Blaise Pascal ( June 19, 1623 – August 19, 1662), was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a Tax Collector in Rouen. Pascal's earliest work was in the natural and applied sciences where he made important contributions to the study of fluids, and clarified the concepts of pressure and vacuum by generalizing the work of Evangelista Torricelli. Pascal also wrote in defense of the scientific method.

In 1642, while still a teenager, he started some pioneering work on calculating machines, and after three years of effort and 50 prototypes[1] he invented the mechanical calculator.[2][3] He built twenty of these machines (called the Pascaline) in the following ten years.[4] Pascal was a mathematician of the first order. He helped create two major new areas of research. He wrote a significant treatise on the subject of projective geometry at the age of sixteen, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science.

Following a mystical experience in late 1654, he abandoned his scientific work and devoted himself to philosophy and theology. His two most famous works date from this period the letter Lettres provinciales and the Pensees. Pascal was in poor health throughout his life and his death came just two month after his 39th birthday.

WHAT IS PASCAL’S PRINCIPLE

Pascal’s principle states that a pressure applied to a confined fluid is transmitted equally in all directions throughout the fluid.

F1 = F2A1 A2

Where F1= The force supplied (N) F2= The force produced (N) A1= The area over which the original force is applied A2= The area over which the produced force is applied

EXAMPLE OF PASCAL’S PRINCIPLE

You need to lift a truck with a loaded mass of 427 000 kg and are designing a hydraulic press to allow you to do so. Knowing that the maximum force that you can provide is 825 N over a contact area of 15cm, calculate the minimum area that will be needed at the point.

SOLUTION

We know F1= 825N, F2= mg 427 000 x 9.8 = 4184600 N, and A1= 15cm = 0.0015m. Substituting into the equation:

= 4184600/825 x 0.0015 = 7.6

So the plate will be applying the produced force will need to have an area of at least 7.6m.

APPLICATION OF PASCAL’S PRINCIPLE

A hydraulic system is a device in which a small applied force can give rise to a large force. The principle of the hydraulic system is widely used in jacks, vehicles braking systems, hydraulic persses, and heavy machinery.

A hydraulic lift for automobiles is an example of a force multiplied by hydraulic press, based on Pascal’s principle. The fluid in the small cylinder must be moved much further than the distance the car is lifted.

For example, if the lift cylinder were 25cm in diameter and the small cylinder were 1.25 cm in diameter, then the ratio of the areas is 400, so the hydraulic press arrangement gives a multiplication of 400 times the force. To lift a 6000 newton car, you would have to exert only 6000N/400 = 15N on the fluid in the small cylinder to lift the car. However, to lift the car 10cm, you would have to move the oil 400 x 10cm = 40 metres. This is practical by pumping oil into this small cylinder with a small compressor.

Hydraulic jack is the basis of Pascal’s principle, named for Blaise Pascal, who lived in the seventeenth century. Basically, the principle states that the pressure in a closed container is the same at all point. Pressure is described mathematically by a Force divided by Area. Therefore if you have two cylinders connected together, a small one and a large one, and apply a small Force to the small cylinder, this would results in a given pressure. By Pascal’s principle, this pressure would be the same in the larger cylinder, but since the larger cylinder has more area, the force emitted by the second cylinder would be greater. This is represented by rearranging the pressure formula P = F/A, to F = PA. The pressure stayed the same in the second cylinder, but Area was increased, resulting in a larger Force. The greater the differences in the areas of the cylinder, the greater the potential force output of the big cylinder. A hydraulic jack is simply two cylinders connected as described above.

Hydraulic brakes are used in cars, lorry and motorcycles. In hydraulic brake system, a liquid, known as brake fluid is used to transmit pressure from the brake pedal to all the wheels of the vehicles. When the brake pedal is pressed, the piston of the control cylinder applies a pressure on the brake fluid and this pressure is transmitted, via a system of pipes, to each cylinder at the wheel. The cylinders at the wheels actuate a pair of friction pads to press against the surface of the brake discs or brake drums. The frictional force between these brake components cause the vehicles to slow down an stop.

HISTORY OF PASCAL’S PRINCIPLE.

The French philosopher and scientist Blaise Pascal (1623-1662) from whom the unit of pressure is named, is credited with the recognition of a property of fluids that we have found to

be very useful. It was observed that the effect of increasing the pressure at one point was to automatically transfer this same increase in pressure to every other point in the fluid.

Best known for Archimedes Principle, he was a legendary genius who invented pulleys and levers for giant ships, built the first water pump, created a planetarium to show motions of planets, and a machine to fire burning pitch at the enemy ships, among others.An inscription of a sphere and cylinder is found in his tomb, a symbol of his proudest discoveries in geometry.

Archimedes was born in 287BC in Syracuse in Sicily, then a Greek colony. He lived during the reign of intellectual rulers King Heiron II and his son King Gelon. He may have been Gelon's tutor. Archimedes went to Alexandria in Egypt, at that time the greatest centre of learning in the ancient world, and the city itself was founded by Alexander the Great half a century earlier that time. According to legend, Archimedes used a series of machines to keep the Romans finally made it into the city, Archimedes was killed by a soldier.

WHAT IS ARCHIMEDES PRINCIPLE? Archimedes principle states that the buoyant force on body immersed in a fluid equal to the weight of the fluid displace by the object.

Mass of fluid displaced = m= vp With v = volume of fluid displaced p = density of fluid

Formula:- Buoyant foce = weight of fluid displaced = pVg

The crown seems lighter under water !

The buoyant force on a submerged object is equal to the weight of the liquid displaced by the object. For water, with a density of one gram per cubic centimeter, this provides the volume of an irregularly shaped object and then to determined its density.

