05 |THE CORIOLIS FORCES

Introduction

In Physics, this name is used for forces created by a body which is moving with a velocity on a rotating system animated by an angular velocity . Since we live on a rotating planet, the Coriolis effect is very common. Nevertheless, the forces are generally weak, and they do not affect our human activities. This is probably the reason why only advanced studies in Physics deal with this phenomenon.
I was always fascinated by Physics, and I made several experiments in connection with the Coriolis effects. As a result, I found that the present science of Physics explains the Coriolis forces erroneously, and is full of confusions.
The confusion starts with introducing the different forces acting in the universe. It is obvious that a force can act only if it is supported by a solid structure. Then, this structure is charged by a force of the same absolute value, but acting in the opposite direction. We call it reaction force. Meanwhile, we experience some forces where the physicists never tried to find and explain the supporting structures and the reaction forces (like the gravitational, the electrostatic and the magnetic forces). The science is satisfied to introduce mathematical formulae in order to calculate them, and it is said that those so-called "fundamental forces" do not obey Newton's law. We can call them non-newtonian forces.
Remark: It was the English philosopher Sir Isaac Newton (1642-1727) who, for the first time, explained the natural law of action-reaction.
Furthermore, another force was also introduced as being non-newtonian. It is called inertia force. In order to understand the Coriolis effect, we are obliged to analyse in detail those inertia forces.

Inertia Forces

They are caused by the accelerating movements of the bodies. This is one of the most marvellous natural law. An acceleration is always caused by a force, but Nature always creates a kind of counterforce in order to assure a perfect dynamic equilibrium for all moving bodies, which is called inertia force in Physics.
Remark: In the universe, everything is always moving and accelerating. A perfect constant linear velocity does not exist. Consequently, each body is always affected by some inertia forces. It is said that an inertia-free point does not exist in the universe. Fortunately, we can deal with most of the problems by neglecting the inertia effects caused by the rotation and revolution of the earth or other celestial units.
The inertia force has an interesting particularity. It affects individually each elementary unit inside the accelerating body. This means that an accelerating force should act individually on each elementary unit. In a rigid body, the elementary units (molecules, atoms) are firmly tied together. The elementary units are accelerating together with the whole body.
When we are sitting in an accelerating car, the acceleration of our body is assured only if we are tied to the rigid structure of the car. In the case of a frontal collision, and without using a seat belt, the acceleration of our body starts only when we hit the front window. (Let us note that a slowing down should be considered as a negative acceleration.)
It is a difficult task to specify exactly the elementary units of a body which are individually affected by the inertia forces. We know that a body (living or inanimate) is an agglomeration built up of a huge quantity of atoms. Inside the atomic structure, there is a nucleus (which I call sun of the atom) and several electrons (which I call planetary units).
If we consider an atom accelerating in the direction , as shown in Figure 05-01, then each inside unit will be affected by an inertia force in the direction as indicated. The acceleration of an atom must be similar to the acceleration of a vehicle. Each unit inside the atomic structure has its special kind of "seat belt". The law of inertia must be the same for all units of the microcosm.
Today, we start to understand that even the members of an atom are complicated structures build up of elementary subatomic particles. So, it is difficult to specify how far we should go in the direction of the infinite little in order to explain the elementary units of an accelerating body. However, to understand the problem, we should imagine a quasi infinite little elementary unit, and we indicate its mass value with the sign _e. Let us indicate the mass value of the whole body by the sign , and the acceleration by . Then, the inertia force acting on the whole body will be
x
It is a resultant vector composed of a huge quantity of elementary inertia forces
_e x
This resultant force acts in the opposite direction to the acceleration, and passes through the centre of inertia of the accelerating body.
Remark: The centre of inertia is practically the same as the centre of gravity. However, there is a basic difference between inertia and gravity.
The gravity affects individually each molecule of a body. We should imagine that each molecule is tied to the earth by an invisible elastic string. The resultant of these elementary forces passes through the centre of gravity.
On the other hand, the elementary units affected by the inertia forces are much smaller than a molecule. The inside of each atom is individually stressed in the case of an acceleration. A human being has a different feeling if he/she strongly accelerates, or he/she is in the outer space without gravitation. A very strong acceleration can even instantaneously kill a living unit. Physicists who try to unify the gravity and the inertia are making a colossal blunder.
We can consider an average mass value of those elementary units (affected by inertia), and indicate them with the sign |_e|. Then, we can introduce a quasi infinite big number , and the inertia force will be
x |_e| x = x
Thus, the value of mass of a body will be
= x |_e|
This means that the value of mass indicates the number of elementary units in the body. In a mass of 2 kilogramme, there are two times as many elementary units as in a mass of 1 kilogramme.
I already mentioned that some physicists are considering the inertia forces without any reaction, like the other non-newtonian forces. The idea is introduced by observing the inertial phenomenon inside an accelerating system (in the framework of an accelerating coordinate system).
In a fast running car, if we suddenly apply the brake, then we are thinking that a mysterious force is pushing us forward. Some books are using the expression "apparent force". This name can be misleading. Precisely, we should use the name "apparent" only for such things or for such phenomena which we can directly observe with our vision. A force is not such a thing. Its existence does not depend upon some moving coordinate systems, and we must always find the supporting structure and the reaction force. Meanwhile, sometimes we observe apparent trajectories, and some relative velocities can be apparent, too.
Remark: In modern science, it became a kind of fashion to introduce relative, apparent entities. I would rather like to call this "scientific mania". By considering the speed of light as a universal constant, some scientists introduced the apparent time value, the apparent mass value, and even the apparent energy value. Sometimes, without any reason, those apparent things are considered to be real. This transformation from apparent to real causes a real scientific confusion.

