Albert Einstein, considering that the free-fall time of bodies in
vacuum,subjected to the gravity force,is independent from the mass (this is the
Galilean law of the freely-falling bodies ),in the context of her theory of the general
relativity (the theory of the gravitation ), was induced to think that the acceleration g acquired by a body subjected to a gravitational field is
physically indistinguishable from the acceleration acquired by the body by means of an
accelerate motion of its reference frame.
The Einstein equivalence principle consists in considering the impossibility of
distinguishing, by physical experiments, the acceleration given to a body by the gravity
from the one produced by an accelerate motion of the reference frame.
The principle of equivalence can be verified by means of a simple experiment, disposing of
a spring balance of the type used to weigh the foods for dietary purposes.
It is enough to observe that the same increase of the displacement of the balance index,
or of the value readable on the display,if an electronic balance is used, can be produced
or by increasing the weight placed on the balance plate or, equally, by a suitable
accelerated motion impressed to the balance.
For example, an increase of n grams of the weight measured by the balance,may be gotten or
by setting a further weight of n grams on the plate, or by leaving unchanged the weight
and lifting the balance with a suitable acceleration, that is equivalent to an additional
n grams weight.
In this example the deformation of the spring of the balance, or, if this is electronic,
the deformation of the force sensor, are identical in both the experimental situations. This
experiment points out that the up acceleration generated by lifting the balance, that
represents the accelerated reference frame, produces the same effect (deformation) of an
increase of weight on the plate.
The equivalence of both the experimental situations depends from the fact that the gravity
force (the weight P = Mg) and the inertial force
F =-Ma that acts on the mass placed on the balance plate,which determine an
increase of the weight measured, are both directly proportional to the mass M, that is, respectively, to the gravitational mass Mg
or to the inertial mass Mi of a body.
The negative sign of the inertial force depends on the fact that the up acceleration
impressed to the balance, acts in the opposite direction in respect to the force of
gravity.
Therefore to the spring (or to the sensor) of the balance is applied an equivalent
additional force , as an effect of the accelerated up motion of the reference frame (that
is of the balance).
A fundamental consequence of the principle of equivalence is the equality
Mg = Mi of the gravitational mass Mg and of the inertial
mass Mi, that is of the aptitude of a body to resist to the
accelerating action of a force.
This equivalence was stated for the first time by any much careful measures performed by
Newton.
Newton observed that, using as a special pendulum-like mass a little empty sphere into
which were inserted time for time any different materials, was measured always the same
period, corresponding to identical weights of the inserted material.
That means that the identity of the gravitational masses of the materials introduced into
the sphere, implicate the identity of their inertial masses.
The identity of value among inertial mass and gravitational mass was verified by Eötvös
in 1909 (with the precision of a part on a billion ), using a special torsion balance, and
in the early 1970s by H.Dicke with a precision of a part on 100 billions.
The experimental verification of the identity among inertial and gravitational mass
provided to Einstein the base on which he built the theory of the general relativity, in
the context of which, in any reference frame a gravitational field is indistinguishable
from an accelerate motion of the reference frame.
Therefore we can say that a gravitational field may be always compensated by a suitable
accelerate motion of the reference frame,in such a mode that an observer that does part of
the considered reference frame, don't realize the existence of a gravitational field.
A further example of application of the principle of equivalence is gotten considering an
astronaut in a space-ship orbiting around the Earth, the Moon or another heavenly body.
In this case the reference frame for the astronaut consists of the space-ship, that is
subjected to a centripetal acceleration anti-parallel to the gravitational one, in every
point of the orbit.
Generally people say that the astronaut operates in gravity absence, while in fact the
gravitational force acting on the astronaut by the heavenly body around which he is
orbiting, is exactly compensated,in every point of the orbit, by the centrifugal force
(the inertial force ) generated by the orbital motion.
Consequently, because there is no resultant force acting on the astronaut, this isn't
accelerated in comparison with the space-ship, and to him can be applied the inertia
principle.
Therefore the astronaut, if he doesn't know to be orbiting around a heavenly body, he
could think to be in a region of the space with no gravitational field.
In an equivalent mode it may be said that, since both the astronaut and spaceship are
submitted to the same gravity acceleration, no relative acceleration
( that is no difference between their accelerations in comparison with the Earth) is
acting on each other;consequently no force is acting between the astronaut and the spaceship,
as when an elevator, for the breakup of the carrying cable, throw headlong in free fall determining
a temporary " gravity absence " for the unlucky persons that are inside it.
The principle of equivalence underlines that the force of gravity is quite different from
the other universal forces (the electromagnetic, subnuclear, weak and subnuclear strong
ones), because it is tightly connected to the space-time bend, that is depending on the
distribution and the motion of the masses that are placed in a certain region of the
universe,as it is stated by the Einstein theory of the general relativity.
Galileo,considering the direct proportionality between the force applied to a body and
its acceleration and moreover the limit case of the inertia principle, that affirms the
persistence of a body in its state of rectilinear and uniform motion, when there is no
resultant force acting on it, he understood that the rectilinear and uniform motion of the
reference frame doesn't modify the laws of mechanics.
