CHAPTER III

THE GYROSCOPE AND THE ROTATION OF THE EARTH

Let us now suppose that the gyroscopic system shown in Fig. 1--without the weight W--is placed on the equator as represented at P (Fig. 7), and that the axle is set pointing due east and west as at B C. At the end of an interval of time, say two hours, the earth will have rotated through some angle α, say 30 deg., and will have carried the gyroscope with it to the position Q. The square frame has thus clearly been inclined relatively to its original position. It has, in fact, suffered the exact equivalent of a direct translation R together with a pure rotation about a horizontal axis through E of amount α. The translation leaves the system unaffected, but the rotation of the frame results in the frame moving relatively to the axle, wheel, and inner ring. The axle, in fact, remains parallel with its original position at P. It is still pointing east and west, but the frame is now inclined to it and, relatively to the horizontal surface of the earth at Q, the axle is dipping at an angle β which is equal to α--or 30 deg. Actually, if we fixed a disc to the square frame and a hand to the inner ring, as indicated in Fig. 8, the system as erected at the equator would form a twenty-four hour clock indicating strictly accurate sidereal time as distinct from mean solar time. To an observer on the earth the hand would appear to travel clockwise round the disc once in twenty-four hours. Actually, however, the hand would not rotate, but would remain constantly parallel with its original position, while the disc would travel anti-clockwise relatively to the hand and would make one complete turn in twenty-four hours. The hand would remain parallel with its original position by virtue of the fact already stated, namely, that the force applied to the inner ring through its all-but-frictionless supports is very small, and in any event does not turn the hand clockwise, but causes the wheel, inner ring, and hand to precess about the vertical axis H J. The rate at which this precession took place would be a measure of the success with which we had eliminated friction at the horizontal axis E F.

It might be thought that the system would without further addition serve as a compass, for if it maintains its axle constantly pointing in one direction it is just as good as a compass which always points its needle towards the magnetic north. In the magnetic compass, however, the needle has a directive force applied to it which enables it to recover its standard direction if it should be accidentally deflected from it. In the gyroscopic system we have been considering there is no such directive force. The axle will remain pointing in one direction, it is true, but the system is indifferent as to what that direction may be. If the axle is accidentally deflected it will remain pointing in the new direction as consistently as it did before in the originally set direction. As a substitute for the ordinary compass, then, the success of the device would depend upon the success with which accidental deflecting forces were prevented from acting on the axle after it had been set in a known direction. In practice, as Dr. Anschütz found in his early investigations, it is excessively difficult, if not quite impossible, to construct a gyroscopic system in which the centre of gravity and the centre of suspension are absolutely coincident. As a result a very minute gravity torque is thrown on the system, and in consequence the axle very slowly precesses away from the original set direction. This fact and the complication of parts required to give practical effect to the idea led Dr. Anschütz to abandon his early attempts at providing a compass substitute of the apparently simple nature described above.

An addition to the system of a very simple kind in itself not only endows the axle with directive force, but makes the direction which it seeks the north and south one, and thus converts the system into a device possessing the familiar property of the compass. This addition consists of a pendulum-like weight S (Fig. 9), attached below the wheel by a stirrup fixed to the inner ring so that the weight, stirrup, inner ring, axle, and wheel may swing as a whole on the horizontal axis E F.

Let us suppose that the system with this addition is set up at the equator and that the axle this time is aligned at right angles to the equator so that the end B, as shown at I (Fig. 10), points due north. In this condition the inner ring is horizontal and the weight S is vertically below the pendulum axis E F. No turning moment is therefore being applied by gravity to the wheel. If through imperfection of workmanship the centre of suspension of the system is not absolutely coincident with the centre of gravity before the weight S and its stirrup are attached, then the minute gravity torque arising from the lack of coincidence will be balanced automatically by the weight, the inner ring taking up some position minutely inclined to the horizontal. There will thus under all conditions be no resultant turning moment applied by gravity to the system as thus set up. In addition, the axle, lying north and south as it does, is aligned parallel with the earth’s polar diameter. Consequently the rotation of the earth can only move it parallel with its original position, and therefore does not tend to cause relative motion between it and the square frame. We conclude, then, that in this north and south position of the axle the system is not acted upon by any force or influence tending to cause the axle to depart from the north and south position.

Now let us suppose that the axle by some agency is forced into parallelism with the equator so that the end B points due east as indicated at II (Fig. 10). Immediately after it takes up this position the tendency of the axle to remain parallel with this, its new original, direction becomes manifested in attempted relative motion between the axle, wheel, and inner ring on the one hand and the square frame on the other. Thus as the earth rotates the axle, etc., tend to set themselves relatively to the frame in the position shown at III. In this position, however, the weight S being rigidly suspended from the inner ring, is no longer vertically below the pendulum axis E F. Gravity acting upon the weight therefore applies a turning force to the wheel, etc., about the axis E F. The system is thus under the same conditions as those represented in Fig. 1, when the weight W is hung on the inner ring at B. Precession about the vertical axis therefore sets in, in the direction M (Fig. 1), so that the end B of the axle swings round from the east towards the north.

