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The forces of friction are resistant forces,acting against the motion or rolling of a
body on the surface of another body which is in contact with.

The contact forces acting between the surfaces of two bodies are determined by the
attractive electric forces by which the molecules interact each other.

If a body is at rest on a horizontal rigid surface,we observe that the least modulus of
the horizontal force Fo _{min} we need to apply to the body to start its motion, is directly
proportional to the modulus of the vertical force F_{v} that
pushes the body against the surface.

F_{v} is given by the weight
P , to which we have to add each possible additional vertical
force Fa pressing the body against the surface:

Fo _{min}
= m_{s} Fv = m_{s} ( P + Fa ), where m_{s} ( P + Fa ) is the static friction
force F_{fs}.

The proportionality coefficient m_{s} (which is less than one)
between the surfaces in contact is the static friction coefficient,that depends on the
nature of the surfaces in contact and decreases if these are lubricated. As soon as the
body starts to move, the coefficient m_{s} is replaced with the
dynamic friction coefficient m_{d} ,that is smaller than the static one. Therefore the
force of dynamic friction F_{fd} acting between two
bodies that are moving being in contact each other, is always smaller of the static
friction force F_{fs}.

Example

If a block with a weight of 100 kg , placed on an horizontal surface, starts to move when
the horizontal force applied to it assumes the value of 40 kg, the coefficient of static
friction is m_{s} = 40/100 = 0,4.

As soon as the intensity of the applied force overcomes the value of 40 kg, the block
accelerates.

To make the body be moving with a constant speed, the applied force must be diminished
shortly after starting, because now the force of static friction is replaced by the one of
dynamic (or kinetic) friction.

If instead, we suppose that the dynamic friction coefficient is 0,25, the intensity of the
horizontal force that has to be applied to maintain the block in an uniform motion is 0,25
* 100 = 25 kg.

The friction between two bodies moving in contact each other is the so-called sliding
friction, and it is smaller than the rolling friction that takes place when a body rolls
on another one.

For this reason in the machines the ball bearings, which are based on the rolling
friction, are more advantageous than the ones founded on the lubricates surfaces.

The internal combustion engines need a continuous lubrication of the cylinders, by means
of pressured oil, to avoid the seizure of pistons.

The gears of some mechanical devices, as the differential and the speed change
device,operate immersed in oil to minimize the friction and so guarantee operation safety.

The presence of the friction forces, that dissipate mechanical energy into heat, isn't
always undesirable:

- The friction clutch and the brakes of a car are based on sliding friction;

- We should not are able to walk neither run without the forces of friction between our
shoes and the ground;

- A railway or road vehicle would not be able to move without the friction force,
respectively among the wheels and the rails and among the tires and the road surface;

- A car or a motor-cycle, to be moving along a curve,need some friction centripetal forces
that must be the greater, the greater is the speed and the smaller is the radius of the
curve.

A body moving in a liquid is liable to the resistant forces produced by the intern
friction of the liquid, that is by the viscosity.

The viscosity of a liquid is produced by the attractive electric forces acting among the
molecules, and it is decreasing with the increasing temperature of the liquid.

The fact that the viscosity of an oil decreases with the increasing temperature, explains
the difficulty that happens in starting the engine of a car when the temperature is very
low.

The intern friction of a liquid and the friction between this and the surface of a moving
body, determine the hydrodynamic (or viscous) resistance to the motion, that is directly
proportional to dimensions and speed of the body.

If the body is moving with a speed greater than 2 m/s, the resistance varies in terms of
the square of the speed ( hydraulic resistance ).

If a body is moving in a gas,it is subjected to a resistant force,that is the
resistance to motion in the medium or the aerodynamic resistance, that depends on the
square of the speed of the body, on the density of the gas, on the surface of the body,
that is on the cross area of the surface projected by the body on a plane perpendicular to
the direction of the speed, and on a form factor that varies in terms of the geometric
characteristics (the aerodynamic profile) of the body.

Examples

In the case of a body with a great surface, for example a parachute, the aerodynamic
resistance is so great that the fall speed stabilizes about a constant value, that is the
smaller, the greater is the surface of the parachute.

Because the aerodynamic resistance R = Cr r V^{ 2} S
is directly proportional to the area S of the parachute, to the
density r of the air
and to the square of the speed V , after some time the fall speed
becomes constant according to the inertia principle.

Then the weight force P = Mg and the aerodynamic resistance

R = Cr r V ^{2} S
are balancing each other, where Cr is a coefficient of
proportionality that depends on the form of the body and it is analogous to the well known
coefficient Cx that determines the aerodynamic resistance
acting on a car.

The constant speed V of the falling body is gotten by
extracting the square root of the ratio [Mg/(Cr S r)].

The aerodynamic resistance is always directed along the perpendicular to the cross section
of the moving body.