Introducing the Coriolis Forces

In Physics, the Coriolis force is introduced with the help of a circular rotating platform, as shown in Figure 05-02 (A).
It rotates with a constant angular velocity . It is assumed that the surface of the platform is perfectly slippery (friction-less). Furthermore, the centre point of the rotation is considered to be inertia-free (neglecting the effects of the rotating earth). A little globe of mass is attached to the centre point with a string of length , and it is following the rotation of the platform. Suddenly, we cut the string and, without a centripetal force, the globe is obliged to change its trajectory. An observer, standing near the platform, will see that the globe moves in a straight line with a velocity
= x
as shown in Figure 05-02 (B). An observer, sitting in the centre point and rotating with the platform, will see that the globe draws a deviated trajectory, as shown in Figure 05-02 (C). He/She attributes this deviation to an apparent force which was introduced as Coriolis force.
The idea of this apparent Coriolis force is misleading. On a perfectly slippery surface, no horizontal force and reaction force can be created. This is just an apparent trajectory observed by a rotating person.
Most physicists are obsessed with handling the inertia and the Coriolis problem in reference to an accelerating or rotating coordinate system. Some spectacular mathematical equations have been introduced and, as a result, the following statement was made.
In a coordinate system, rotating with an angular velocity (like our rotating platform), we have to consider two kinds of apparent forces acting on each body (with mass value ).
1. The so-called centrifugal force. Its value is given by the formula
  _ce = x x ²
  where is the distance between the body and the centre of rotation. Its direction is radial, pointing away from the rotation's axis.
2. If the body moves with a velocity on the rotating platform, then we also should consider a Coriolis force, and its value can be calculated by the formula
  _co = 2 x x ( x )
  where the sign ( x ) means the vectorial product of the vectors and . The resulting force is perpendicular to both and . If we define the direction of the , then we can find the direction of the Coriolis force with the well-known right-hand or left-hand three fingers rule.
According to this formula, the Coriolis force does not depend on the direction of the movement. The only condition is that the velocity vector should be on the surface of the rotating platform, which means perpendicular to the rotation's axis. And this is not true. We can prove it with a simple experiment.
Let us consider a large rotating platform on which a person can easily walk or run, as shown in Figure 05-03. Suppose that he/she runs with a velocity
= x
perpendicular to the radius and in the opposite direction of the rotation as shown. According to the formulae, this person should be affected by a centrifugal force
_ce = x x ²
and a Coriolis force
_co = 2 x x x = 2 x x x ²
because
= x
Those two forces are acting in the opposite direction. As a result, we should feel a force
_co - _ce = x x ²
which should push our body in the direction of the rotation's axis. Everybody can find out that this is not true. Anyway, if we examine this situation in a standing (assumed to be inertia-free) system, then it is obvious that the person does not move at all, and no movement means no acceleration and no inertia forces.
If we start to run in the radial direction, then the situation is different. We feel a force which tries to overturn our body in the right direction (if the direction of the rotation is as indicated in the sketch). We must work harder with our right foot in order to keep our equilibrium.
