The overall effect will be to a change in speed ball and / or inclination of Axis of rotation. Will depend on the side surface of the sphere which will apply the moment of forces.
If such forces is applied so that its result is parallel to the rotation, in this case only vary the speed of rotation of the sphere.
Conversely, if the result of the moment of forces formed a certain angle with the axis of rotation, you have the either the inclination of a of the rotation ball axis either a change in its speed of rotation. And again, if the result of the moment of forces were applied orthogonal to the axis of rotation, we would have a sole effect, the tilt of the rotation axis. And this is not a fantasy, Physics says it.
The movement of bodies in rotation is governed by the second law of dynamics applied to extended bodies, which is:
M = L '
the apex ' is to indicate the derivative (ie the change) compared to time. This equation means that the sum of moments of external forces M causes a proportional variation of angular momentum L. In this equation all sizes, except the moment of inertia I, are magnitudes vector, that is equipped with intensity (form), to and direction.
In a sphere that rotates around a particular axis (the main axis of inertia), the angular momentum L is parallel to the rotation and has a magnitude that is equal to Iw.
-- I is the moment of inertia and its value depends on the mass of the body and how this mass is distributed around the axis of rotation. It expresses the "resistance" of the body to a rotation. For a sphere of mass m, radius R, the moment of inertia applies 3/5mR2.
-- w is the angular velocity in radians per second and measure the angle that is covered by the rotation of the body in a second.
Imagine having a homogeneous sphere that rotates but not traslate around an axis, without having to act any moment of forces: in this case its angular momentum L is constant, and thus the sphere rotate with constant angular velocity. (drawings by ing. Joseph Marino, from Savoia di Lucania)
ball that spins around its axis with a constant angular velocity, wAt a certain time we apply a moment of M. If M is parallel or antiparallel to L, the only change that will be is a change in the magnitude of vector L, so the ball will continue to rotate around the same axis but w will change in magnitude, namely related to the the Earth it mean that the days are longer or shorter depending on the angular velocity decreases or increases).
on the ball a moment of forces parallel …
and / or antiparallel to axis of rotation is determined, in which case not vary the tilt of the rotation but his angular velocity, wIf M is perpendicular to L there is a constant rotation of the direction of L and thus of the rotation, without change of the magnitude of angular momentum, so the angular velocity around the rotation axis will remain the same, but the axis will no longer be in the starting position (the Earth would continue to have the same length of day of today but with hours of light of different duration and also would vary the impact of solar heating, and this will depend on what is the inclination of varied axis of rotation).
on the ball rotation a moment M of forces orthogonal to the axis of rotation is determined …
…which cause the tilt of the rotation axis around which the ball continues to spin with the same angular velocity wFor inclinations of M compared to L intermediate to the previous cases, you have a combined effect of variation of the rotation angle and of angular velocity, since the component of M parallel to L cause changes in the magnitude of w, while the orthogonal component to L a change of direction of rotation Axis (on Earth would change both the duration of the day that daylight hours of the concerned emisphere and therefore the impact of global warming due to sunlight).
if the momentum M of forces applies on the intermediate surface of the sphere, so that forming a certain generic angle with the axis of rotation…
its components act simultaneously, determining an inclination of the rotation axis and a change in angular velocity wWhen the moment of forces ceases to act, the angular momentum remains unchanged in magnitude and direction and the ball (and then the Earth) will remain to rotate around an axis different from that of departure. The starting assumption was that on the sphere external forces act, and this, related to the the Earth, would mean that those forces must necessarily come from space. Question, we must hope that a giant meteorite impacts with the Earth and cause the displacement of the earth? But neither in your dreams! We have seen first as the earthquake in Sumatra has produced a variation of the inclination of rotation axis of the Earth.
A change in the distribution of masses produced by the earthquake has changed the moment of inertia. But L must remain constant, not acting on Earth any external force. Since L = Iw , the effect of the change of I was offset by the change of w in magnitude and to and including removal of axis of rotation. An equivalent interpretation of the phenomenon can be given in terms of interactions between subsystems. Let me explain better.
In any Cabinet of physics of scientific schools you can do a little experiment. Take a rotating stool. Let sit on this stool a student who supports a bicycle wheel in motion with some speed. After the student rotate 180 ° the axis of the wheel.(drawings by Angelica Caggianese, from Savoia di Lucania)
What happens? The student and the stool (which were until then still) take a turn with a certain speed. What happened? Since the system (student stool) + wheel is isolated, namely M of external forces = 0, follows that system L = constant. Since then L system must be unchanged after the wheel was overturned on, the student stool turns in the opposite direction towards the rotation of the wheel (and at speeds twice that of wheel) to compensate for the variation of L system induced by this one.
The phenomenon can be analyzed in an equivalent manner even splitting the system (student stool) + wheel in two subsystems: (student stool) and wheel. The two subsystems are not isolated. The student, in turn axis of the wheel, holding a moment of Ms force on the wheel, and for the principle of action and reaction, the wheel exerts a moment Mr, equal and opposite on the student, forcing it to rotate. Ms and Mr moments are considered external, that is produced by a subsystem on the other.
