I just want to announce that the lecture I give today is unlike the others, in that I’ll talk about a large number of subjects which are only for your own entertainment and interest, and if you don’t understand something because it’s too complicated, you can just forget about it; it’s absolutely unimportant.

Every subject that we study could, of course, be studied in greater and greater detail—certainly in greater detail than would be warranted for a first approach—and we could continue to pursue the problems of rotational dynamics almost forever, but then we wouldn’t have time to learn much else about physics. So we’re going to take leave of the subject here.

Now, someday you may want to return to rotational dynamics, each in your own way, whether as a mechanical engineer, or an astronomer worrying about the spinning stars, or in quantum mechanics (you have rotation in quantum mechanics)—however it comes back to you again, that’s up to you. But this is the first time that we will leave a subject unfinished; we have a lot of broken ideas, or threads of ideas, that go out and aren’t continued, and I’d like to tell you where they go, so that you get some better appreciation of what you know.

In particular, most of the lectures up until now have been, to a large extent, theoretical—full of equations, and so on—and many of you with an interest in practical engineering may be longing to see a few instances of the “cleverness of man” in making use of some of these effects. If that’s so, our subject today is ideally suited to delight you, because there’s nothing more exquisite in mechanical engineering than the practical development of inertial guidance over the last few years.

This was dramatically illustrated by the voyage of the submarine Nautilus under the polar ice cap: no stars could be observed; maps of the bottom of the sea, under the ice cap, were practically non-existent; inside the ship there was no way to see where you were—and nevertheless they knew at any moment exactly where they were.¹ The trip would have been impossible without the development of inertial guidance, and I would like to explain to you today how it works. But before I get to that, it will be better if I explain a few of the older, less sensitive devices in order for you to more fully appreciate the principles and problems involved in the more delicate and marvelous developments that came later.

In case you haven’t seen one of these things, Figure 4-1 shows a demonstration gyroscope, set in gimbals.

Once the wheel is set spinning it stays in the same orientation even if the base is picked up and moved around in an arbitrary direction—the gyroscope remains with its spin axis, AB, fixed in space. For practical applications, where the gyro must be kept spinning, a small motor is used to compensate for friction in the gyro’s pivots.

If you try to change the direction of axis AB by pushing downward on point A (creating a torque on the gyro around axis XY), point A does not move downward but actually moves sideways, towards Y in Figure 4-1. Applying a torque to the gyro around any axis (other than the spin axis) produces a rotation of the gyro around an axis that is mutually perpendicular to the applied torque and the gyro’s spin axis.

I start with the simplest possible application of a gyroscope: if it’s in an airplane which is turning from one direction to another, the gyro’s axis of rotation—set horizontally, for example—stays pointed in the same direction. This is very useful: as the airplane goes through various motions, you can maintain a direction—it’s called a directional gyro. (See Fig. 4-2.)

You say, “That is like a compass.”

It is not like a compass, because it doesn’t seek north. It’s used like this: when the airplane is on the ground, you calibrate the magnetic compass and use it to set the gyro’s axis in some direction, say north. Then as you fly around, the gyro maintains its orientation, and so you can always use it to find north.

“Why not just use the magnetic compass?”

It’s very difficult to use a magnetic compass in an airplane because the needle swings and dips from the motion, and there’s iron and other sources of magnetic fields in the airplane.

On the other hand, when the airplane quiets down and goes in a straight line for a while, you’ll find that the gyro doesn’t point north anymore, because of friction in the gimbals. The airplane has been turning, slowly, and there has been friction, small torques have been generated, the gyro has had precessional motions, and it is no longer pointing in exactly the same direction. So, from time to time it’s necessary for the pilot to reset his directional gyro against the compass—every hour, or perhaps more, depending on how perfectly frictionless the thing is made.

The same system works with the artificial horizon, a device to determine “up.” When you’re on the ground, you set a gyro with its axis vertical. Then you go up in the air, and the airplane pitches and rolls; the gyro maintains its vertical orientation, but it also needs to be reset every once in a while.

What can we check the artificial horizon against?

We could use gravity to find out which way is up, but as you can well appreciate, when you’re going in a curve, the apparent gravity is off at an angle, and it’s not so easy to check. But in the long run, on average, the gravity is in a certain direction—unless the airplane ultimately ends up flying upside down! (See Fig. 4-3.)

And so, consider what would happen if we added a weight to the gimbals at point A of the gyro shown in Figure 4-1, then set the gyro spinning with its axis vertical, and $A$ down. When the plane flies straight and level, the weight pulls straight down which tends to keep the spin axis vertical. As the airplane goes around a corner, the weight tries to pull the axis off vertical, but the gyroscope resists through the precession and the axis drifts away from vertical only very slowly. Eventually the airplane stops its maneuver, and the weight pulls straight down again. In the long run, on average, the weight tends to orient the axis of the gyro in the direction of gravity. This is much like the comparison of the directional gyro to the magnetic compass, except instead of being done every hour or so, it’s done perpetually, all during the flight, so that in spite of the gyro’s tendency to drift very slowly, its orientation is maintained by the average effect of gravity over long periods of time. The slower the gyro drifts, naturally, the longer the period of time over which this average is effectively taken, and the better the instrument is for more complex maneuvers. It’s not unusual to make maneuvers in an airplane that throw gravity off for half a minute, so if the averaging period were only half a minute, the artificial horizon wouldn’t work right.

