpendicularly on the surface on which it acts, so it is desirable to introduce the concept of mean pressure *P _{m}* and the force exerted perpendicularly

*F*

*per area*

_{⊥}*A*, the unit of SI is pascal:

*P _{m} = F *

_{⊥}/ APressure increases with depth. It can be seen isolating a small portion of fluid in static equilibrium and applying Newton’s second law:

Horizontal forces are canceled in pairs, but in the vertical direction, forces are grouped as follows:

*∑F _{y} = F _{2} – F _{1} – mg = 0 → *

*F*

_{2}– F_{1}= mgExpressing mass in terms of density ρ = mass / volume:

*P _{2} .A- P _{1} .A = ρ *

*x volume xg*

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

*A. (P _{2} – P _{1} ) = ρ *

*x A xhxg*

*ΔP = ρ **.gh **Fundamental theorem of hydrostatics*

__applications__

__applications__

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

-The hydraulic press

-The car’s brake system

-Mechanical blades and mechanical arms

-The hydraulic jack

-Trucks and elevators

Next, we’ll look at how Pascal’s principle makes small forces become big forces to do all of these jobs. The hydraulic press is the most characteristic example and will be discussed below.

**Hydraulic press**

To build a hydraulic press, the same device as in figure 3 is used, that is, a U-shaped container, from which we already know that the same force is transmitted from one piston to another. The difference will be the size of the pistons and that’s 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 small and large pistons:

Figure 5. Hydraulic press diagram. Source: Wikimedia Commons.

p = F _{1} / S _{1} = F _{2} / S _{2}

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

F _{2} = (S _{2} / S _{1} ). F _{1}

As S _{2} > S _{1} , results in F _{2} > F _{1} , so the output force was multiplied by the factor given by the ratio between the areas.

**Examples**

This section presents application examples.

**hydraulic brakes**

Car brakes make use of the Pascal principle through a hydraulic fluid that fills the tubes connected to the wheels. When it needs to stop, the driver applies force by pressing the brake pedal and generating pressure in the fluid.

At the other end, pressure pushes the brake pads against the drum or brake discs that rotate together with the wheels (not the tires). The resulting friction causes the disc to stop, slowing down the wheels.

**Mechanical advantage of hydraulic press**

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

Input force **F **_{1} causes the piston to travel a distance d _{1} when lowering, while output force **F **_{2} allows a displacement d _{2} of the rising piston. If the mechanical work performed by both forces is the same:

*F _{1} .d _{1} = F _{2} . d _{2}*

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

*M = F _{2} / F _{1} = d _{1} / d _{2}*

And, as shown in the previous section, it can also be expressed as the ratio between the areas:

*F _{2} / F _{1} = S _{2} / S _{1}*

It appears that it is possible to do free work, but in fact no energy is being created with this device as the mechanical advantage is gained at the expense of the displacement of the small piston d _{1} .

Therefore, to optimize performance, a valve system is added to the device so that the output piston rises thanks to short pulses on the input piston.

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

__Exercise solved__

__Exercise solved__

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

a) The mechanical advantage of this printer.

b) The force required to lift a load of 1 ton.

c) The distance at which the input force must act to raise said load by 1 inch.

Express all results in British System and International SI System units.

**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 is equal to 2000 lb-force. The force required 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 necessary:

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 inThe required conversion factor is: 1 in = 2.54 cm

*d _{1} = 127 cm = 1.27 m*