The aerodynamic resistance acting on the aileron of a F1
race-car or on the wings of an airplane, because the mobile surface is disposed sideways
in comparison with the direction of motion, it may decomposed in two components: the
vertical component, in the case of the F1 race-car
pushes it against the ground, increasing the adherence among the tires and the road
surface; in the case of the airplane instead, it is up directed and constitutes the
so-called carrying capacity (force) of the wings, that has to overcome the weight of the
airplane so that it is able to take off.

The horizontal component acts against the motion in both cases, and consequently against
the propulsive force.

In the case of a car, the aerodynamic resistance in relation to the cross-section of the
car body, determines the maximal speed corresponding to the maximal power generated by the
engine.

In fact, with the increasing speed, the aerodynamic resistance, added to the rolling
resistance of the car, that is directly proportional to both the weight and the
coefficient of rolling friction between the tires and the road surface, is balancing the
propulsive force , till the acceleration is reduced to zero and the speed remains
constant,according to the inertia principle.

If the cross-section of the car body is S = 1,5 m^{2} ,
and we consider the air density r = 1,29 kg/m^{3}, the maximal speed Vmax =
130 km/h ~= 36,11 m/s, the value Cx = 0,2 , the weight P = 1000 kg and the coefficient of rolling friction Cv = 0,01 , we get the following values:

Ra =(aerodynamic resistance) = Cx r S V ^{2}
= 0,2 x 1,29 x 1,5 x (36,11)^{2 }= 504, 62 N = (504,62/9,8) kg = 51,49 kg.

The resistance to the rolling is instead:

Rr = Cv P = 0, 01 x 1000 = 10 kg.

Therefore, at the maximal speed of 130 km/h the engine generates the maximal power Pmax, which is given by the product of the total resistant force:

Rt = Ra + Rv = 51,49 + 10 = 61,49 kg at the speed Vmax:

Pmax = Rt Vmax = 61,49 x 36,11 (kgm/s) = 2220,4/75 HP = 29, 6 HP

( 1 Hp = 75 kgm/s ).

The power that has to be generated to overcome the aerodynamic resistance is:

Pa = Ra Vmax = 51,49 x 36,11 ~= 1859,3 ~= 24,79 HP, while the
power ncessary to overcome the resistance to the rolling is

Pv = RvVmax = 10 . 36,11 ~= 361,1 kgm/s ~= 4,814 HP.

The simulations by computer and the aerodynamic tests that are performed in the the wind
tunnels of the car-factories, are requested to optimize the form of the car-body to make
the engine generate the optimum power in relation to the maximal speed.

The Pascal principle (a French philosopher and mathematician , 1623- 1662) is
fundamental to study the statics of fluids ( liquids and gases ), that are
characterized by a greatest mobility of the molecules, because the electric forces by
which these interact each other, are much reduced in comparison with the ones acting inside
crystals.

To enunciate the Pascal principle we need to remember the fundamental properties of
fluids.

While a liquid has the minimal compressibility and assume the form of its container, with
a constant volume, a gas instead is easily compressible and, because of the very weak
attractive forces among the molecules (the Van Waals forces), that are moving with high
thermal motion speeds, they are expanding till take the whole volume of the container,
acting a pressure on the walls of it.

How can be defined the pressure of a fluid?

It is the force acting along the perpendicular to any unitary surface that is in contact
with it, and it is measured by the force per surface unit:

1) N/m^{2 }= Pa (Newton/meters^{2}) = Pascal;

2) kg/cm^{2} ~= 9,81 N/cm^{2} = 98100 N/m^{2 }= 98100 Pa;

3) atmosphere = 1,033 kg/cm^{2}.

Since the pressure is definite by the relationship p = F/S
between the force F acting along the perpendicular to a surface
and the area S of it, if we know the value of the pressure in an
assigned point of the liquid, we may calculate the force F = pS
acting along the perpendicular to the surface S.

If in fact we consider a tank containing a liquid in the equilibrium condition (that is
when its free surface is horizontal), the pressure forces the liquid acts on all the tank
walls, produces, according to the III law of dynamics, some pressure forces that are equal
and acting in opposite directions with respect to the ones the tank walls effect on the
liquid.

These reaction forces, if weren't directed along the perpendicular to the tank walls,
would make the liquid flow parallelly to them, because of the great molecular mobility.

Therefore it is deduced that in a fluid the pressure forces act always along the
perpendicular to the tank walls and to any surface placed inside the fluid, independently from the orientation of the surface.

The pressure of the gas in a container is the effect of the continuous collisions of the
molecules against the walls, and increases with the increasing temperature,because the
molecular kinetic energy is directly proportional to the absolute temperature ( in Kelvin
degrees ) of the gas.

The electric forces acting among the high mobility molecules of a fluid make it possible
that the pressure we act in any point of its mass,is transmitted along every direction and
with the same value to whole the fluid mass, without any diminution.

Therefore the Pascal principle states that, if a fluid is inside a container and the
pressure value at any point of it is increased, this increase of pressure is transmitted
unchanged to whole the fluid and to the walls of the container.