The devices I have just described—the artificial horizon and the directional gyro—are the machinery used to guide automatic pilots in airplanes. That is, information taken from these devices is used to steer the airplane in a certain direction. If, for example, the airplane turns away from the axis of the directional gyro, electrical contacts are made which, working their way through a lot of things, result in some flaps being moved, steering the airplane back on course. Automatic pilots have at their heart such gyroscopes.

Another interesting application of gyroscopes that is no longer used today, but was once proposed and built, is to stabilize ships. Of course everybody thinks you do this just by spinning a big wheel on an axle affixed to the ship, but that’s not right. If you were to do that with the spin axis vertical, for example, and a force pitched the front of the ship up, the net result would be to make the gyro precess to one side, and the ship would flip over—so that doesn’t work! A gyroscope doesn’t stabilize anything by itself.

What’s done instead illustrates a principle used in inertial guidance. The trick is this: somewhere in the ship there is a very small, but beautifully built, master gyroscope, with its axis, say, vertical. The moment the ship rolls a little bit out of vertical, electrical contacts in the master gyro operate a tremendous slave gyro that is used to stabilize the ship—these were probably the biggest gyroscopes ever built! (See Fig. 4-4.) Ordinarily the slave gyro’s axis is kept vertical, but it is gimbaled so it can be swiveled around the pitch axis of the ship. If the ship starts to roll right or left, then to straighten it out, the slave gyro is wrenched back or forward—you know how gyros are always obstinate and go the wrong way. The sudden rotation around the pitch axis produces a torque about the roll axis that opposes the roll of the ship. The pitching of the ship is not corrected by this gyro, but of course the pitching of a big ship is relatively small.

I’d like now to describe another device used on ships, the “gyrocompass.” Unlike the directional gyro, which always drifts away from north and must be reset periodically, a gyrocompass actually seeks north—in fact, it’s better than the magnetic compass because it seeks true north, in the sense of the axis of the earth’s rotation. It works as follows: suppose that we look at the earth from above the North Pole, going around counterclockwise, and we have set up a gyroscope somewhere, say on the equator, with its axis pointing east-west, parallel to the equator, as shown in Figure 4-5(a). For the moment let’s just take the example of an ideal free gyroscope, with lots of gimbals and whatnot. (It could be in a ball floating in oil—however you want it so that there is no friction.) Six hours later, the gyroscope would be still pointing in the same absolute direction (because there are no torques on it from friction), but if we were standing next to it on the equator, we would see it slowly turning over: six hours later it would be pointing straight up, as shown in Figure 4-5(c).

But now imagine what would happen if we put a weight on the gyroscope as shown in Figure 4-6; the weight would tend to keep the spin axis of the gyroscope perpendicular to gravity.

As the earth rotates, the weight will be lifted, and the weight lifting up, of course, will want to come back down, and that will produce a torque parallel to the earth’s rotation which will make the gyroscope turn at right angles to everything; in this particular case, if you figure it out, it means that instead of lifting the weight up, the gyro will turn over. And so it turns its axis around toward the north, as shown in Figure 4-7.

Now, suppose the gyro’s axis is finally pointing north: will it stay there? If we draw the same picture with the axis pointing north, as shown in Figure 4-8, then as the earth rotates, the arm swings around the gyro’s axis and the weight stays down; there are no torques on the axis from the weight being lifted, and the axis is still pointing north later.

So, if the gyrocompass has its axis pointing north, there’s no reason why it can’t stay that way, but if its axis is pointing even slightly east-west, then as the earth rotates, the weight will turn the axis toward the north. This, therefore, is a north-seeking device. (Actually, if I built it just this way, it would seek north and pass, coast on the other side, and go back and forth—so a little bit of damping has to be introduced.)

We have made an artificial gyrocompass kind of a gadget, which is shown in Figure 4-9. The gyroscope unfortunately hasn’t got all the axes free; it’s got two of them free, and you have to do a little thinking to figure out that that’s almost the same. You turn the thing around to simulate the motion of the earth, and gravity is imitated by a rubber band tied to the gyro, analogous to the weight on the end of the arm. When you start turning the thing around, the gyro precesses for a while, but if you’re patient enough, and keep the thing going, it settles down. The only place where it can stay without trying to turn in some other direction is parallel to the axis of rotation of its frame—the imaginary earth in this case—and so it settles down, very nicely, pointing toward the north. When I stop the rotation, the axis drifts, because there are various frictions and forces in the bearings. Real gyros always drift; they don’t do the ideal thing.

The best gyros that could be made about ten years ago had a drift of between $2$ and $3$ degrees in an hour—that was the limitation of inertial guidance: it was impossible to determine your direction in space more accurately than that. For instance, if you went on a trip in a submarine for $10$ hours, the axis of your directional gyro could be off by as much as $30$ degrees! (The gyrocompass and the artificial horizon would work all right, because they are “checked” by gravity, but the free-rotating directional gyros wouldn’t be accurate.)

