How much thought have you ever given to gyroscopes? A gyroscope is a device that is made to operate by spinning a mass (generally, a round slightly tubular or wheel-like object) at a constant speed on a stable axis. We use them for a number of items: toy tops (of course), flywheels, centrifuges, bicycle and motorcycle wheels, but most importantly, as inertial guidance systems.
A rocket, or any system undergoing linear acceleration or even cruising at a constant velocity in general space, cannot rely on gravity for determining its orientation. In many macroscopic moving systems (ships, trains and automobiles) we count on the gravity of the Earth to provide the basis, the foundation, of our measurements. This is less obvious in a car (which is designed to operate only in a system dominated by gravity and having a hard surface to operate on), but more so in a craft such as an airplane or a submarine. These latter vehicles need to control their relation to the Earth, and thereby need to know their angular relationship to gravity and the surface, to navigate and avoid crashing. And typically, they cannot determine this relationship by mere visual observation, especially if that relationship needs to be known with some precision.
In the earlier days, this was done by utilizing a bubble in a chamber of fluid. Due to the difference between the density of a gas (air) and a fluid (generally, oil), a bubble introduced into a chamber that is otherwise filled with fluid can easily provide that measurement. This is the basic principle behind a ‘level’ which has a small, slightly curved, tubular vial with markings equidistant from the center; that contains oil and a small bubble. Since the bubble, because of gravity, always goes to the very highest point, this device can show the angular relation to the surface of the source of the gravity, that is, the Earth. This device works well for systems at rest, like for building houses and setting up surveying equipment, but not so well for things that are accelerating, especially if that acceleration is varying with time (like in a rocket or a torpedo).
So it was that the first rocket pioneer, the famous Dr. Robert Goddard, came up with an outstanding, and frankly, underappreciated idea. Use a gyroscope as the basis for your navigation system. Since a gyroscope resists change in its orientation (to the gravitational field, or is it space in general?) regardless of the outside conditions in regard to velocity or acceleration, it can be used to provide a reference for the determination of the craft’s orientation and thereby, to provide system (inertial) guidance.
What a wonderful idea! This is a much more accurate and universal tool than a compass, the singular device that allowed exploration of the planet to a degree undreamed of at the time that it was first invented. A gyroscope not only allows you to know your orientation to the planet and the earth beneath your feet, but can be used as a constant reference anywhere in the known universe! (As far as we know, anyway, because, as mentioned in the last chapter, the only space that we can truly know is one that we live in which is within the gravitational field created by the Sun. It will be a long time before we ever penetrate interstellar space and can definitively determine that a gyroscope works in the complete absence of gravity, but all of the evidence accumulated to date indicates that it will.)
The gyroscope is the space compass, the tool that allows you to carefully navigate your spacecraft and to get where you have intended to go, without instructions from home. And like the early explorers and map makers of the Earth, this tool has set us free, and provides the measurement that allows us to explore the universe. This, quite simply, would not be possible without the gyroscope.
And it was Dr. Goddard who gave us this marvelous invention. It was the gyroscope, as developed by the Germans under Dr. Werner Von Braun that allowed the V2 Rockets to be launched against England at the close of WW II. It was a gyroscope, developed by the Americans under Dr. Werner Von Braun (do you see a pattern here? Dr. Goddard died in 1945, and never got to help with this task) that allowed the launch and control of a successful manned mission (round trip!) to the moon in the late 1960’s. Gyroscopes are now an integral part of every non-land based vehicle known to man, and without them, frankly, we just couldn’t get around.
And yet, Dr. Goddard is mostly recognized for his fundamental work on rockets, which, while important, does not rise to the level of his truly unique invention: gyroscopic inertial guidance.
So why all this talk about gyroscopes in the section on Dimension?
Because a gyroscope perfectly illustrates the characteristics of the Rotational Dimension, and we can use it to understand and describe them.
The Rotational Dimension is very different from the Linear Dimension. In it, every velocity creates acceleration; the two are never separate entities like they are in the Linear Dimension. Weight can change. Gravity exhibits an inverse curve and inertia is a variable. Oh, and it has an inside character and an outside one. Let’s start with the inside, and work our way out.
We’ll start by putting Dr. Einstein back in his little elevator, and start whirling him around as in the illustration below. This time, we’ll make the radius of rotation sufficiently large so that he doesn’t notice the edge effects that he did in the previous chapter. We’ll spin him in deep space so that the rate will create an internal gravity that is just equal to that of Earth, and the radius of the rotation shall remain constant.