Buoyancy.

Buoyancy arises from the fact that fluid pressure increases with dept and from the fact that the increased pressure is exerted in all direction so that there is an unbalanced upward force on the bottom of a submerged object.

APPLICATION OF PASCAL’S PRINCIPLE

A submarine can be made to surface or sink in water as required by controlling its buoyancy or overall density. This done by filling its ballast with air (to surface) or with water ( to sink or dive)

In order to surface, the upper vents are closed and air is pumped from the air tank into the ballast tanks. This has a effect of reducing the overall density of the submarine, thus causing it to rise (positive buoyant).

In order to sink or dive, the vent on the air tanks are closed and the upper vents opened. Water enters into the ballast tanks. The overall density of the submarine is increased, enabling it to sink.

Hot air balloon floats and rise upwards because the buoyant force acting on it, due the surrounding air, is greater than its weigh. The balloon descend when the buoyant force is less than its weigh. When its weight is equal to the buoyant force, it remains stationary in the air.

The density of water varies with its temperature and salinity. These two factors affect the depth to which a ship will float on water. The Plimsoll line is mark on the hull or a side of a ship used to indicate the maximum permissible (and safe) loading levels of the ships in different regions around the world ( above the figure ). It was introduced in Great Britain in 1875 is named after Samuel Plimsoll, a politician and shipping merchant.

The Greek scientist Archimedes (287-212 BC) first conducted experiments to measure the buoyant force of a liquid. Archimedes was asked by the king to determine whether a crown was pure gold or was alloyed with silver. He knew the pure gold had a greater density than an alloy, so all he needed to do was compare the density of the crown with the density of gold. Although he had measured the mass of the crown he couldn’t find an accurate way to determine its volume.

While stapping into a bath, Archimedes noticed that the water level rose and conclude that this change in volume was the same as the volume of the object now immersed. He had just discovered an easy way to find the volume of the crown and thus determine whether or not it was made of gold.

He found thet the king’s crown displaced more water than an equal weight of pure gold. From this he was able to show that the crown was made of an alloy of gold with silver or some another metal less dense than gold.

Daniel Bernoulli (1700-1782), Dutch-born Swiss scientist, who discovered the basic principles of fluid behavior. He was the son of Johann Bernoulli and the nephew of Jakob

Bernoulli, both of whom made major contribution to the early development of calculus. Bernoulli was born in Groningen, the Netherlands, on January 29, 1700, and took an early interest mathematics. Although he earned a medical degree in 1721, he became a Professor of Mathematics at the Russian Academy in St Petersburg in1725. He later taught experimental philosophy, anatomy, and botany at the universities of Groningen and Basel, Switzerland.

Bernoulli pioneered in Europe the acceptance of the new physics of the English scientist Isaac Newton. He studied the flow of fluids and formulated the principle that the pressure exerted by a fluid is iversely proportional to its rate of flow (see Bernoulli’s Principle). He used atomistic concepts in trying to develop the first kinetic theory of gases, accounting for their behavior under conditions of changing pressure and temperature in probabilistic terms. This work, however, did not gain wide notice at the time. Bernoulli died in Basel on March 17, 1782.

WHAT IS BERNOULLI’S PRINCIPLE.

Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid

occurs simultaneously with a decrease inpressure or a decrease in the fluid's potential energy.[1]

[2] Bernoulli's principle is named after the Dutch-Swiss mathematician Daniel Bernoulli who

published his principle in his book Hydrodynamica in 1738.[3]

Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely

denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for

different types of flow. The simple form of Bernoulli's principle is valid for incompressible

flows (e.g. most liquid flows) and also forcompressible flows (e.g. gases) moving at low Mach

numbers. More advanced forms may in some cases be applied to compressible flows at

higher Mach numbers (see the derivations of the Bernoulli equation).

Bernoulli's principle can be derived from the principle of conservation of energy. This

states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along

a streamline is the same at all points on that streamline. This requires that the sum of kinetic

energy and potential energy remain constant. If the fluid is flowing out of a reservoir the sum of

all forms of energy is the same on all streamlines because in a reservoir the energy per unit mass

(the sum of pressure and gravitational potential ρ g h) is the same everywhere.[4]

Fluid particles are subject only to pressure and their own weight. If a fluid is flowing

horizontally and along a section of a streamline, where the speed increases it can only be because

the fluid on that section has moved from a region of higher pressure to a region of lower

pressure; and if its speed decreases, it can only be because it has moved from a region of lower

pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the

highest speed occurs where the pressure is lowest, and the lowest speed occurs where the

pressure is highest.

APPLICATIONS OF BERNOULLI’S PRINCIPLE.

A streamline design of an object reduces resistance to the flow of fluid. This enables faster speed and the ease of motion for the object in the fluid.

An aerofoil besides offer such advantages, it also gives the most favourable ratio of lift to drag in flight.

The figure shows that the upper region of an aerofoil has higher air velocity than the lower region of the aerofoil. By Bernoulli’s principle, the lower region has a higher pressure than the upper region of the aerofoil. This causes a lifting force on the aerofoil.

The carburetor is a device used in cars to mix petrol with air before the mixture is sent to combustion chamber. When the engine is switched on, air is sucked into the Venturi tube. The

reduced pressure at the narrow section of the tube causes petrol to withdrawn into the tube to be mixed with the incoming air. This air petrol mixture is then sucked into the combustion chamber.

In the Bunsen burner, a region of low pressure is produce where the jet of gas flows out from the nozzle. As a result, the surrounding air is sucked in and mixed with the gas. The mixture of gas and air (containing oxygen) allows more complete combustion of the gas.