The development of inertial guidance required the development of much better gyroscopes—gyroscopes in which the uncontrollable frictional forces that tend to make them precess are at an absolute minimum. A number of inventions have been made to make this possible, and I’d like to illustrate the general principles involved.

In the first place, the gyros we’ve been talking about so far are “two-degrees-of-freedom” gyroscopes, because there are two ways that the spin axis can turn. It turns out to be better if you only need to worry about one way at a time—that is, it is better to set up your gyros so that you only need to consider the rotations of each one about a single axis. A “one-degree-of-freedom” gyroscope is illustrated in Figure 4-10. (I have to thank Mr. Skull of the Jet Propulsion Laboratory for not only lending me these slides, but also explaining to me everything that’s been going on the last few years.)

The gyro wheel is spinning around a horizontal axis (“Spin axis” in the figure), which is only allowed to turn freely around one axis (IA), not two. Nevertheless, this is a useful device for the following reason: imagine that the gyro is being turned around the vertical input axis (IA), because it’s in a car or a ship which is turning. Then the gyro wheel will try to precess around the horizontal output axis (OA); more accurately, a torque will be developed about the output axis, and if the torque is not opposed, the gyro wheel will precess about that axis. So, if we have a signal generator (SG) which can detect the angle through which the wheel precesses, then we can use it to discover that the ship is turning.

Now, there are several features to be taken into account here: the delicate part is that the torque around the output axis must represent the result of rotation around the input axis with absolute accuracy. Any other torques about the output axis are noise, and we have to get rid of them to avoid confusion. And the difficulty is that the gyro wheel itself has some weight, which has to be supported against the weight of the pivots on the output axis—and those are the real problem, because they produce a friction which is uncertain and indefinite.

So the first and main trick that improved the gyroscope was to put the gyro wheel in a can and float the can in oil. The can is a cylinder completely surrounded by oil, and free to turn about its axis (“Output axis” in Figure 4-11). The weight of the can, with the wheel and air in it, is exactly the same as the oil it displaces (or as near as it can be made) so that the can is neutrally balanced. That way there’s very little weight to support at the pivots, so very fine jewel bearings can be used, like the ones inside a watch, consisting of a pin and a jewel. Jewel bearings can take very little sideways force, but they don’t have to take much sideways force in this case—and they have very little friction. So that was the first great improvement: to float the gyro wheel, and use jewel bearings at the pivots that support it.

The next important improvement was to never actually use the gyroscope to create any forces—or very great forces. The way we’ve been talking about this thing so far, the gyro wheel precesses about the output axis and we measure how far it precesses. But another interesting technique for measuring the effect of rotation about the input axis is based on the following idea (see Figures 4-10 and 4-11): suppose we have a device carefully built, so that by giving it a definite amount of electric current we can, very accurately, generate a certain torque on the output axis—an electromagnetic torque generator. Then we can make a feedback device with tremendous amplification between the signal generator and the torque generator, so that when the ship turns around the input axis, the gyro wheel starts to precess around the output axis, but as soon as it moves a shade, a hair—just a hair—the signal generator says, “Hey! It’s moving!” and the torque generator immediately puts a torque on the output axis that counteracts the torque making the gyro wheel precess, and holds it in place. And then we ask the question, “How hard do we have to hold it?” In other words, we measure the amount of juice going into the torque generator. Essentially, we measure the torque making the gyro wheel precess, by measuring how much torque is needed to counterbalance it. This feedback principle is very important in the design and development of gyroscopes.

Now, another interesting method of feeding back, which is in fact used even more often, is illustrated in Figure 4-12.

The gyro is the little can (“Gyro” in Figure 4-12) on the horizontal platform (Platform) in the center of the supporting framework. (You can ignore the accelerometer (Accel) for the moment; we’ll just worry about the gyro.) Unlike the previous example, this gyro’s spin axis (SRA) is vertical; however, the output axis (OA) is still horizontal. If we imagine that the framework is mounted in an airplane traveling in the indicated direction (“Forward motion” in Figure 4-12), then the input axis is the airplane’s pitch axis. When the airplane pitches up or down, the gyro wheel starts to precess around the output axis and the signal generator makes a signal, but instead of balancing it by a torque, this feedback system works as follows: as soon as the airplane starts to turn around the pitch axis, the framework which supports the gyroscope in relation to the airplane is turned the opposite way, so as to undo the motion; we turn it back, so that we get no signal. In other words, we keep the platform stable via feedback, and we never really move the gyroscope! That’s a heck of a lot better than having it swinging and turning, and trying to figure out the airplane’s pitch by measuring the output of the signal generator! It’s much easier to feed the signal back like this, so that the platform doesn’t turn at all, and the gyroscope maintains its axis—then we can just see the pitch angle, by comparing the platform to the floor of the airplane.