Now, we’ll ever so slightly increase the speed of rotation until it is twice what it was before. As you can well imagine, poor Dr. Einstein has fallen on the floor, and can’t get up. Gravity in the elevator has increased by a factor of four.
Since the gravitational effects of this situation are felt only inside the system, and are associated with the velocity, physicists call this a ‘fictitious force’. Tell that to poor Dr. Einstein. The formula for the force created on the inner surface of a gyroscopic system is:
F = mω2R
Where: F = the force created
ω = the rate of the rotation, in radians
R = the radius of the arm
This ‘fictitious force’ stresses the entire system, and puts everything in tension. As the outer band gets ‘heavier’ it puts an increasing ‘stretching’ force on the spokes or the structure that restrains the outer band. The band itself also feels this force in the form of an attempted stretching of its outer diameter, since the mass wants to fly out, away from the center. Engineers call this tension. The whole system is put in tension by the force (gravity) that is directed radially, away from the center. It is also perpendicular to both the line of motion and that of the disc and the surface that helps to create it.
But in this case, as opposed to one in the Linear Dimension, the force on the surface and the line of motion are perpendicular to each other.
Rotational motion, as long as it is associated with a discernible radius, is, to use the old vernacular, 2-D, since a circle can and does describe a plane.
Things are quite different on the outside.
Outside of the rotating system, none of this force is felt. The gyroscope does not get any heavier to the static observer outside. Dr. Einstein’s experience with the crushing internal forces of the system has not affected the overall static mass of the gyroscope, whatsoever.
And yet, there is a difference to the outside between the rotating and the non-rotating system, and that has to do with inertia.
A rotating system has more inertia than a non-rotating one, and not just in the Rotational Dimension. It exhibits this characteristic in the Linear Dimension as well. And it is not just that it has more inertia, which it certainly does, but it is directional inertia.
Inertia is the quality of an object that makes it want to continue at its current velocity, or conversely, its resistance to change. Inertial mass and gravitational mass have been shown to be so close to the same thing as to be indistinguishable, but that doesn’t take the gyroscopic situation into consideration.
It is the rotational inertia, and its specific character, or quality, that makes a gyroscope useful, and also uniquely distinguishes it from the other velocity dimensions.
Get yourself a gyroscope. Start it up in an orientation where the axis is horizontal, parallel to the surface of the Earth. Now get a string with a loop on one end, and put the end of the gyroscope axis into the loop and hang the gyroscope from that end. A miracle occurs! Although totally overbalanced, the gyroscope does not fall down, but instead begins to slowly rotate around the supporting string.
As illustrated above, the string will be almost vertical. This is a cute trick, and everyone has probably seen this, or at least a variation of this using a stand, done it and watched the gyro supported only on one end, spinning around the support point. Physicists have a description for this phenomenon. They say that the precession (we’ll talk a little more about this in a few minutes) of the gyroscope works in opposition to the force of gravity, and that the cross product of the rotation around the support point and the precession satisfies the equation of motion and allows the gyro to stay horizontal.
The equation for this is as follows (from Wikipedia):
The torque-free precession rate of an object with an axis of symmetry, such as a disk, spinning about an axis not aligned with that axis of symmetry can be calculated as follows:
where is the precession rate, is the spin rate about the axis of symmetry, is the angle between the axis of symmetry and the axis about which it precesses, is the moment of inertia about the axis of symmetry, and is moment of inertia about either of the other two perpendicular principal axes. They should be the same, due to the symmetry of the disk.
Why doesn’t the gyro fall down? The slow rate of rotation about the support should not be able to counter the force of gravity, which acts instantaneously, at all times. Furthermore, it you’ve done this by supporting the gyroscope with a string, you can see that the string is almost vertical. It’s not skewed to one side like it is carrying an unbalanced load.
Where is the center of gravity of the gyro from the perspective of the string (or the support stand)?
The answer is: that the center of gravity is (and must be) at the contact point, where the end of the shaft meets the support. The string cannot tell that the gyro is internally rotating – that motion is invisible to it. It doesn’t have any more mass than a non-spinning gyro. All that the string knows is that there is an almost balanced weight on the end. From the string’s perspective, the situation where the gyro is standing out horizontally is equivalent to hanging a non-rotating gyro in an upside down condition, as shown below.
The gyroscope should fall over. But it doesn’t.