Question: if the student was in rotation on the stool and instead of bicycle wheel had a rotating flywheel (in this case consists of a mass-shaped metal ring, in fact, a gyro) rotating around an axis perpendicular to the system student stool and, suddenly, with his hands the student activated the brake abruptly to block the rotation of badminton, what would happen? The system student stool would continue to turn but the student would no longer be sitting comfortably, but tilted.
The magnitude of angular momentum would not change (the system would continue to rotate as the first with the same speed) but would change the direction of which continues to rotate because this would in the meantime changed its initial position, sloping. The system student stool is equivalent to a sphere in rotation. Certainly a sphere with uneven mass, but the principle is the same.
The Earth planet is not a perfect sphere but a geoide, with a difference of about 45 km diameter measured between the equator and that measured at the poles. The Earth also consists of a thin crust and a huge mass "pasty." In the event of impact with an asteroid significant in scale, after the enormous destruction that would follow, there would be over time the effect produced by the new position available under Earth's tilt. If the Earth were a rigid moving mass the precession movement of its axis would take it back, over time, in its initial position. Our planet has an uneven nor rigid mass, and once the location of its axis of rotation varied, it tend to remain in the new conditions. Whatever artifice put in by man to change the angle of inclination of the Earth's rotation should reckon with an experimental stage, the computer simulation. This is because the laws of dynamics that we know refer to bodies with mass homogeneous. It 'true that the Earth's crust is very thin and therefore size negligible compared to internal dimensions of the planet, yet because of the "viscosity" of the inner mass, a simulation must be done. This is because we can not calculate easily as the laws of dynamic acting on a ball rolling mass smooth and uneven. Not only that, the simulation should take into account that the Earth is a "ball" that not only rotates on itself but is also subject to translation and that interacts with the gravity system of the Sun and the Moon. Then the simulation should take this into consideration, especially the Moon has so much influence on the stability of Earth's rotation.
In fact, the gravitational force of the moon stabilizes the slope of the earth, which varies by only 1.3 degrees around a mean value of 23.3 degrees.
Thanks to the stabilizing of the Moon, even catastrophic events such as the impact of our planet with a large meteorite could not move the Earth's axis: could be that only an impact with a body of almost planetary dimensions.
People will say: but then the axis will move or not? Be patient, move, move!
We have said that the Earth is a sphere that almost always turns on its axis and that traslate around the sun.
What would happen if on a sphere rotating on its axis is applied a moment of forces direct into a general direction? And above all, how do you create a moment of forces that generate such a strong result that they can affect the direction of rotation, rotation speed, or both simultaneously? We said that if this time the forces applied so that its results were parallel to the rotation, in this case only vary the speed of rotation of the sphere (in technical terms only the magnitude of vary). Conversely, if the result of the momentum of forces form an angle with the axis of rotation, it cause either the inclination of the rotation axis of the ball either a change in its speed of rotation.
And again, if the moment of forces were applied orthogonal to the axis of rotation, we would simply tilt the angle of the rotation, as already mentioned above. All this means that if we apply the forces artificially on the surface of the Earth, depending on where it realizes, we have a variation in the speed of rotation, the variation of the inclination of earth’s axis or both phenomena. You can create the moment by putting loads of vehicles on a large surface in the shape of the circle (comparable to "flyweel" in rotation in the experiment of the student stool + flyweel mentioned above). There are so many camions and tir and if necessary the worldwide car industry would be able to build the right number in a very short period of time. Once in motion “the flywheel tir" acquire an angular momentum directly proportional to the speed with which moves and the moment of inertia I determined by the mass of vehicles (which is known) and their distance from the centre of the "flywheel" . This angular momentum would be transferred to Earth as it would produce a moment of "acceleration" forces that act by the "driving force of tir" to the Earth subsystem, by tilting the Earth's axis in a certain direction. If at some point, after a certain time, the "flywheel of tir" in circular motion on the surface of the "sphere" Earth was stopped, it transfer its angular momentum to the Earth in rotation since it would produce a momentum of forces opposed to the previous ( "deceleration"), bringing the axis in the original direction.
Remember the previous example of the student on the stool and bicycle wheel or flywheel? Well, in parallel with the case of the student on the stool, it were the same as if the student put the rotation of flywhhel and stop it, sloping in one direction when the flywheel accelerate and in the opposite direction when it stop, remaining to rotate in the original direction because the momentum acting on him is zero, being the sum of two opposites ones.
Question: What happens if the truck, during their acceleration, lose mass (i.e. losing the material with which they were loaded…) and are holding back when they are empty? Well, the moments of acceleration and deceleration would be different, their sum would not be zero and therefore the total force transmitted by the "flywheel of tir" to the Earth will cause the tilt of axis in a certain direction!
In the student case, it is as if he had speeded up a flywheel and had braked a mass different one.