Figure 4-13 is a cutaway drawing that shows how an actual “one-degree-of-freedom” gyroscope is built. The gyro wheel looks very big in this picture, but the entire apparatus fits in the palm of my hand. The gyro wheel is inside of a can, which is floating in a very small amount of oil—it’s all in a little crevice around the can—but it’s enough so that no weight needs to be supported by the minuscule jewel bearings at each end. The gyro wheel is spinning all the time. The bearings it spins on need not be frictionless, because they are opposed—the friction is opposed by the engine, which turns a little motor, which turns the gyro wheel around. There are electromagnetic coils (“Signal-torquer dualsyn” in Figure 4-13) which detect the very slight motions of the can, and those provide the feedback signals which are used either to produce a torque on the can around the output axis, or to turn the platform that the gyro’s standing on around the input axis.

There is a technical problem here of some difficulty: to power the motor that makes the gyro wheel go around, we have to get electricity from a fixed part of the apparatus into the turning can. That means wires have to come in contact with the can, yet the contacts must be practically frictionless, which is very difficult. The way it’s done is as follows: four carefully made springs in semicircular form are connected to conductors on the can, as shown in Figure 4-14; the springs are made of very good material, like watch spring material, only very fine. They are balanced so that when the can is exactly in the zero position they make no torque; if the can is even slightly rotated, they make a little torque—however, because the springs are so perfectly made, that torque is exactly known—we know the right equations for it—and it’s corrected for in the electrical circuits of the feedback devices.

There’s also plenty of friction on the can from the oil, which creates torque around the output axis when the can rotates. But the law of friction for liquid oil is very accurately known: the torque is exactly proportional to the speed of the can’s rotation. And so it can be completely corrected for in the calculational parts of the circuit that make the feedback, same as the springs.

The big principle of all the accurate devices of this kind is not so much to make everything perfect, but to make everything very definite and precise.

This device is like the wonderful “one-horse shay”:² everything is made at the absolute limit of mechanical possibilities at the present time, and they’re still trying to make it better. But the most serious problem is this: what happens if the gyro wheel’s axle is a little off-center in the can, as shown in Figure 4-15? Then the can’s center of gravity won’t coincide with the output axis, and the weight of the wheel will turn the can around, creating plenty of unwanted torque.

To fix that, the first thing you do is drill little holes, or put weights on the can, to make it as balanced as possible. Then you measure very carefully what remaining drift there is, and use that measurement for calibration. When you’ve measured a particular device you’ve built, and find that you can’t reduce the drift to zero, you can always correct that in the feedback circuit. The problem in this case, though, is that the drift is indefinite: after the gyro runs for two or three hours, the position of the center of gravity moves slightly because of wear in the axle’s bearings.

Nowadays, gyroscopes of this kind are over a hundred times better than the ones made $10$ years ago. The very best ones have a drift of not more than of a $1/100^\text{th}$ degree per hour. For the device shown in Figure 4-13, that means the gyro wheel’s center of gravity cannot move more than $1/10^\text{th}$ of one-millionth of an inch from the center of the can! Good mechanical practice is something like 100 millionths of an inch, so this has to be a thousand times better than good mechanical practice. Indeed, this is one of the most serious problems—to keep the axle bearings from wearing, so that the gyro wheel moves no more than $20$ atoms to either side of the center.

The devices we’ve been talking about can be used to tell which way is up, or to keep something from turning around an axis. If we have three such devices set on three axes, with all kinds of gimbals, and so on, then we can keep something absolutely stationary. While the airplane goes around, the platform inside stays horizontal, it never turns to the right or the left; it doesn’t do anything. That way we can maintain our north, or east, or up and down, or any other direction. But the next problem is to find out where we are: how far have we gone?

Now, you know you can’t make a measurement inside an airplane to find out how fast it’s going, so you certainly can’t measure how far it’s gone, but you can measure how much it’s accelerating. So, if we initially measure no acceleration, we say, “Well, we have zero position and no acceleration.” When we start going we have to accelerate. When we accelerate we can measure that. And then, if we integrate the acceleration with a calculating machine, we can calculate the speed of the airplane, and, integrating again, we can find its position. Therefore, the method of determining how far something has gone is to measure the acceleration and integrate it twice.

How do you measure the acceleration? An obvious device for measuring acceleration is shown schematically in Figure 4-16. The most important component is just a weight (“Seismic mass” in the figure). There’s also a kind of weak spring (Elastic restraint) to hold the weight more or less in place, and a damper to keep it from oscillating, but these details are unimportant. Now, suppose this whole device is accelerated forward, in the direction indicated by the arrow (Sensitive axis). Then, of course, the weight starts to move back, and we use the scale (Scale of indicated accelerations) to measure how far back it moves; from this we can find the acceleration, and by integrating it twice we get the distance. Naturally, if we make a little error in measuring the position of the weight, so that the acceleration we find is slightly off at some point, then after a long time, over which we integrate twice, the distance is going to be way off. So, we have to make the device better.

The next stage of improvement, shown schematically in Figure 4-17, uses our familiar feedback principle: when this device accelerates, the mass moves, and the motion causes a signal generator to output a voltage proportional to the displacement. Then, instead of just measuring the voltage, the trick is to feed it back through an amplifier to a device that pulls the weight back, to find out how much force is needed to keep the weight from moving. In other words, rather than letting the weight move and measuring how far it goes, we measure the reaction force needed to balance it, and then, by $\FLPF = m\FLPa$, we find the acceleration.