Let us reset the system in position I, and then by some agency cause the axle again to align itself parallel with the equator, this time, however, with the end B pointing due west as shown at IV (Fig. 10). As before, the rotation of the earth combined with the tendency of the axle to remain parallel with the new west and east position results in attempted relative motion between the axle, etc., and the square frame, so that in a little time the system would adopt the configuration shown at V. In this configuration, however, gravity as before applies through the weight S a turning force about the pendulum axis E F. Now, comparing the two configurations III and V, it will be seen that, mere reference letters or similar distinguishing marks being washed out, they are indistinguishable except for one fact: the wheel is rotating in opposite directions. If with the system as arranged in Fig. 1 we reverse the direction of spin of the wheel without reversing the direction of the applied force W, then, as we know already, the direction of the precession will be reversed. Precession about the vertical axis will take place in the direction opposed to the arrow M. Hence in the configuration shown at V (Fig. 10), the precession induced by the action of gravity on the weight S causes the end B of the axle to swing up from due west towards the north.

We are thus led to identify the end B of the axle as the north-seeking end and the end C as the south-seeking. With B pointing due north as at I, there is no force acting on the system tending to make the axle depart from the north and south direction. If B is swung over to the east or west--or intermediately, as may be taken for granted--a force is called into play tending to move the end B back towards the north. It follows, therefore, that the resting position of the axle is the north and south one with the end B pointing north.

It may be said, perhaps, that we have neglected to discuss what happens if from the position I the wheel is turned by some agency right round until the end B of the axle points due south. In this condition there is no resultant gravity torque on the pendulum axis, and the axle is lying parallel with the earth’s polar axis, so that the rotation of the earth does not cause relative motion between the wheel, etc., and the frame. Just as in the original configuration I, there is thus in this condition no force applied to the system tending to make the end B swing away from the pole. But as the reader may readily establish for himself by reversing the arrows and reference letters in the five diagrams of Fig. 10, the slightest departure of the end B of the axle from the south-pointing direction towards either side of the meridian will call into play a force which will cause the end B to precess up towards the _north_. With the wheel spinning in the direction we have shown throughout our illustrations the only _stable_ position of equilibrium for the axle is the north and south with the end B pointing north. It may be pointed out that the magnetic needle can, like the gyro-compass axle, assume a position of unstable equilibrium with the north-seeking end pointing south.

A point of very great practical importance into which to inquire is the magnitude of the directive force, the existence of which, when the axle is deflected from the north and south position, we have just demonstrated. This directive force or restoring moment, as will have been gathered from our explanation, increases with the deflection from the north, being a maximum when the axle is lying east and west or west and east. Its magnitude in any position of the axle depends upon (_a_) the speed of rotation of the earth on its polar axis, (_b_) the speed of the spinning wheel on its axle, and (_c_) the mass, or, more correctly, the moment of inertia, of the wheel. The first item is small--0.0007 of a revolution per minute--and is quite beyond our control. The second factor is consequently made as large as possible, while the third is also made large, but a limit is placed to our choice by questions of safety and temperature rise at the high speeds adopted for the spin of the wheel. In the following table we give the values of these factors for three of the types of gyro-compass to be described later.

Compass Wheel diam. Wheel weight Speed in. lbs. r.p.m. Anschütz[1] 6 10 20,000 Sperry 12 45 8,600 Brown 4 4¼ 15,000

The value of the directive force for the same three gyro-compasses and for an ordinary magnetic compass is given in the next table, (1) for the axle--or needle--lying due east and west, and (2) for the axle--or needle--deflected 1 deg. east or west of north--true or magnetic.

_Directive Force at Equator._

Axle (or needle) Axle (or needle) E. and W. 1 deg. E. or W. of N. Force Leverage Force Leverage Grains in. Grains in. Anschütz 145 1 2 1 Sperry. 1140 1 20 1 Brown 12 1 ⅕ 1 Magnetic[2] 40 1 ⅔ 1

We have now to explain how the gyro-pendulum system manifests its compass-like property when it is transferred from the equator to some degree of latitude north or south. In Fig. 11 we represent the system as set up in the latitude of the British Isles. The axle is horizontal and the end B is pointing due east. In this configuration the earth’s rotation is, through the action of gravity on the pendulum bob S, trying to make the wheel turn round the earth’s polar axis once every twenty-four hours. As before, in accordance with the fundamental property of the gyroscope, the wheel will try to set its axle into coincidence with or parallel with the axis about which it is being forced to rotate. In other words, the wheel will endeavour to turn in such a way as to align its axle along V U with the end B towards U. This movement can be effected by a rotation about the vertical axis H J through a right angle combined with a rotation about the horizontal axis E F through an angle θ equal to the latitude of the station at which the system is set up. The rotation about the vertical axis H J does not result in deflecting the weight S away from the plumb line, and therefore can be completely fulfilled. The rotation about the horizontal axis does, however, affect the bob. The axle, having executed the horizontal portion of its movement, is pointing its end B due north, but this end, unlike its behaviour at the equator, manifests a desire to rise vertically so as to align the axle along V U. Its desire to do so is resisted by the bob S, and the axle therefore fails to complete the full movement.