We can continue our experiment by running in another direction on the rotating platform. We will still feel this force, but it will always become weaker as we approach the direction perpendicular to the radius. This means that, by evaluating the Coriolis force, only the radial component of the velocity should be considered. In other words, we should consider the speed at which we are approaching the rotating axis or moving from it.
Remark: We should not deviate too much from the radial direction because then the centrifugal force dominates. Anyway, moving perpendicularly to the radius, the Coriolis effect completely disappears.
This experiment shows that there is a real Coriolis force. Due to the rotation, the trajectory of the moving body is always curved, and this causes a kind of inertia force.
In order to better understand the problem, it is useful to realise another experiment.
On a rotating platform, we fix two rails in the radial direction, as shown in Figure 05-04. On each of the rails, there is a kind of vehicle (like a little toy train). Vehicle moves towards the rotating axis, and Vehicle moves away from it. We can draw the trajectories of the vehicles in the standing (inertia-free) system, and we find the curves and . A curved trajectory means an acceleration, and the vector of this acceleration is pointed towards the centre of curvature.
It is interesting to draw a section - through the vehicle as shown in Figure 05-05.
We can observe four different forces.
1. The force is causing the acceleration. It obliges the vehicle to follow a curved trajectory. This force is supported by the rail, and the rail must be fixed to the rotating platform. Otherwise, there is no Coriolis acceleration.
2. The force _r is the reaction force of . It proves the validity of Newton's third axiom, and it charges the same structure as the force .
3. The force _co is a kind of inertia force. It is the result of a quasi infinite number of elementary inertia forces. The resultant force vector passes through the centre of inertia of the vehicle.
4. Finally, we should mention the force . It increases the weight load of the vehicle, and counterbalances the overturning moment
  _co x
  Consequently, there is an additional load on the right wheels of the vehicle (by observing it in the direction of the movement).
Remark: It was observed that, in the northern hemisphere, the right-side wheels and rails are more loaded in places where a train moves in the north-south (or south-north) direction. In the southern hemisphere, the left-side wheels and rails are more loaded. However, the forces caused by the Coriolis effect are weak because of the low angular velocity value of the earth's rotation.
When a vehicle moves in a circle (circular motion), the picture of the forces is the same as shown in Figure 05-05. This time, the inertia force has a value of
x x ²
and it is called centrifugal force. In most of the Physics books, there is a kind of confusion because there is no distinction between the inertia force and the reaction force _r. Our Figure 05-05 shows the important difference between them.
We should conclude that the Coriolis force is a kind of inertia force, and as such it is a perfectly newtonian force. This becomes obvious if we consider the elementary inertia forces of an accelerating body. This time, there is no more difference between the force _co and the force _r. It is the reaction force of that force which causes the acceleration. Of course, there must be a supporting structure which ties the elementary unit to the accelerating body. Without this structure, the elementary unit just precipitates until it finds its support. Such a precipitation is sometimes useful in human technology. For a living unit, like a human body, the precipitation of the elementary unit is fatal.