One embodiment of this device is shown schematically in Figure 4-18. Figure 4-19 is a cutaway drawing that shows how the actual device is built. It’s much like the gyro in Figures 4-11 and 4-13, except the can looks empty: instead of a gyroscope, there’s just a weight attached to one side, near the bottom. The whole can is floating so that it is entirely supported and balanced by liquid oil (it’s on perfectly beautiful, fine, jewel pivots), and, of course, the weighted side of the can stays down, due to gravity.

This device is used to measure horizontal acceleration in the direction perpendicular to the axis of the can; as soon as it accelerates in that direction, the weight lags behind and slops up the side of the can, which turns on its pivots; the signal generator immediately makes a signal, and that signal is put on the torque generator’s coils to pull the can back to its original position. Just as before, we feed torque back to straighten things out, and we measure how much is needed to keep the thing from shaking, and that torque tells us how much we’re accelerating.

Another interesting device for measuring acceleration, which, in fact, automatically does one of the integrations, is shown schematically in Figure 4-20. The scheme is the same as the device shown in Figure 4-11, except that there’s a weight (“Pendulous mass” in Figure 4-20) on one side of the spin axis. If this device is accelerated upward, a torque is generated on the gyroscope, and then it’s the same as our other device—only the torque is caused by an acceleration, instead of by turning the can. The signal generator, the torque generator, and all the rest of the stuff are the same. The feedback is used to twist the can back around the output axis. In order to balance the can, the upward force on the weight must be proportional to the acceleration, but the upward force on the weight is proportional to the angular velocity at which the can is twisted, so the can’s angular velocity is proportional to the acceleration. This implies that the can’s angle is proportional to velocity. Measuring how far the can has turned gives you the velocity—and so one integration is already done. (That doesn’t mean this accelerometer is better than the other one; what works best in a particular application depends on a whole lot of technical details, and that’s a problem of design.)

Now, if we build some devices like these, we can put them together on a platform as shown in Figure 4-21, which represents a complete navigational system. The three little cylinders (Gx, Gy, Gz) are gyroscopes with axes set in three mutually perpendicular directions, and the three rectangular boxes (Ax, Ay, Az) are accelerometers, one for each axis. These gyroscopes, with their feedback systems, maintain the platform in absolute space without turning in any direction—neither yaw, nor pitch, nor roll—while the airplane (or ship, or whatever it’s in) goes around, so that the plane of the platform is very accurately fixed. This is very important for the accelerating-measuring gadgets because you’ve got to know precisely which directions they’re measuring in: if they’ve gotten cockeyed, so the navigational system thinks they’re turned one way when they’re actually turned some other way, then the system will go haywire. The trick is to keep the accelerometers in a fixed orientation in space so it’s easy to make the displacement calculations.

The outputs of the $x$, $y$, and $z$ accelerometers go into integrating circuits, which make the displacement calculations by integrating twice in each direction. So, assuming that we started at rest from a known position, we can know at any moment where we are. And we also know in what direction we’re headed, because the platform is still in the same direction it was set when we started (ideally). That’s the general idea. However, there are a few points I’d like to make.

First, when measuring acceleration, consider what happens if the device makes an error of, say, one part in a million. Suppose it’s in a rocket, and it needs to measure accelerations up to $10$ g. It would be hard to resolve less than with a device that can measure up to $10^{-5}$ g (in fact, I doubt you could). But it turns out that a error in acceleration, after you integrate it twice for an hour, means an error in position of over half a kilometer—after $10$ hours, it’s more like $50$ kilometers, which is way off. So this system won’t just keep on working. In rockets it doesn’t matter because all the acceleration happens at the very beginning and afterwards they coast free. However, in an airplane or a boat you need to reset the system from time to time, just like an ordinary directional gyro, to make sure it is still pointed the same way. This can be done by looking at a star or the sun, but how do you check it inside a submarine?

Well, if we have a map of the ocean, we can see if we went over a mountain top or something that was supposed to pass underneath us. But suppose we don’t have a map—there’s still a way to check! Here’s the idea: the earth is round, and, if we have determined that we’ve gone, say, $100$ miles in some direction, then the gravitational force should no longer be in the same direction as it was before. If we don’t keep the platform perpendicular to gravity, the output of the acceleration-measuring devices will be all wrong. Therefore we do the following: we start with the platform horizontal, and use the accelerating-measuring devices to calculate our position; according to the position we figure out how we should turn the platform so that it remains horizontal, and we turn it at a rate predicted to keep it horizontal. That’s a very handy thing—but it’s also the device which saves the day!

