A.D.--"Can you account for the crass ignorance of our forefathers in not
being able to see from England what their friends were doing in
Australia?"[13] Or this--"Messages are being received every minute from our
friends on the planet Mars, and are now being answered: how do you account
for our ancestors being utterly ignorant that these messages were
occasionally sent to them?" Or this--"What metal is as strong compared with
steel as steel is compared with lead? and explain why the discovery of it
was not made in Sheffield."
But there is one question that our descendants will never ask in accents of
jocularity, for to their bitter sorrow every man, woman, and child of them
will know the answer, and that question is this--"If our ancestors in the
matter of coal economy were not quite as ignorant as a baby who takes a
penny {132} as equivalent for a half-crown, why did they waste our coal?
Why did they destroy what never can be replaced?"
My friends, let me conclude by impressing upon you the value of knowledge,
and the importance of using every opportunity within your reach to increase
your own store of it. Many are the glittering things that seem to compete
successfully with it, and to exercise a stronger fascination over human
hearts. Wealth and rank, fashion and luxury, power and fame--these fire the
ambitions of men, and attract myriads of eager worshippers; but, believe
it, they are but poor things in comparison with knowledge, and have no such
pure satisfactions to give as those which it is able to bestow. There is no
evil thing under the sun which knowledge, when wielded by an earnest and
rightly directed will, may not help to purge out and destroy; and there is
no man or woman born into this world who has not been given the capacity,
not merely to gather in knowledge for his own improvement and delight, but
even to add something, however little, to that general stock of knowledge
which is the world's best wealth.
* * * * *
{133}
ARGUMENT.
1. _Introduction_, pages 9-14, showing the importance of the study of
spinning-top behaviour.
2. _Quasi-rigidity induced even in flexible and fluid bodies by rapid
motion_, 14-21.
Illustrations: Top, 14; belt or rope, 14; disc of thin paper, 14; ring
of chain, 15; soft hat, 16; drunken man, 16; rotating water, 16; smoke
rings, 17; Thomson's Molecular Theory, 19; swimmer caught in an eddy,
20; mining water jet, 20; cased gyrostat, 21.
3. _The nature of this quasi-rigidity in spinning bodies is a
resistance to change of direction of the axis of spinning_, 21-30.
Illustrations: Cased gyrostat, 21-24; tops, biscuits, hats, thrown into
the air, 24-26; quoits, hoops, projectiles from guns, 27; jugglers at
the Victoria Music Hall, 26-30; child trundling hoop, man on bicycle,
ballet-dancer, the earth pointing to pole star, boy's top, 30.
4. _Study of the crab-like behaviour of a spinning body_, 30-49.
Illustrations: Spinning top, 31; cased gyrostat, 32; balanced gyrostat,
33-36; windage of projectiles from {134} rifled guns, 36-38; tilting a
hoop or bicycle, turning quickly on horseback, 38; bowls, 39; how to
simplify one's observations, 39, 40; the illustration which gives us
our simple universal rule, 40-42; testing the rule, 42-44; explanation
of precession of gyrostat, 44, 45; precession of common top, 46;
precession of overhung top, 46; list of our results given in a wall
sheet, 48, 49.
5. _Proof or explanation of our simple universal rule_, 50-54.
Giving two independent rotations to a body, 50, 51; composition of
rotations, 52, 53.
6. _Warning that the rule is not, after all, so simple_, 54-66.
Two independent spins given to the earth, 54; centrifugal force, 55;
balancing of quick speed machinery, 56, 57; the possible wobbling of
the earth, 58; the three principal axes of a body, 59; the free
spinning of discs, cones, rods, rings of chain, 60; nodding motion of a
gyrostat, 62; of a top, 63; parenthesis about inaccuracy of statement
and Rankine's rhyme, 63, 64; further complications in gyrostatic
behaviour, 64; strange elastic, jelly-like behaviour, 65; gyrostat on
stilts, 66.
7. _Why a gyrostat falls_, 66, 67.
8. _Why a top rises_, 67-74.
General ignorance, 67; Thomson preparing for the mathematical tripos,
68; behaviour of a water-worn stone when spun on a table, 68, 69;
parenthesis on technical education, 70; simple explanation of why a top
rises, 70-73; behaviour of heterogeneous sphere when spun, 74.