Action of the Coriolis Reaction Force

There is another important difference between the Coriolis and the centrifugal force. The vector of the Coriolis force never passes through the centre of rotation! This means that the Coriolis reaction force influences the angular velocity of the rotating platform.
We can make the following statement. When a body moves towards the axis of a rotating structure, it has the tendency to accelerate the rotation. If it moves away from the axis, then it slows down the rotation. This is a very interesting characteristic of the Coriolis forces. Here are some typical examples.
1. In a well-known experiment, a person sits on a rotating chair and holds weight in his/her hands, as shown in Figure 05-06. Bringing the weights closer to his/her body, the chair accelerates. The school books introduce this fact as the principle of angular momentum conservation. This principle is correct if the whole system is closed (not affected by outside forces). But an acceleration can be caused only by a force and, this time, the force is a Coriolis reaction force.
2. I made an interesting experiment by constructing a kind of windmill which I could call "Coriolis Windmill". In this engine, the air molecules are obliged to move towards the rotating axis, as shown in Figure 05-07. The rotating blades are the supporting structures causing the Coriolis effect. We can construct the running blades with a certain curvature in order to start the rotation. Once the rotation has been started, we can observe a gradual acceleration, which is obviously caused by the Coriolis reaction forces.
3. Another example is the function of some hydraulic turbines with vertical axis, as shown in Figure 05-08. Generally, they are called reaction type turbines. (Sometimes the name "Francis turbine" or "Kaplan turbine" is also used.) In those turbines, the entering water flows in radial direction towards the rotating axis. The running blades deviate the direction of the water flow. This creates inertia forces and reaction forces which rotate the turbine.
  
  Furthermore, the flow in the radial direction also creates the Coriolis effect, and the reaction force _r helps the rotation. The vertical axis reaction turbine works with a relatively high efficiency. It is adapted world-wide in the hydraulic power stations.
4. My last example is in connection with the earth's rotation. The rotating globe of our planet is surrounded by a rotating (or revolving) atmosphere. Near the earth's surface, the air molecules are following the earth's rotation. Molecules, which are not following this regular trajectory, are causing the phenomenon which we call wind. Winds are generally caused by the action of the sunshine and the gravitational forces. Most of the winds are local, and they are blowing in various directions. Sometimes, even human activities can create irregular air molecule movements.
  However, near the equator, there is a steady and very regular wind activity. The air molecules are moving in the direction as shown in Figure 05-09. Near the equator, the molecules are considerably heated, and a hot air molecule has a strong tendency to move upwards. This creates a kind of vacuum, and the gravitational forces are pushing in cooler air molecules from the subtropical regions. This regular wind system is sometimes called "passat wind".
  
  With those winds, huge air masses are continuously moving in a direction away from the earth's axis. Such a movement causes a Coriolis effect, and the reaction force _r brakes the earth's rotation. It was observed that in the high altitudes, there is no such air movement which could compensate for this braking force.
The rotation of the earth and other planets is still a mystery for the science of Physics. Especially delicate is to explain the revolving motion of the atmosphere. Even more mysterious is the fact that the atmosphere of the planet Venus moves in the opposite direction to the globe's rotation.
We are taught in the schools that the rotating earth follows the principle of angular momentum conservation. This principle works only for a perfectly closed system, and such a thing does not exist in the real world. If the action of the sunshine and the gravitational forces can affect the earth's rotation, then the system is no more closed. Therefore, the momentum conservation principle can not be applied.
In another study entitled "The Living Planet Earth", I tried to evaluate the effect of this Coriolis type braking force. I found that the earth's rotation would slow down in such a way that each year, we would have our days 6 minutes longer. Such a slowing down does not exist, and I must conclude that the earth's rotation is "motorised". There must be a Nature-created organ which balances the braking force, and keeps the angular velocity constant.