But it can be even weirder than that. Set up the gyro again, hanging on a string. Then hang a separate weight on the unsupported end.
The gyro (if it has sufficient speed) still will not fall down! Clearly, the system is overbalanced. The added weight is only connected to the system by a string, and serves to add more torque to the system. And the string supporting the gyro still hangs vertically. Where is the center of gravity for this system?
If one is observing the situation from the perspective of the string, it would appear that the axis of the gyro has zero length, and that the weight of the gyro and the added mass are hanging directly below.
There is only one logical conclusion that one can reach from this example, and this statement pretty much puts the author at odds with the present theories of physics.
The center of gravity has changed, and is no longer a point. It is an axis.
It doesn’t really matter how long the shaft (axis) of the gyro is (except that, because of the force of gravity acting on it, there is a minimum threshold for the internal rate of rotation). As long as the center shaft is rigid and a part of the rotating system, the entire shaft is the center of gravity, and the gyro will react to any external force applied to it at any point on that shaft.
This is why a gyroscope works so well as an inertial guidance reference. It has an inertial vector, not a point like most common (non-rotating) objects do. Since the whole axis is the center of gravity, and since it has length, the overall rotating system can have torque; and can therefore resist any change in direction
In the Rotational Dimension, the center of gravity of an object is its axis, within the bounds of the system.
And this axis is simultaneously perpendicular to the line of motion (the rotational velocity vector), the line of action of the acceleration (the centrifugal force) and the surface of the rotating mass.
On a table are two apparently identical boxes, each having the same mass. Box One is solid having a uniform density. Box Two contains the most perfect gyroscope ever, that has no vibration or friction, and that maintains a constant rotational speed. It is oriented so that its axis of rotation is vertical. You, the experimenter, have no experience with gyroscopes and wouldn’t know one if you saw it. You are asked to experiment with the boxes and determine if they are the same, or different, and if different, how.
You start by picking each of the boxes up, in turn. You’ll notice that box number two seems a little heavier when you first start to pick it up, but not any heavier when you hold it. You weigh each. They weigh the same. You try picking each up and twisting them around the vertical axis that runs through the center of each. No difference.
You decide to try rotating each one about the other axes. Whoa! That freaking box number two almost practically took off! Not only was it harder to rotate in those other directions but the whole time that it being rotated it tried to push back and veer off 900 from the path that you intended. When brought back to a standstill however, it acted just like Box One. What is going on with this thing?
After these tests, and others, you would have to conclude that while the boxes each shared a number of characteristics, like weight and rotational inertia in the plane of the table, Box Two had another special sort of inertia that only is exhibited when that box is accelerated in any direction except ‘twist’, and that the ‘special inertia’ exhibits torque as a reaction to the applied movements and force.
This ‘special inertia’ only comes into play when Box Two is accelerated or changes orientation, and seems to have a selective quality, that is, it seems to know what movement is being imposed, and only reacts differently from Box One when motions in particular directions are implemented. The special inertia must be both directional, and present only in cases having some sort of acceleration.
This is a very curious feature of the Rotational Dimension. But there is one more, as we have alluded to in the example of the gyroscope, and it is called: Precession.
Precession is the (spooky) tendency for a gyroscope to exhibit a perpendicular reaction to an applied off axis force. Before moving on to other discussions, a little better definition of Precession is required.
One of Sir Isaac Newton’s most famous and fundamental observations is: “For every action there is an equal but opposite reaction”. Even if you are not a physicist or an engineer, you’ve probably heard that before. This was a very important basic assumption, in many respects similar to his observations regarding gravity. Gravity was around a long time before Newton, and many of its aspects had been explored by the researchers and scientists before him. But it was not until Newton used his keen observational powers to record, describe and quantify this force, that people could truly understand how it acted (we still don’t know what causes it, though). And so it is with the previous statement, which came to be known as Newton’s Third Law of Motion.
Newton’s Third Law of Motion, like his theory of gravity, quantifies and details effects that were known and accounted for, but never accurately described. A good example of the Third Law is the recoil that results from a cannon when it is fired. Ship builders and army generals had known for years that when using cannons, the weapon ‘kicks back’ when the shot is fired, just like the butt of a rifle will to your shoulder. For this reason, shipbuilders chained their heavy cannons to the ship’s framework, or anchored them securely to the deck for use. Modern ship weaponry has automatic recoil absorbers built into the weapon itself to avoid impacting the ship and its trajectory when its big guns are fired. Similarly, the mobile weaponry of the army needs to be anchored before use.