9. _Precessional motion of the earth_, 74-91.
Its nature and effects on climate, 75-80; resemblance of the precessing
earth to certain models, 80-82; tilting forces exerted by the sun and
moon on the {135} earth, 82-84; how the earth's precessional motion is
always altering, 85-88; the retrogression of the moon's nodes is itself
another example, 88, 89; an exact statement made and a sort of apology
for making it, 90, 91.
10. _Influence of possible internal fluidity of the earth on its
precessional motion_, 91-98.
Effect of fluids and sand in tumblers, 91-93; three tests of the
internal rigidity of an egg, that is, of its being a boiled egg, 93,
94; quasi-rigidity of fluids due to rapid motion, forgotten in original
argument, 95; beautiful behaviour of hollow top filled with water, 95;
striking contrasts in the behaviour of two tops which are very much
alike, 97, 98; fourth test of a boiled egg, 98.
11. Apology for dwelling further upon astronomical matters, and
impertinent remarks about astronomers, 99-101.
12. How a gyrostat would enable a person living in subterranean regions
to know, _1st, that the earth rotates_; _2nd, the amount of rotation_;
_3rd, the direction of true north_; _4th, the latitude_, 101-111.
Some men's want of faith, 101; disbelief in the earth's rotation, 102;
how a free gyrostat behaves, 103, 104; Foucault's laboratory
measurement of the earth's rotation, 105-107; to find the true north,
108; all rotating bodies vainly endeavouring to point to the pole star,
108; to find the latitude, 110; analogies between the gyrostat and the
mariner's compass and the dipping needle, 110, 111; dynamical
connection between magnetism and gyrostatic phenomena, 111.
13. How the lecturer spun his tops, using electro-motors, 112-114.
14. _Light_, _magnetism_, _and molecular spinning tops_, 115-128.
Light takes time to travel, 115; the electro-magnetic {136} theory of
light, 116, 117; signalling through fogs and buildings by means of a
new kind of radiation, 117; Faraday's rotation of the plane of
polarization by magnetism, with illustrations and models, 118-124;
chain of gyrostats, 124; gyrostat as a pendulum bob, 126; Thomson's
mechanical illustration of Faraday's experiment, 127, 128.
15. _Conclusion_, 129-132.
The necessity for cultivating the observation, 129; future discovery,
130; questions to be asked one hundred years hence, 131; knowledge the
thing most to be wished for, 132.
* * * * *
{137}
APPENDIX I.
THE USE OF GYROSTATS.
In 1874 two famous men made a great mistake in endeavouring to prevent or
diminish the rolling motion of the saloon of a vessel by using a rapidly
rotating wheel. Mr. Macfarlane Gray pointed out their mistake. It is only
when the wheel is allowed to _precess_ that it can exercise a steadying
effect; the moment which it then exerts is equal to the angular speed of
the precession multiplied by the moment of momentum of the spinning wheel.
It is astonishing how many engineers who know the laws of motion of mere
translation, are ignorant of angular motion, and yet the analogies between
the two sets of laws are perfectly simple. I have set out these analogies
in my book on _Applied Mechanics_. The last of them between centripetal
force on a body moving in a curved path, and torque or moment on a rotating
body is the simple key to all gyrostatic or top calculation. When the spin
of a top is greatly reduced it is necessary to remember that the total
moment of momentum is not about the spinning axis (see my _Applied
Mechanics_, page 594); correction for this is, I suppose, what introduces
the complexity which scares students from studying the vagaries of tops;
but in all cases that are likely to come before an engineer it would be
absurd to study {138} such a small correction, and consequently calculation
is exceedingly simple.
Inventors using gyrostats have succeeded in doing the following things--
(1) Keeping the platform of a gun level on board ship, however the ship may
roll or pitch. Keeping a submarine vessel or a flying machine with any
plane exactly horizontal or inclined in any specified way.[14] It is easy
to effect such objects without the use of a gyrostat, as by means of spirit
levels it is possible to command powerful electric or other motors to keep
anything always level. The actual methods employed by Mr. Beauchamp Tower
(an hydraulic method), and by myself (an electric method), depend upon the
use of a gyrostat, which is really a pendulum, the axis being vertical.