In fact, if they are performed any mechanics experiments ( for example free fall of
bodies, measure of the oscillation period of a pendulum, etc ...) aboard a car, a ship or
an airplane that is moving on a straight line with a constant speed,provided there are no
irregularities of the motion like oscillations or abrupt deviations from the rectilinear
trajectory, no difference can be evidenced in comparison with the experiments performed in
a reference frame at rest in comparison with the ground.
That is equivalent to say that by performing mechanics experiments it isn't possible
for an observer to evidence the rectilinear and uniform motion of its reference frame
(car,ship or airplane).
If instead the motion of the observer's reference frame is accelerated, its acceleration
can be measured by means of mechanics experiments.
For example, if we measure the oscillation period of an ideal (simple) pendulum,
constituted by a little mass hung to a thin wire of length L, in
a lift which is raising with a constant acceleration a, we
get the value
Ts = 2p square root of [L /(g + a)] ,that is shorter
than the value
T = 2p square root of ( L/g ) that we should get if the lift were at rest or were moving with
a rectilinear and uniform motion.
From the formula a = (4 p
2 L / Ts2) - g it would be possible to get
the value of the acceleration of the non-inertial reference frame ( the lift).
In fact, for the equivalence principle, the accelerate motion of the lift would modify the
motion of the pendulum as if were produced an increase of the gravity acceleration ,from
the g value to the value g + a.
If instead the lift is going down with an acceleration a,
we can measure
a Td period = 2p square root of [L/( g - a )], that is greater than the one measured with the lift at
rest.
The Galileo relativity principle affirms therefore that the laws of the mechanics remain
unchanged in transiting from a reference frame fixed in comparison with the ground to
another frame which is in rectilinear and uniform motion in respect to the first, or,
generally, in transiting from an inertial reference frame to another, if we define an
inertial reference frame is a frame in which a body, not subjected to any forces, if it is
initially at rest, remains at rest, or if it is moving on a straight line with a constant
speed, it maintains indefinitely its motion state, till isn't applied to it a force that,
accelerating the body ,modifies its motion.
When a force F is applied to a body having a mass m and being at rest or moving with a variable speed, for example
when we push on an horizontal road a car initially at rest, with its gear in neutral, the
body acquires an acceleration a = F/m, for effect of which its
speed increases with continuity from the initial value to the final one, till the body is
subjected to accelerating forces.
Therefore, during the acceleration time, we expense on a body some work which equates the
loss of an amount of biochemical energy stored in our muscles.
If, for example, we push a car having a mass m along a road
with a lenght s, applying the constant force F, we make the work L = F x s.
If the force we apply is F = 50 kg = 50*9,8 N (newton) =
49 N and the car is moving of 10 meters, a work of 50 x 10 kgm (kilogram.meters) = 500 kgm
has been made, which is equivalent to
9, 8 *500 J (joule) = 4900 J.
The work made by the force F makes the speed of the body
increase from the initial value V0 to the final value V, and it is equivalent to the variation of the kinetic energy (that is
the motion energy) of the body, where the kinetic energy K (which
is known also as the "live force" ) of a body with a mass m,
that is moving with the speed V ,
is the half-product of the mass for the square of the speed: K = ( 1/2
)M V2 .
The theorem of the "live forces" ( or the work-energy theorem ) states that the
work made by one or more forces on a body equates the variation ( increase or diminution )
of the kinetic energy of the body:
L ( work ) = F (resultant force) x s (displacement)
= K (
final kinetic energy ) - Ko ( initial kinetic energy ).
If the applied forces favor the motion, they are defined engine forces, because they
determine an increase of the speed and of the kinetic energy of the body; if instead the
forces act in the opposite direction in respect to the motion, as in the case of the
aerodynamic resistance that slows down a moving vehicle, or in the case of the friction
forces that are opposing to the motion and transform the kinetic energy into heat,we speak
of resistant forces, because they determine a diminution of the kinetic energy of the
body, as it happens during the braking operation of a vehicle.
The third principle of dynamics implicates as a consequence that in the systems
consisting of two or more bodies is worth the maintenance law of the linear momentum,
provided is zero the resultant of the external forces applied to the system.
If F (the resultant of the external forces ) = 0, there is no variation of the total linear momentum (dPtotal) of the system per a time unit:
d (Ptotale) /dt = F = 0 .
Therefore the total linear momentum P = P1 + P2 +... Pn of the
system maintains a constant value, while the linear momenta of the single bodies can vary,
because the pairs of forces inside the system produce equal and opposite variations of the
single linear momenta, that are balancing two by two each other.
If consider, for example, the dynamic phenomena that take place in a fire-weapon during
the expulsion of the bullet, we realize that the only effective forces are the inside
forces generated by the pressure of the expanding gases,produced by the shot.
They are action-reaction forces with the same modulus and opposite directions: if we
define as action force the force throwing the bullet across the weapon barrel, the
reaction force is the one effected by the expelled bullet on the weapon recoiling.