Let us return now to tir: Even if vehicles could be united with each other to form a single homogeneous mass rotating there would be the problem of arrest them all simultaneously, but it is a problem not insurmountable. In any case, the time it takes to stop the truck in the rotation could be much less than that used to accelerate them, allocateing the tirs with an adequate equipment of brakes.
The effect produced by their rotary motion would depend on which side of the surface of Earth would be interested by the location of "flywheel to tir" (made up of tens of thousands of vehicles). From a purely theoretical reason, according to the second cardinal equation of mechanics, that time can cause both the tilt of the Earth's direction axis (the known 23.27 °) and the variation of rotating speed, because in general has a component orthogonal to the axis and a component parallel or antiparallel to the direction of rotation (depending on which side of the planet will operate the truck in a circle).
Example: imagine for simplicity that the axis of rotation of the earth is vertical; Well if the "driving force of tir" was placed on the equator, it would produce only a momentum orthogonal to the axis of rotation and thus only the variation of the rotation angle of the earth’s axis.
If the truck in a circle were made to act in a generic area in the north or south, that we would have a momentum acting on Earth directed in a direction where the generic components (parallel and orthogonal to the axis) would give life to the inclination of the axis of rotation and a variation in the speed of rotation. In the actual case being the axis of rotation inclined to 23.27 °, also a moment applied perpendicularly the equator would not be perpendicular to the direction of axis, but would also have a component parallel to it. Then the condition of squareness of the momentum produced by the flywheel of tir on Earth would be roughly in the Earth's surface between the equator and the Tropic of Cancer and / or Capricorn.
Obviously the diameter of the flywheel of tir, which has to produce the momentum needed to obtain the desired inclination of axis, is easy to calculate. Just know the moment of inertia, mass and angular velocity of the Earth, some astronomical parameters and all that.
In summary, everything (in the laboratory reference system) is based on the conservation of angular momentum L of a system when the total external forces and total M is zero: if the flywheel is blocked, the system stool + student buys an angular momentum ( which is the axis tilt system stool + student) that L of the whole system stool+ student + fly remains constant.
With regard to tir, (on the surface of the Earth) probably should be kept under control the effect that the truck produce when moving, along the meridians, and this why we are particularly interested in seeing the torque of internal forces to study their effect on Earth subsystem.
Because the effect is perceptible angular momentum of two subsystems Earth and tir must be comparable, for example, the ratio of the masses must not be too small, even if in principle an effect it might be still despite one of two subsystems is much more "light" of the other.
In "mass variable tirs" the mass deportation was part of the overall system and helped to determine the initial angular momentum. While in the first version you put in motion the "flywheel tir" and their next stop does not cause any change in overall (the system regains the angular momentum possessed before the truck begin to move), now the total angular momentum of the Earth + tir + expelled mass mass remains constant, while that of subsystem Earth + tir varies. The latter is the subsystem that interests us and therefore should be studied the forces exerted on it from the rest of the system (mass deportation).
To model all this would be quite long and complex, and other parameters should be set.
Put simply: the Earth rotates on itself, I call axis Z0 the axis NS which is also the axis of rotation.
I have a row of truck that move along a meridian (if they move along a parallel, the only effects might affect the magnitude of speed but not its direction).
To take account of the rotating system we must resort to Euler equations that describe the change in angular momentum in the reference system solidarity of the Earth. If we call x-axis orthogonal to z contained in the plan of the meridian, the moments of inertia of (Earth + tir) compared with axes x and z, Ix and Iz, coincide and are slightly smaller than Iy.
(The forces that carry between Earth and tirs have obvious component tangential to the meridian, but as the Earth rotates on itself the truck,to keep their route along the meridian, must also exercise a force parallel to the tangent, to counter the Coriolis force.)
The expulsion of the masses (at a rate dm / dt) can be described by taking an equal "push" av (speed of expulsion) for dm / dt, and directed tangentially to the meridian.
If all pushed agree (each tir throws away its cargo in the same side), twisting momentums due to all the push cause a total momentum direct along the y-axis (Ny) that vary the angular momentum of the subsystem Earth + tir along the same axis.
The Euler equations become in this case:
which mean precisely (inter alia) that the y component of angular velocity, initially nothing, grows over time (and consequently the same thing occur for the y component of angular momentum) ... with an incredible result .... tilt of the Earth's axis.
But abruptly changing the axis of rotation what happens to our homes, mountains, lakes, oceans etc? If you do it gradually, on next steps, "nothing" happen, see the earthquake in Sumatra (not tzumami). In reality, what may occur, since the composition of Earth uneven is the "slippage" of hard crust on the underlying pasty mass, It would thus a shifting of seismic faults that could act as happens when you experience the great earthquakes on the movement of the earth.
A simulation is prior, as is often quite necessary.
If we were to resort to geoingegneria mean that humanity has come to its end.
But perhaps my daughter Laura ( among the five winners of the Student Contest competition in the International Year of Planet Earth promoted by the United Nations) is right, the humanity can find the solution only rediscovering its roots…
Nessun commento:
Posta un commento