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.
[Illustration: FIG. 7. Elementary Gyroscope at Equator.]
[Illustration: FIG. 8. Gyroscopic Clock.]
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.
[Illustration: FIG. 9. Elementary Gyro-Compass.]
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.
[Illustration: FIG. 10. Elementary Compass at Equator.]
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
[Illustration: FIG. 11. Elementary Compass at 55 deg. N. Lat.]
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.
[Illustration: FIG. 12. Compass at Equator and near North Pole.]
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.