And while people knew about this effect, before Newton, it was not quantified and the force imparted back to the weapon could not be calculated or fully understood.
But Sir Isaac was not thinking about gyroscopes when he wrote the Third Law, because if he had been, he would have added a qualifier at the opening that said: “For every action there is an equal but opposite reaction in the Linear Dimension,.”
Because, in the Rotational Dimension, this is obviously not the case.
In the Rotational Dimension, any action out of the plane of rotation creates an equal but perpendicular reaction, and we call this reaction, Precession.
Precession is not as intuitively obvious as Newton’s Third Law, because we don’t have as much direct experience with rotating things as we do with things operating primarily in the Linear Dimension, but a good common experience example comes from a bicycle. A bicycle provides a wonderful demonstration of all of the principles that have been discussed in this chapter. Those big rimmed wheels make very effective gyroscopes, that frankly, make staying upright possible (I know, you figured that you could ride one based on your skill alone). The resistance to out of plane movement provided to the bicycle frame by the shaft of the wheels is the major stabilizing force that allows you to ride. It’s so effective that a rider-less bicycle whose steering is locked can travel a good distance without falling over, the travel distance depending mostly on the straightness of the start and the velocity.
But if you are riding a bike and you attempt to turn the front wheel, something mysterious happens: you automatically lean into the turn. This does not happen because you intuitively lean for better balance or traction (although this is undeniably the effect), you lean because of the Precession created by turning the shaft that restrains the bicycle wheel. It is the wheel that does the leaning as a result of the force (the turning of the handlebars) that has been applied to it, and it takes you and the rest of the bike with it. Conversely, if you are riding with ‘no hands’, that is, balanced only on the seat without holding the handlebars (I know that your mother told you never to do this) and you lean to one side, the front wheel will pivot in the direction that you lean. This too is Precession, and it is equal and perpendicular to the applied force.
It is for all of these reasons, and actually a few more that we’ll discuss later, that Rotation should be considered as a separate dimension. It not only has a form that is completely different and distinct from Linear motion, but it operates under its own separate rules, with different lines of action and reaction.
In the Rotational Dimension:
- Velocity (even constant velocity) always creates Acceleration, (unlike in the Linear Dimension where the velocity of the system does not affect the objects within it)
- Weight within the system, and only within the system, can vary, and depends on the specific character of the system.
- Inertia is relative, axial (not point like) and directional.
- The line of action of the acceleration is perpendicular to the velocity (versus parallel in the Linear Dimension).
- The line of the inertia is perpendicular to the plane of the rotation
- And as a rotating system interfaces with the larger world, every action creates an proportional and perpendicular reaction.
But before moving on to the next topic, there is one more very important point to emphasize.
In the Rotational Dimension, there is no required linear component. That does not mean that the linear aspects of the system do not affect how it will behave, or the characteristics that it will have. It means that the motion itself, and the interactions that characterize its interface with the larger universe depend primarily on the rotational aspects of the system. There is no theoretical lower limit to the diameter of the rotational system, excepting the limit of existence itself, somewhere on the order of the Planck Length, which is defined as being 10-35 meters.
From the content in these last few chapters, you can begin to see why it is both meaningful and important to parse our descriptions of the behavior of matter by the velocities that dominate their state of being. The laws that govern the Rotational dimension are different from those for the Linear dimension. In the Rotational dimension, the properties of the bodies of matter are primarily dependent on the magnitude of the rotation, the rotational velocity of the body, quite unlike the properties that they have when obeying the rules of the Linear.
My real breakthrough in this regard came about when I finally realized that in the Rotational dimension, every action creates a proportional and perpendicular reaction. Newton’s Third Law of Motion turned sideways. While I had come to understand beforehand that the natures of these velocities were sufficiently distinct to allow them independently defined, until I really began to analyze their characteristics, I didn’t fully comprehend that each velocity creates different rules that control the interactions of matter and space.
But clearly, they do.
We’re going to explore those thoughts and more of the differences between the properties of the dimensional velocities further in the next few chapters, along with a related but somewhat off track discussion about phi, the Golden Ratio and finally, a discussion about hurricanes and Coriolis.
It still seems somewhat strange to me that, starting from a completely different perspective, we still ended up with three, and only three dimensions (velocities). I could be missing something, but as far as I’ve been able to discern so far, the three described in this book can summarize all motion.
But we’ll talk more about that later. Go play with your gyroscope.