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Relativity and the GPS( Last checked 2017/01/15 - The estate of Paul Marmet )
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Relativity and the GPS( Last checked 2017/01/15 - The estate of Paul Marmet )
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Source of Energy Between the Pole and the Equator.
Let us examine how
the principle of energy conservation is applied to the two
experiments above. In the case of the orbiting satellite,
it is well known that the total energy of the satellite is
conserved because the sum of potential plus kinetic energy
everywhere along the orbit is equal to zero. There is no
exchange of energy between the orbiting satellite and the
Earth.
However, we have
seen above that when a mass moves from the pole to the rotating
equator, there is a net increase of energy (potential plus
kinetic), given to that moving clock. Since it requires no
external energy to carry a mass slowly between the Earth pole
and the equator (neglecting friction), some people believe that
there is no difference of energy. That is an error, as we
will see now. Let us calculate where that energy
comes from.
When a mass is at
the Earth pole, its angular momentum around the Earth axis of
rotation is zero. Let us slowly move the clock away from
the Earth pole. When that clock, moving along a meridian,
gets to an increasing distance from the pole, the rotating Earth
makes it move faster and faster in a direction perpendicular to
the meridian. It is the inertia of the rotating Earth,
which accelerates the moving clock in a transverse direction,
until the clock reaches the equator and attains the transverse
velocity of the equator, which is 1670 Km/hr.
At the same time the clock
moves away from the Earth pole, another phenomenon takes
place. Since the equator is further away from the Earth
center, a radial force is required to maintain the effortless
motion of the clock along the meridian, which becomes larger
near the equator. That effortless motion is natural, since
the Earth rotation produces a centrifugal force away from the
pole in the direction of the equator. Those two forces
compensates naturally. In fact, the exact shape of the
Earth is the natural equilibrium between these forces.
Let us calculate
first the total increase of energy DE(clock)
given
to the clock moving from the pole to the equator. It is
equal to the potential energy plus the kinetic energy.
DE(clock) = (kinetic energy) + (potential
energy)
1
Taking into account the flattened shape of the Earth and the
centrifugal force, it is well known that the potential energy is
equal to the kinetic energy. Therefore the total increase
of energy of the clock is:
DE(clock) =2 (kinetic
energy)
2
The kinetic energy
of the clock at the equator is
K.E. (clock)=½mV2
3
Where m is the mass
of the clock and V is the velocity of the Earth equator.
Equation 3 in 2
gives that the total increase of energy DE(clock)
of
the clock moving from the Earth pole to the equator.
DE(clock) =2 (½mV2)=mV2.
4
The clock at Earth
pole has zero angular momentum “P” with respect to the Earth
axis (because R=0). At the equator, the increase of
angular momentum DP of the clock
around the Earth axis is:
DP(clock)=mVR
5
Where R is the
equatorial radius of the Earth.
We know that the
Earth is an inertial frame in space and its total angular
momentum remains constant even when a mass (like a clock) moves
on its surface. Since there is no external force acting on
the Earth, its total angular momentum is constant. We know
that the moment of inertia “I” of a spheroid of revolution (as
the Earth) along the polar axis is
6
We have M is the
mass of the Earth. The angular momentum “P” around its
axis is:
P=Iw
7
The angular velocity
w is defined as:
8
Using equations 6, 7
and 8, we find that the total Earth angular momentum “P” before
the clock moves to the equator is:
9
When the clock is
initially at the Earth pole, its mass brings no contribution to
the Earth angular momentum (R=0). However, after the mass of the
clock reaches the equator, its mass gives a new contribution to
the angular momentum. However, since Newton’s laws
requires that the total Earth angular momentum is constant, the
new angular momentum of the clock at the equator appears at the
detriment of the total initial Earth angular momentum.
Equation 5 gives the
change of angular momentum due to the displacement of the
clock. The relative change of angular momentum of the
Earth (minus the clock) due to the displacement of the clock is
obtained using equations 5 and 9. This gives:
10
Taking the
derivative of equation 9 for the same Earth mass M at the same
distance R (equator) we find:
11
Equation 11 in 10
gives
12
Let us calculate the
energy corresponding to that change of Earth velocity. The
energy of a rotating body is
E(Earth)=½ Iw2
13
Equations 6 and 7 in
13 give:
14
The variation of the
Earth rotational energy is given by the derivative of equation
14. This gives:
15
Using equation 15
let us calculate the total loss of the Earth rotational energy
due to the displacement of the clock from the pole to the
equator. Let us substitute equation 12 in 15. This
gives:
16
We find that
equation 16 is identical to equation 4. Therefore the
energy acquired by the clock at the equator is equal to the
energy lost due to the slowing down of the Earth rotation.
Therefore,
17
Equation 17 shows
that the increase of energy of the clock moving from the pole to
the equator is exactly equal to the lost of kinetic energy of
the rotating Earth. Therefore, as expected, supplementary
energy is really needed to move the clock from the pole to the
equator. However, that energy comes from the kinetic
energy of the rotating Earth. Therefore there is
mass-energy conservation and also angular momentum conservation
in the Earth frame. Therefore is no need to use any force
to carry the clock from the pole to the equator, because the
internal Earth energy of rotation transfers that energy
naturally.
The whole Earth is
slowed down when a mass is moved from the pole to the equator as
demonstrated here. For that reason, the height of the sea
level is in a natural equilibrium, higher at the equator than at
the pole. Unfortunately, Hatch paper neglects the fact
that the mass of the clock has absorbed energy when passing from
the pole to the equator while there is no change of energy
between perigee and apogee. We have seen that the increase
of kinetic energy slows down the clock at the same time as an
increase of potential energy speeds it up. This explains
the fact that the clock rate does not change when we move on the
Earth surface remaining at sea level. No new theory is
required. Conversely, in the case of an orbiting clock,
there is no change of total energy between the perigee and
apogee so that this leads naturally to a change of clock rate
between perigee and apogee of an orbiting body. Those two
completely different experiments must not be confused.
References
1- Natural Length Contraction Mechanism Due to Kinetic
Energy. On the web at:
newtonphysics.on.ca/kinetic/index.html
2- Natural Physical Length Contraction Due to Gravity. On the
Web at:
newtonphysics.on.ca/gravity/index.html
3 - A Detailed Classical Description of the Advance of the
Perihelion of Mercury. On the web at:
newtonphysics.on.ca/mercury/index.html
4 – Einstein’s Theory of Relativity Versus Classical Mechanics
P. Marmet, Book, (1997) Ed. Newton Physics Books, Ogilvie Rd.
Ottawa, Canada, K1J 7N4
On the Web at:
newtonphysics.on.ca/einstein/index.html
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Paul Marmet, April 4, 2002
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