Consider what would happen if there was an error. Suppose the machine was just standing in a room, not moving, and after some time, because it was built imperfectly, the platform was not horizontal, but rotated slightly, as shown on Figure 4-22(a). Then the weights in the accelerometers would be displaced, corresponding to an acceleration, and the positions calculated by the machinery would indicate motion to the right, towards (b). The mechanism which tries to keep the platform horizontal would rotate it slowly, and eventually, when the platform was level again, the machine would no longer think it’s accelerating. However, because of the apparent acceleration, the machine would still think it had a velocity in the same direction, and so the mechanism which tries to keep the platform horizontal would continue to rotate it, very slowly, until it was no longer horizontal, as shown in Figure 4-22(c). In fact, it would go through the zero of acceleration, and then it would think it was accelerating in the opposite direction. So we’d have an oscillatory motion which is very small, and the errors would only accumulate over one of these oscillations. If you figure out all the angles and turnings and whatnot, it takes $84$ minutes for one of these oscillations. Thus, it is only necessary to make the device good enough to give the right accuracy within a period of $84$ minutes, because it will correct itself in that time. It is much like what is done in an airplane where the gyrocompass is checked against a magnetic compass from time to time, but in this case the machine is checked against gravity as in the case of the artificial horizon.

In roughly the same manner, the azimuth device on a submarine (which tells you which way is north) is set from time to time against a gyrocompass, which is averaging over long periods, so that the motions of the ship don’t make any difference. Thus, you can correct the azimuth against the gyrocompass, and you can correct the accelerometers against gravity, and so the errors do not accumulate forever, but only for about an hour and a half.

In the Nautilus submarine there were three monstrous platforms of this type, each in a great big ball, hung right next to each other from the ceiling of the navigator’s room, all completely independent, in case any of them broke down—or, if they didn’t agree with each other, the navigator would take the best two out of three (which must have made him pretty nervous!). These platforms were all different when they were built, because you can’t make anything perfect. The drift caused by slight inaccuracies had to be measured in each device, and the devices had to be calibrated to compensate for it.

There’s a laboratory at JPL where some of these new devices are tested. It’s an interesting laboratory, if you consider how you would check such a device: you don’t want to get in a ship and move around; no, in this laboratory they check the device against the rotation of the earth! If the device is sensitive, it will turn because of the rotation of the earth, and it will drift. By measuring the drift, corrections can be determined within a very short time. This laboratory is probably the only one in the world whose fundamental feature—the thing that makes it go—is the fact that the earth is turning. It wouldn’t be useful for calibration if the earth didn’t turn!

The next thing I want to talk about is effects of the rotation of the earth (besides the effects on the calibration of inertial guidance devices).

One of the most obvious effects of the rotation of the earth is on the large-scale motion of the winds. There’s a famous tale, which you hear again and again, that if you have a bathtub, and you pull out the plug, the water goes around one way if you’re in the Northern Hemisphere, and the other way if you’re in the Southern Hemisphere—but if you try it, it doesn’t work. The reason it’s supposed to go around one way is something like this: suppose we have a plug in a drain at the bottom of the ocean, under the North Pole. Then we pull the plug out, and the water starts moving down the drain. (See Fig. 4-23.)

The ocean has a large radius, and the water is slowly turning around the drain because of the earth’s rotation. As the water comes in toward the drain it goes from a larger radius to a smaller radius, and so it has to go around faster to maintain its angular momentum (like when the spinning ice skater pulls her arms in). The water goes around the same way the earth is turning, but it has to turn faster, so somebody standing on the earth would see the water swirling around the drain. That’s right, and that’s the way it should work. And that’s the way it does work with the winds: if there’s a place where there’s low pressure, and the surrounding air is trying to move into it, then instead of moving straight in, it gets some sideways motion—in fact, ultimately, the sideways motion becomes so great, that instead of moving in at all, the air is practically rotating around the low pressure area.

So this is one of the laws of weather: if you face downwind in the Northern Hemisphere, low pressure is always on the left, high pressure on the right (see Figure 4-24), and the reason has to do with the rotation of the earth. (This is nearly always true; from time to time, under certain crazy circumstances, it doesn’t work, because there are other forces involved besides the rotation of the earth.)

The reason it doesn’t work in your bathtub is as follows: what causes this phenomenon is the initial rotation of the water—and the water in your bathtub is rotating. But how fast is the earth’s rotation? Once around a day. Can you guarantee that the water in your bathtub hasn’t got a little bit of motion equivalent to one swash around the bathtub in a day? No. Ordinarily, there’s a lot of swishing and swashing in the tub! So this only works on a big enough scale, like a great big lake, where the water’s pretty quiet, and you can easily demonstrate that the circulation is not so great as to correspond to once around the lake in a day. Then, if you make a hole in the bottom of the lake and let the water run out, it’ll turn in the correct direction, as advertised.

There are a few other points about the rotation of the earth which are interesting. One of them is that the earth is not exactly a sphere; it’s a little bit off as a result of its spinning—the centrifugal forces, balancing against gravity, make it oblate. And you can calculate how oblate, if you know how much the earth gives. If you assume it’s like a perfect fluid that oozes into its ultimate position and ask what the oblateness should be, you’ll find that it agrees with the actual oblateness of the earth within the accuracy of the calculations and the measurements (an accuracy of about $1$ percent).