The only external force acting on the system is the gravitational one, that however doesn't
affect significantly the phenomenon during the expulsion of the bullet, because the
impulsive forces generated by the shot are much more intense than the gravitational ones.
During and right after the shot, the motion of the bullet, if we consider only the initial
part of its parabolic trajectory, has such a great speed that is negligible the
simultaneous effect of its free fall.
Then the bullet acquires a forward linear momentum as an effect of the impulse of the gas
expansion forces in the weapon barrel, while this, for effect of the reaction force
effected by the bullet, acquires an equal and opposite linear momentum in respect to the
bullet momentum.
Therefore the sum of the two linear momenta, which is zero before the shot, mantains a
value equal to zero during the shot time, in which we can consider neglectable the effect
of the gravity.
Even the friction forces between the bullet and the inside of barrel are opposite forces
with the same modulus, that doesn't change the total linear momentum of the system, since
they determine equal variations of the opposite momenta of the bullet and the fire weapon.
In a system consisting of two or more bodies, for example in the solar system, it is
necessary to consider as inside forces the force of mutual attraction among all the
possible pairs of bodies: Sun-planet, planet-planet, planet-satellite, Sun-satellite and
satellite-satellite.
Since all this forces are equal in modulus and opposite two by two each other, they
determine equal variations of the opposite momenta; then we realize that the sum of the
momenta of all the bodies in the system mantains a constant value.
The solar system is moving in the space as a point-like body having a mass equal to the
sum of the masses of all the bodies:the Sun,the planets and the satellites,that are
subjected to all the possible attractive inside forces, pairs of action and reaction
forces, that annihilate two by two each other, giving a zero resultant force, and moreover
to all the forces of gravitational attraction effected on the Sun, its planets and their
satellites by every other heavenly body outside the solar system.
Since from the astronomic measures it results that the solar system is moving with a
rectilinear and uniform motion toward the constellation of Hercules, with the speed of
about 20Km/sec, it is necessary to deduce that is effectively equal to zero the average
value of the resulting gravitational forces effected by every other heavenly body outside
it, that is the galaxies, quasars and black holes, that have to be considered, in a first
approximation, to be disposed symmetrically in comparison with the solar system.
From the reduction to zero of the resultant of the external forces acting on the solar
system, the maintenance of its total linear momentum can be deduced.
The position of the above-mentioned point-like body identifies with that of the mass
center of the solar system, that may be considered, in first approximation, with the
center of the Sun, because the solar mass is over 700 times greater than the one of all
together the other bodies of the solar system .
The mass center of a system of bodies is the point whose motion is effected only by the
external forces applied to the system, and in which it can be thought is concentrated the
whole mass of the system.
The position of the center of mass depends on the mass and the position of each body, that
is on the mass distribution of the bodies in the space.
For example, if we consider a metallic bar having in its extrems two equal masses, the
mass center, for symmetry,is placed in the middle point of the bar.
If instead one of the two masses is twice the other one, the mass center is placed nearer
to the double mass,at a distance from it equal to one third of the bar length.
If, in particular,we consider the motion of the Earth in its elliptic orbit around the
Sun, the Earth linear momentum, that is the product of the Earth mass for its velocity,
that is always tangent to the orbit, increases near the perihelion, when the Earth is
approaching to the Sun.
The Earth is indeed accelerated toward the Sun for effect of gravitational attraction
force,as when a body is accelerated falling freely on the Earth ground.
Instead the linear momentum decreases with the increasing Earth-Sun distance,when the
Earth,going toward the aphelion, is subjected to a decreasing force of gravitational
attraction, as when a body is launched up vertically.
According to the third principle of dynamics,also the Sun is accelerated or decelerated by
a gravitational attraction force equal in modulus and in opposite direction in respect
with the one acting on the Earth; therefore also the linear-momentum variations of the Sun
and the Earth,are equal in modulus and opposite, without that the total linear momentum of
the solar system would be changed.
Yet, because the mass of the Sun is over 300000 times greater of the Earth mass, then the
centripetal acceleration and consequently the speed variation of the Sun for effect of the
Earth attraction,are over 300000 times smaller than those of the Earth for effect of the
Sun attraction,so that the elliptical orbit of Sun-sphere center about the mass center of
the solar system, is inside the Sun,and it is reduced to an ellipse whose greater axle (
about 1000 km ) is over 300000 times smaller than that of the Earth orbit (about
300.000.000 Km).
Therefore,the center of the Sun-sphere, whose diameter is about 1.400.000 Km, for effect
of the only Earth attraction, provided we don't consider, for simplicity, the masses and
the attractive forces of all other the bodies of the solar system, is describing an
elliptical orbit,which is assimilable, in first approximation, to a circumference having a
radius of about 500 Km and the center in the mass center of the Earth-Sun system, that is
inside the Sun-sphere.
Analogous considerations could be done about the other planets of the solar system.
If we finally consider that the universe is the greatest conceivable isolated system,
because we cannot think, for logical coherence, that it is effected by external forces,
then,for all we said before, we can deduce that its total linear momentum is constant, as
all the forces acting on the system are forces inside it.