The axle is thus held substantially horizontal with its end B pointing to the north. As the earth rotates the desire of the axle to align itself parallel with the polar axis persists. In attempting continuously to fulfil this desire it acquires a slight upward tilt, which is sufficient to bring the pendulum weight into action. With the weight thus slightly deflected towards the north a moment is applied to the wheel which tends to turn the wheel about the horizontal axis E F in such a way as to bring the end B down again to the horizontal plane. Such a moment, as we know from the fundamental rule of the gyroscope, will actually produce precession about the vertical axis H J, the direction of this precession being such as to cause the end B to move away from the north towards the _west_.

The fact, then, that the axle is prevented from aligning itself completely parallel with the earth’s polar axis thus apparently results, once it has found the north, in making it wander, in northern latitudes, towards the west. This is not so. Once the axle has _found_ the north a steady uniform precession towards the west is required to _maintain_ it on the north. Thus in Fig. 12 let A be the wheel and axle of the compass when it finds the north. If the axle maintained this alignment then some time later it would assume the position shown dotted at D; that is to say, it would be pointing towards east of north. To maintain it on the north it must rotate westwardly through the angle ϕ in the interval between A and D. As this angle ϕ grows with the interval the required rotation is really equivalent to a steady uniform precession towards the west.

If the compass in Fig. 12 is practically at the north pole it is clear that to hold the end B of the axle on the north the axle has to precess about the vertical axis H J of the compass mounting at the rate of practically one complete revolution per twenty-four hours--that is, 0.0007 revolution per minute. At the equator the required rate of precession about H J is zero, for any movement about this axis will carry the axle away from the north. At intermediate latitudes the precession required to hold the axle on the north has an intermediate value. Its value for any latitude θ is, in fact, 0.0007 × sin θ in revolutions per minute.

Thus as the latitude is changed the required rate of precession also changes. So, too, does the angle θ (Fig. 11), by which the axle fails to reach parallelism with the earth’s polar axis, and consequently so does the strength with which the axle desires to reach this alignment. As the equator is left farther and farther behind, then, the axle comes to rest pointing north with a greater and greater upward tilt from the horizontal. The applied moment of the weight S thus increases. It increases just at the rate necessary to give the required rate of westerly precession for the particular latitude in which the compass is at any moment stationed. Should anything prevent the axle from acquiring the tilt appropriate to the latitude, or should the westerly precession on the vertical axis caused by this tilt be opposed and reduced in any way, the axle will fail to keep on the north and will lag behind the meridian with an easterly deviation. We shall see later on that the precession about the vertical axis is in some designs of gyro-compass unavoidably opposed, and that as a consequence these compasses exhibit a latitude error.

We have thus shown that the effort of the compass to set its axle parallel with the earth’s polar axis, combined with the action of gravity on the pendulum weight, is necessary to the compass if the axle once having found the north is to remain on it, and that this effort of the axle increases in strength the farther north--or south--the compass is moved from the equator. What, however, this effort gains in strength as the angle of latitude increases the effective directive force on the compass loses. Thus in Fig. 11 the directive force may be represented as at _f_ by a line parallel with the earth’s polar axis. This line represents the magnitude of the compass’s effort to set its axle parallel with the polar axis. The speed of the spinning wheel and its moment of inertia have not been altered by moving the compass away from the equator, nor has the angular speed of the compass round the earth’s axis, for although the compass is moving in a circle of reduced radius T B, and therefore is travelling with less linear velocity than at the equator, it is still making one turn per twenty-four hours round the polar axis of the earth. Thus the three factors fixing the magnitude of the “directive force” are unaltered. The force _f_ is thus the same as that exerted on the compass at the equator. It does not, however, act as before, purely about the vertical axis H J, but partly about H J and partly about the horizontal axis E F. It may be regarded as a force applied at the end B of the axle and therefore as tending to turn the wheel about an imaginary axis _a b_. We may resolve it into two components _p_ and _q_, _p_ being at right angles to the axis H J and _q_ at right angles to E F. The component _q_ represents the magnitude of the upwardly tilting effect applied to the axle by the rotation of the earth. The component _p_ represents the effective directive force tending to restore the axle from the deflected position represented towards the north in the horizontal plane. The angle between this effective component and the full force _f_ is θ, the angle of latitude of the station at which the compass is set up. The effective directive force is thus _f_ cos θ, and therefore diminishes from the value _f_ at the equator towards zero as the north--or south--pole is approached.