This is not true of the moon. The moon is more lopsided than it ought to be, for the speed at which it’s turning. In other words, either the moon was turning faster when it was liquefied, and it froze strong enough to resist the tendency to get into the right shape, or else it was never molten, but was formed by throwing together a bunch of meteors—and the god who did it didn’t do it in a perfectly precise and balanced manner, so it’s a little lopsided.

I also want to talk about the fact that the oblate earth is spinning around an axis which is not perpendicular to the plane of the earth’s rotation around the sun (or the moon’s rotation around the earth, which is almost the same plane). If the earth were a sphere, the gravitational and centrifugal forces on it would be balanced with respect to its center, but because it’s a little lopsided, the force is not balanced; there’s a torque due to gravitation which tends to turn the earth’s axis perpendicular to the line of force, and so, like a great gyroscope, the earth precesses in space. (See Fig. 4-25.)

The axis of the earth, which today points to the North Star, is actually drifting slowly around, and in time it will point to all the stars in the heavens on a big cone subtending an angle of $23\frac12$ degrees. It takes $26,000$ years for it to come back to the pole star, so if you are reincarnated $26,000$ years from now, you may have nothing new to learn, but if it’s any other time, you’ll have to learn another position (and maybe another name) for the “pole” star.

In the last lecture (see FLP Vol. I, Ch. 20, “Rotation in Space”) we discussed the interesting fact that the angular momentum of a rigid body is not necessarily in the same direction as its angular velocity. We took as an example a disk that is fastened onto a rotating shaft in a lopsided fashion, as shown in Figure 4-26. I’d like to explore this example in further detail.

First, let me remind you of an interesting thing that we’ve already talked about: that for any rigid body, there is an axis through the body’s center of mass about which the moment of inertia is maximal, there is another axis through the body’s center of mass about which the moment of inertia is minimal, and these are always at right angles. It’s easy enough to see this for a rectangular block as shown in Figure 4-27, but surprisingly it’s true for any rigid body.

These two axes, and the axis which is perpendicular to them both, are called the principal axes of the body. The principal axes of a body have the following special property: the component of the body’s angular momentum in the direction of a principal axis is equal to the component of its angular velocity in that direction times the body’s moment of inertia about that axis. So, if $\FLPi$, $\FLPj$, and $\FLPk$ are unit vectors along the principal axes of a body, with respective principal moments of inertia $A$, $B$, and $C$, then when the body rotates about its center of mass with angular velocity $\FLPomega = (\omega_i,\omega_j,\omega_k)$, its angular momentum is \begin{equation} \label{Eq:TIPS:4:1} \FLPL = A\omega_i\FLPi+B\omega_j\FLPj+C\omega_k\FLPk. \end{equation}

For a thin disk of mass $m$ and radius $r$, the principal axes are as follows: the main axis is perpendicular to the disk, with maximal moment of inertia $A = \frac12 mr^2$; any axis perpendicular to the main axis has the minimum moment of inertia $B = C = \frac14 mr^2$ The principal moments of inertia are not equal; in fact, $A = 2B = 2C$. So, when the shaft in Figure 4-26 is rotated, the disk’s angular momentum is not parallel to its angular velocity. The disk is statically balanced because it is attached to the shaft at its center of mass. But it is not dynamically balanced. When we turn the shaft, we have to turn the disk’s angular momentum, so we must exert a torque. Figure 4-28 shows the disk’s angular velocity $\FLPomega$ and its angular momentum $\FLPL$, and their components along the principal axes of the disk.

But now, consider this interesting, additional thing: suppose we put a bearing on the disk, so that we can also spin the disk around its main axis with angular velocity $\FLPOmega$ as shown in Figure 4-29.

Then while the shaft is turning, the disk would have an actual angular momentum which is the result of the shaft turning and the disk spinning. If we spin the disk in the direction opposite to the way the shaft is turning it, as shown in the figure, we will reduce the component of the disk’s angular velocity along its main axis. In fact, since the ratio of the disk’s principal moments of inertia is exactly $2\!:\!1$, Eq. (4.1) tells us that by spinning the disk backwards at exactly half the speed the shaft turns it around (such that $\Omega = -(\omega_i/2)\FLPi$, we can put this thing together in such a miraculous manner that the total angular momentum is exactly along the shaft—and then we can take the shaft away, because there are no forces! (See Fig. 4-30.)

And that is the way a free body turns: if you throw an object into space alone, like a plate³ or a coin, you see it doesn’t just turn around one axis. What it does is a combination of spinning around its main axis, and spinning around some other cockeyed axis in such a nice balance, that the net result is that the angular momentum is constant. That makes it wobble—and the earth wobbles, too.

From the period of the earth’s precession—$26,000$ years—it’s been shown that the maximum moment of inertia (around the pole) and the minimum moment of inertia (around an axis in the equator) differ by only $1$ part in $306$—the earth is almost a sphere. However, since the two moments of inertia do differ, any disturbance of the earth could result in a slight rotation around some other axis, or, what amounts to the same thing: the earth nutates as well as precesses.

