The Pascal's principle, or Pascal's law, establishes that a change in the pressure of a fluid confined in any of its points is transmitted without alteration 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..
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 / mtwo.
Article index
To test his principle, Pascal devised a fairly forceful 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.
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 of computing.
And it is that at the age of 19, Pascal created a mechanical calculator for his father to use in his work in the French tax system: the pascaline.
Also, together with his friend and colleague the great mathematician Pierre de Fermat, they shaped the theory of probabilities, indispensable in Physics and Statistics. Pascal passed away in Paris, at the age of 39.
The following experiment is quite simple: a U-tube is filled with water and plugs are placed at 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 below).
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 equal in magnitude and direction to 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 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 an extra pressure that would make the liquid flow.
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 jets of water come out perpendicularly to the sphere.
The fluids distribute the force perpendicularly on the surface on which it acts, that is why it is convenient to introduce the concept of average pressure Pm as the perpendicular force exerted F⊥ By area TO, whose SI unit is the pascal:
Pm = F⊥ / TO
Pressure increases with depth. It can be seen by isolating a small portion of fluid in static equilibrium and applying Newton's second law:
The horizontal forces cancel out in pairs, but in the vertical direction the forces are grouped like this:
∑FY = Ftwo - F1 - mg = 0 → Ftwo - F1 = mg
Expressing mass in terms of density ρ = mass / volume:
Ptwo.A- P1.A = ρ x volume x g
The volume of the fluid portion is the product A x h:
A. (Ptwo - P1) = ρ x A x h x g
ΔP = ρ.g.h Fundamental theorem of hydrostatics
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:
-The hydraulic press
-The car braking system
-Mechanical shovels and mechanical arms
-The hydraulic jack
-Cranes and lifts
Next, let's see how Pascal's principle turns small forces into large forces to do all of these jobs. The hydraulic press is the most characteristic example and will be analyzed below.
To build a hydraulic press, the same device as in the figure above 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:
p = F1 / S1 = Ftwo / Stwo
The magnitude of the force transmitted to the large piston is:
Ftwo = (Stwo / S1). F1
Like Stwo > S1, results in Ftwo > F1, therefore the output force has been multiplied by the factor given by the quotient between the areas.
This section presents application examples.
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 a 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 stop, also slowing down the wheels.
In the hydraulic press in the figure below, the input work must be equal to the output work as long as friction is not taken into account..
Input force F1 makes the piston travel a distance d1 going down, while the output force Ftwo allows a tour dtwo of the rising piston. If the mechanical work done by both forces is the same:
F1.d1 = Ftwo. dtwo
The mechanical advantage M is the quotient between the magnitudes of the input force and the output force:
M = Ftwo/F1 = d1/ dtwo
And as demonstrated in the previous section, it can also be expressed as the quotient between the areas:
Ftwo/F1 = Stwo / S1
It seems that work can be done for free but in reality energy is not being created with this device, since the mechanical advantage is obtained at the expense of the displacement of the small piston d1.
So, to optimize performance, a valve system is added to the device in such a way that the outlet piston rises thanks to brief impulses on the inlet piston..
In this way, the operator of a hydraulic garage jack pumps several times to gradually lift a vehicle..
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 required 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.
a) The mechanical advantage is:
M = Ftwo/F1 = Stwo/ S1 = 25 intwo / 0.5 intwo = 50
b) 1 ton equals 2000 lb-force. The force required is F1:
F1 = Ftwo / 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 = d1/ d2 → d1 = M.dtwo = 50 x 1 in = 50 in
The necessary conversion factor is: 1 in = 2.54 cm
d1 = 127 cm = 1.27 m
Yet No Comments