Examples

If a liquid fills a cylinder furnished with a piston effecting a pressure on the liquid,
pushing down of the piston determines a pressure increase that is transmitted unchanged to
whole the liquid and to the walls of the cylinder.

An important application of this principle concerns the braking system of a car, that
permits to apply to the front and rear brakes the force acted by the foot of the driver,
across a pressured liquid that transmits the pressure increase.

We remember besides that on the Pascal principle is based the operation of the
oil-pressured systems used in excavators, bulldozer and so on, and the operation of
hydraulic presses, that apply to a mechanical piece that has to be moulded, forces
of several tons by means of much more little engine forces.

In an oil-filled cylinder a piston with a surface of 10 cm^{2 }is pushed down
of 100 cm,under the action of a force of 10 kg, transmitting a pressure increase of 10
kg/10 cm^{2} = 1 kg/cm^{2} to another piston with a surface of 1 m^{2}
= 10000 cm^{2}.

On the great piston is acting a force of 10 tons (10000/10 = 1000 times greater than 10
kg), whereas its displacement is 1000 times smaller ( 1 mm ) than the one of the little piston.

The work made on the little piston is L = 10 kg x 1 meter = 10 kgm (kilogram.meters),and
it is about equal to the work made by the great piston on the object that has to be
moulded (this is the principle of operation of the hydraulic press ), provided it is
negligible the work that is transformed into heat because of the friction forces and the
viscosity of oil.

The principle of Archimedes is a direct consequence of the fact that the hydrostatic
pressure a liquid produces because of the gravity force, depends on both the density and
the height of the liquid inside a container.

If we consider in fact a liquid which is in static equilibrium in a container, the
hydrostatic pressure p at any point of the liquid at a depth h under the free surface of the liquid, is given by the formula:

p = p_{o}
+ r g h ( this
is the principle of Stevino), in which r is the density of the liquid, g is the gravity acceleration and p_{o} is the atmospheric pressure acting on the free surface
of the liquid.

The product r g h is
the weight of a liquid column with the density r , the base of 1 cm^{2} and the height h,
and is numerically equivalent to the hydrostatic pressure, that depends only on the depth h under the free surface of the liquid and produces always forces
(pressure forces) directed along the perpendicular to the surface that is considered.

The Archimedes principle states that a body immersed in a liquid or in a gas with the
density r , is
subjected to the force S , the so-called hydrostatic thrust,
which is equal to the weight of the liquid volume V removed by
the body.

If the body is partially immersed in the liquid,(if the body floats in the liquid), the
hydrostatic thrust equates the weight of the volume of the body part immersed in the
liquid.

The Archimedes thrust S = r
gV is generated by the resultant of all the forces the fluid
produces on the surface of the body by means of the hydrostatic or aerostatic pressure.

In gravity absence conditions, for example in a spaceship orbiting around a planet or a
satellite, there is no hydrostatic pressure and then there is no thrust of Archimedes.

A body immersed in a liquid floats if the Archimedes thrust overcomes its weight.

In this case the resultant of the hydrostatic forces pushes the body toward the liquid
free surface, till the thrust produced by the immersed part doesn't equate the weight of
the body.

Ships, submarine, hot-air balloons and dirigibles operate according to the Archimedes
principle.

A submarine,pumping some water into or out its watertight compartments, is able to control
the difference,up or down directed, between the resultant of the hydrostatic pressure
forces and its weight, respectively when it is going up again to the sea surface or when
it is submerging.

When the weight of the submarine equates the resultant of the hydrostatic pressure forces,
once it has reached the pre-established depth, can remain at this constant depth.

An hot-air balloon is subjected to an ascensional force Fa which
is given by the difference between the aerostatic thrust S and
the weight P:

Fa = S-P = r_{c}Vg- r_{h}Vg,
where r_{c}
and r_{h}
are, respectively, the density of the external cool air and the one of the hot air inside
the ballon.

The mass conservation principle,formulated in 1789 by the French chemist Lavoisier and
already expressed by Lucretius in the poem
"De rerum natura", states that the whole amount of matter in Nature is constant
and indestructible.

Therefore, in any system and, by extension, in the universe,the total amount of matter is
always the same.

The matter is subjected to physical or chemical transformations, but it is never
destroied.

In Nature nothing is destroying, everything is transforming .

Example

The combustion of coal (C) or methane (CH_{4})
is a chemical reaction between carbon and oxygene or between methane and oxygen.

The combustion products of the chemical reaction consist, in the case of coal,of carbon
monoxide (CO) and carbon dioxide (CO_{2});

instead,in the case of methane,they consist of carbon dioxide and steam

( CH_{4} + 2 O_{2} = CO_{2
}+ 2H_{2} O ).

If we weighted the fuel ( coal or methane ) , the oxygene and the combustion gases, we could
verify that the total mass of the reaction products is always equal to the total mass of
the reagents, provided we neglect, in the first approximation, the extremely little mass
of the reagents that is transformed into heat according to the relativistic equivalence
principle between mass and energy.