You can calculate the nutation frequency of the earth: it turns out, in fact, to be $306$ days. And you can measure it very accurately: the pole wobbles in space by $50$ feet measured at the earth’s surface; it wobbles around, and back and forth, rather irregularly, but the major motion has a period of $439$ days, not $306$ days, and therein lies a mystery. However, this mystery is easily resolved: the analysis was made for rigid bodies, but the earth is not rigid; it’s got liquid goop on the inside, and so, first of all, its period is different from that of a rigid body, and secondly, the motion is damped out so it should stop eventually—that’s why it’s so small. What makes it nutate at all, despite the damping, are various irregular effects which jiggle the earth, such as the sudden motions of winds, and ocean currents.

One of the most striking characteristics of the solar system, discovered by Kepler, is that everything goes around in ellipses. This was explained, ultimately, by the law of gravitation. But there are a whole lot of other things about the solar system—peculiar simplifications—which are harder to explain. For example, all the planets seem to go around the sun in roughly the same plane, and, except for one or two, they all rotate around their poles the same way—west to east, like Earth; almost all the planetary moons go around in the same direction, and so with few exceptions, everything turns the same way. It’s an interesting question to ask: How did the solar system get that way?

In studying the origin of the solar system, one of the most important considerations is that of angular momentum. If you imagine a whole lot of dust or gas contracting as a result of gravitation, even if it only has a small amount of internal motion, the angular momentum must remain constant; those “arms” are coming in and the moment of inertia is going down, so the angular velocity has to increase. It’s possible that the planets are merely the result of a necessity the solar system has to dump its angular momentum from time to time in order to be able to contract still further—we don’t know. But it is true that $95\%$ of the angular momentum in the solar system is in the planets, and not in the sun. (The sun is spinning, all right, but it’s only got $5\%$ of the total angular momentum.) This problem has been discussed many times, but it is still not understood how a gas contracts or how a pile of dust falls together when it is rotating slightly. Most discussions pay lip service to the angular momentum at the beginning; then, when they make the analysis, they disregard it.

Another serious problem in astronomy has to do with the development of the galaxies—the nebulae. What determines their form? Figure 4-31 shows several different types of nebulae: the famous ordinary spiral (much like our own galaxy), barred spiral nebulae (whose long arms extend from a central bar), and elliptic nebulae (which don’t even have arms). And the question is: How did they become different?

It could be, of course, that the masses of different nebulae are different, and that if you start with different amounts of mass, you come out with different results. That’s possible, but because the spiral character of nebulae almost certainly has something to do with angular momentum, it seems more likely that differences from one nebula to another are explained by differences in the initial angular momentum of the original masses of gas and dust (or whatever you assume they start with). Another possibility, which some people have proposed, is that the different types of nebulae represent different stages of development. That would mean that they are all different ages—which, of course, would have dramatic implications for our theory of the universe: Did it all explode at one time, after which the gas condensed to form different types of nebulae? Then they would all have to be the same age. Or, are the nebulae perpetually being formed from debris in space, in which case they could have different ages?

A real understanding of the formation of these nebulae is a problem in mechanics, one involving angular momentum, and one which is still not solved. The physicists should be ashamed of themselves: astronomers keep asking, “Why don’t you figure out for us what will happen if you have a big mass of junk pulled together by gravity and spinning? Can you understand the shapes of these nebulae?” And nobody ever answers them.

In quantum mechanics the fundamental law $\FLPF = m\FLPa$ fails. Nevertheless, some things remain: the law of conservation of energy remains; the law of conservation of momentum remains; and the law of conservation of angular momentum also remains—it remains in a very beautiful form, very deep in the heart of the quantum mechanics. Angular momentum is a central feature in the analyses of quantum mechanics, and that’s in fact one of the main reasons for going so far into it in mechanics—in order to be able to understand the phenomena in atoms.

One of the interesting differences between classical and quantum mechanics is this: in classical mechanics, a given object can have arbitrary amounts of angular momentum by spinning at different speeds; in quantum mechanics, the angular momentum along a given axis cannot be arbitrary—it can only have a value that is an integral or half-integral multiple of Planck’s constant over two pi ($h/2\pi$, or $\hbar$)and it must jump from one value to another in increments of $\hbar$. This is one of the deeper principles of quantum mechanics associated with angular momentum.

Finally, an interesting point: we think of the electron as a fundamental particle, as simple as it can be. Nevertheless, it has an intrinsic angular momentum. We picture the electron not simply as a point charge, but as a point charge that is a sort of limit of a real object that has angular momentum. It is something like an object spinning on its axis in the classical theory, but not exactly: it turns out that the electron is analogous to the simplest kind of gyro, which we imagine to have a very small moment of inertia, spinning extremely fast about its main axis. And, interestingly, the thing that we always do in the first approximation in classical mechanics, which is to neglect the moments of inertia around the precession axis—that seems to be exactly right for the electron! In other words, the electron seems to be like a gyroscope with an infinitesimal moment of inertia, spinning at infinite angular velocity, so as to have a finite angular momentum. It’s a limiting case; it’s not exactly the same as a gyro—it’s even simpler. But it’s still a curiosity.

I have here the insides of the gyro shown in Figure 4-13, if you want to look at it. That’s all for today.