Blaise Pascal

Short Description of Topic

Mathematical and Scientific Discoveries

Blaise Pascal (June 19, 1623 - August 19, 1662) was a French inventor, physicist, and mathematician who went on to make several discoveries in both mathematics and physics. Pascal's earliest work was in natural and applied sciences, where he made important contributions to the study of fluids, as well as better explaining the ideas of pressure and vacuum. In his teenage years, Pascal started working on calculating machines. Over the course of several years, he developed fifty prototypes, and completely built twenty finished machines. These machines were called Pascal's calculators and established him as one of the first inventors of the mechanical calculator. These machines were capable of addition and subtraction, and even though they failed in becoming a commercial success, they were catalysts in the next 400 years of development of mechanical methods of calculation. His most well known mathematical contribution is what people refer to as Pascal's Triangle. This "triangle" refers to Pascal's 1653 "Treatise on the Arithmetical Triangle" which describes a convenient tabular presentation for binomial coefficients.

This shows that if you add the two numbers directly above each spot of the next row, the values you get are the order of every coefficient for each increasing degree equation.

Pascal also contributed to the field of study of hydrodynamics and hydrostatics, centered on the principles of hydraulic fluids. His main inventions in these areas were the hydraulic press, which involved using hydraulic pressure to multiply force, as well as the syringe. Pascal proved that hydrostatic pressure does not depend on the weight of the fluid but rather on the elevation difference. Pascal also experimented some with barometers, and in 1646 he replicated Evangelista Torricelli's experimentation with barometers. Torricelli's experiment consisted of placing a mercury-filled tube upside down in a bowl of mercury. This is where he made a discovery about vacuums because he questioned what force kept some mercury in the tube and what filled the space above the mercury when the tube of mercury was upside down. This went against Aristotle's declaration of "Everything that is in motion must be moved by something", which caused Pascal to conduct further experimentation. In 1647 Pascal introduced "New Experiments with the Vacuum" which explained basic rules as to what degree liquids could be supported by air pressure, as well as providing reasons why it was a vacuum above the column of liquid in a barometer tube.

Since Pascal made so many contributions to the scientific knowledge of pressure, his last name was used for the SI unit of pressure. The pascal (symbol: Pa) is the derived unit for multiple measurements, including: pressure, internal pressure, stress, Young's Modulus, and ultimate tensile strength.

The Pascal is defined as one newton per square meter. Common units of the pascal are the hectopascal(hPa) = 100 Pa, which is also equal to 1mbar, as well as the kilopascal (1kPa = 1000Pa).

Another unit of measurement called standard atmosphere (atm) is equivalent to 101.325 kPa and is an approximation to the average pressure at sea level at 45° N.

A Mathematical Model

[math]\displaystyle{ {\rm 1~Pa = 1~\frac{N}{m^2} = 1~\frac{kg}{m \cdot s^2}} }[/math]

where N is the newton,m is the meter, kg is the kilogram, and s is the second.

[math]\displaystyle{ {\rm P = \frac{F}{A}} }[/math]

where P is Pressure in pascals(Pa), F is the force in newtons (N), and A is area in meters squared([math]\displaystyle{ {\rm {m^2}} }[/math]).

A Computational Model

This a model applicable in Vpython for calculating pressure a box with sides (l & w) puts on the ground it sits on from gravity plus an external force(Fy):

Area = l*w

Fgrav = 9.8

Fnet = Fgrav + Fy

Pressure = Fnet / Area

Examples

Here are a couple examples of calculating pressure in pascals.

Simple

A golf club with a flat face supplies a force of 700 N. The club face has an area of 7.1 x 10-4 m^2. What is the pressure?

                                               P  =  F/A
                                               P  =  (700 N/7.1 x 10-4 m^2)
                                               P  =  9.86 x 105  N/m^2
                                               P  =  9.86 x 105  Pa

Middling

A force of 100N is applied to one end of a tube with a radius of 8cm. What is the force resulting at the other end of the tube with a diameter of 6cm?

P1 = P2

(F1)/A1 = (F2)/A2

F2 = (F1*A2)/A1

A1 = pi*(0.08m)^2 = 0.02 m^2

A2 = pi*(0.06m/2)^2 = 2.83e-3 m^2

F2 = (100N*2.83e-3 m^2)/(0.02 m^2)

F2 = 14.15 N

Difficult

Calculate the pressure at a depth "h" in a cylinder with an area A and a height h filled with a fluid.

P = pressure, Po = pressure due to the air (atmospheric pressure)

The fluid is at rest, So we can write:

                                           Fnet  =  0
                                  PA – mg – PoA  =  0

However we know that: Mass = density x volume = density x (area x height) , r = density

    M  =  rV  =  rAh

So we can substitute:

                       PA – (rAh)g - PoA  =  0

cancel out the Area, A:

                               P – (rh)g - Po  =  0

and the pressure at any depth will be:

                               P  =  Po + rgh

In words this says: the pressure at a depth ‘h’ will be the atmospheric pressure(14.7 lbs/in2) + (rgh)

Connectedness

Pressure and the Pascal relate to some of the classes I will be taking down the road, like fluids. I find fluids and their characteristics to be intriguing, and since I have a co-op with an HVAC company, it helps to have more knowledge on how fluids and pressure affect the HVAC systems.

This is very connected to my major because my major is mechanical engineering, which has several classes pertaining to the understanding of fluids and how they work, as well as pressure and how the two relate.

There are many industrial applications to pressure because there are many reactions that can only take place under high pressure conditions. Some of these are found in the areas of polymerization, separations, oil and gas recovery, and food processing.

History

-In 1631, Pascal's father moved him and his two sisters to Paris, and already at his early age, Blaise showed an amazing aptitude for mathematics and science.

-In 1642, when Pascal was not even 19, he made his first calculator in an effort to help is father with his commissioner of taxes work.

- In 1646, Pascal replicated Evangelista Torricelli's experiment with a tube, bowl, and mercury which led to his discoveries about vacuums.

-Pascal created his Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle") in 1653 while he was living in Paris, France.

- In 1654, Pascal developed a probabilistic argument called Pascal's Wager, to justify God and a virtuous life. This work done by Pascal into the calculus of probabilities helped the formation of calculus.

- Later in 1654, Pascal endured a religious experience which led to him mostly giving up work in mathematics.

- In late 1654, Pascal began writing influential works on philosophy and theology.

- Pascal had poor health from the time he turned 18, and in 1662 Pascal's illness became more violent along with his worsening emotional condition from his sister's death.

- Pascal died on August 19th, 1662 at the young age of 39.