The Partial Coriolis Forces

I already mentioned that the Coriolis forces caused by the earth's rotation are very weak. When a vehicle moves on a rail which is fixed to the rigid surface of the earth, we can approximately calculate the value of the force with the help of the formula
_co = 2 x x _r x
For example, if a 10 ton locomotive runs in the north-south direction with a velocity of 100 kilometre/hour, then a Coriolis force of about 50 newton will be created. That corresponds to about a 5 kilogramme weight, and such a force is quasi negligible. We should note that this train must run as close as possible to the north or south pole. Near the equator, a vehicle does not create Coriolis forces. (The velocity does not have radial component.)
The situation gets even more delicate if a body moves in the water, or if it flies in the air. The atmosphere is a kind of rotating structure, and it is able to support some forces. Otherwise, the birds and the airplanes could not fly. However, if the force (which causes the Coriolis acceleration) is supported by the air molecules, then this support is imperfect. The situation is similar if a body (or a ship) moves in the water.
Because of this imperfect support, the value
2 x x _r x
can not be reached. We should multiply it by a factor which is a number between 0 (zero) and +1. Then, we should consider the Partial Coriolis Forces. It is regrettable that I could not find a scientific textbook which mentions these obvious natural phenomena.
The partial Coriolis forces are very weak indeed. If we calculate a force of 1 x 10^-6 newton, then it is better to say that the force is quasi nil.
There are numerous movements in the air, which are apparently deviated and, in Physics, the deviation is attributed to the Coriolis forces. Let us mention the most typical example.
In 1851, the French physicist Foucault attached a 67 metre long pendulum on the ceiling of the Pantheon building in Paris. The attached 28 kilogramme mass described, in 24 hours, an amazing trajectory on the floor. This trajectory perfectly proves the rotation of our planet. Surprisingly, this trajectory is attributed to the Coriolis forces in all Physics books which I ever read. This is a scientific blunder. They introduced a kind of apparent Coriolis force by observing the movement in a rotating coordinate system. Without any reason, the apparent force is considered to be real. I call this a real confusion.
Remark: Of course, in the period of time when the pendulum swings in the north-south direction, there is a quasi infinite little partial Coriolis force. The trajectory is very regular, and it shows that this partial Coriolis force does not affect it.
The movement of the Foucault pendulum creates a typical apparent trajectory as observed on our rotating planet.

The Gyrocompass

In spite of the very weak partial Coriolis forces, there is an application which proves that such a force really exist.
Figure 05-10 shows the rotating earth (out of proportion indeed) and a fast running gyroscope. The direction of the gyroscope's axis differs from the direction of the earth's axis by an angle . One part of the molecules in the gyroscope (indicated by ) are moving towards the earth's axis, and another part (indicated by ) are moving away from it. Two partial Coriolis forces _co are created, and the accelerating force is supported by the revolving air molecules of the atmosphere. Furthermore, there is a couple
_co x
which acts on the gyroscope's axis. This couple vanishes only when the two axes are in the same direction.
I tried to calculate the forces _co. Suppose that the gyroscope is a kind of electric motor with a rotor of 2 kilogramme mass. Its diameter is 0.2 metre, and it runs with 20000 revolutions/minute. I also assumed a 0.5 factor for the partial Coriolis force effect. I found that the force _co is about 0.015 newton, which corresponds to about 0.0015 kilogramme weight. Such a force is able to move the gyroscope's axis if it is mounted in precisely working gimbal rings. (Generally, they use bearings working in liquid mercury.)
It is obvious that a gyrocompass works only in the revolving atmosphere near the earth's surface. It does not work in the very high altitudes or in a vacuum. It is a very useful instrument for the navigation.
I can also say that if we want to explain the function of the gyrocompass with the help of the apparent Coriolis forces, then the mathematical equations are so complicated that nobody can understand them.

Conclusion

All along in this research paper, I tried to explain how important it is to find the supporting structures when we deal with the phenomena of forces. I am convinced that all the forces of the universe must have supporting structures, and they must follow Newton's action-reaction principle.
It is very much regrettable that physicists never tried to find the supporting structures for the gravitational, electrostatic and magnetic forces.
I made huge efforts trying to introduce and explain the supporting structures for those "fundamental" forces. Without this knowledge, we will never understand how Nature works.
Furthermore, in the schools and universities, we indoctrinate the future generation with false scientific principles. The Coriolis force problem is just an example of this.
A confused natural science is very harmful to the whole humankind.

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