(2) Greatly reducing the rolling (or pitching) of a ship, or the saloon of
a ship. This is the problem which Mr. Schlick has solved with great
success, at any rate in the case of torpedo boats.
(3) In Mr. Brennan's Mono-rail railway, keeping the resultant force due to
weight, wind pressure, centrifugal force, etc., exactly in line with the
rail, so that, however the load on a wagon may alter in position, and
although the wagon may be going round a curve, it is quickly brought to a
position such that there are no forces tending to alter its angular
position. The wagon leans over towards the wind or towards the centre of
the curve of the rail so as to be in equilibrium.
(4) I need not refer to such matters as the use of gyrostats for the
correction of compasses on board ship, referred to in page 111.
{139}
[Illustration: FIG. 1.]
{140} Problems (2) and (3) are those to which I wish to refer. For a ship
of 6,000 tons Mr. Schlick would use a large wheel of 10 to 20 tons,
revolving about an axis E F (fig. 1) whose mean position is vertical. Its
bearings are in a frame E C F D which can move about a thwart-ship axis C D
with a precessional motion. Its centre of gravity is below this axis, so
that like the ship itself the frame is in stable equilibrium. Let the ship
have rolled through an angle R from its upright position, and suppose the
axis E F to have precessed through the angle P from a vertical position.
Let the angular velocity of rolling be called [.R], and the angular
velocity of precession [.P]; let the moment of momentum of the wheel be m.
For any vibrating body like a ship it is easy to write out the equation of
motion; into this equation we have merely to introduce the moment m [.P]
diminishing R; into the equation for P we merely introduce the moment m
[.R] increasing P. As usual we introduce frictional terms; in the first
place F [.R] (F being a constant co-efficient) stilling the roll of the
ship; in the second case f [.P] a fluid friction introduced by a pair of
dash pots applied at the pins A and B to still the precessional vibrations
of the frame. It will be found that the angular motion P is very much
greater than the roll R. Indeed, so great is P that there are stops to
prevent its exceeding a certain amount. Of course so long as a stop acts,
preventing precession, the roll of the ship proceeds as if the gyrostat
wheel were not rotating. Mr. Schlick drives his wheels by steam; he will
probably in future do as Mr. Brennan does, drive them by electromotors, and
keep them in air-tight cases in good vacuums, because the loss of energy by
friction against an atmosphere is proportional to the density of the
atmosphere. The solution of the equations to find the nature of the R and P
motions is sometimes tedious, but requires no great amount of mathematical
knowledge. In a case considered by me of {141} a 6,000 ton ship, the period
of a roll was increased from 14 to 20 seconds by the use of the gyrostat,
and the roll rapidly diminished in amount. There was accompanying this slow
periodic motion, one of a two seconds' period, but if it did appear it was
damped out with great rapidity. Of course it is assumed that, by the use of
bilge keels and rolling chambers, and as low a metacentre as is allowable,
we have already lengthened the time of vibration, and damped the roll R as
much as possible, before applying the gyrostat. I take it that everybody
knows the importance of lengthening the period of the natural roll of a
ship, although he may not know the reason. The reason why modern ships of
great tonnage are so steady is because their natural periodic times of
rolling vibration are so much greater than the probable periods of any
waves of the sea, for if a series of waves acts upon a ship tending to make
it roll, if the periodic time of each wave is not very different from the
natural periodic time of vibration of the ship, the rolling motion may
become dangerously great.
If we try to apply Mr. Schlick's method to Mr. Brennan's car it is easy to
show that there is instability of motion, whether there is or is not
friction. If there is no friction, and we make the gyrostat frame unstable
by keeping its centre of gravity above the axis C D, there will be
vibrations, but the smallest amount of friction will cause these vibrations
to get greater and greater. Even without friction there will be instability
if m, the moment of momentum of the wheel, is less than a certain amount.
We see, then, that no form of the Schlick method, or modification of it,
can be applied to solve the Brennan problem.
{142}
[Illustration: FIG. 2.]
{143} Mr. Brennan's method of working is quite different from that of Mr.
Schlick. Fig. 2 shows his model car (about six feet long); it is driven by
electric accumulators carried by the car. His gyrostat wheels are driven by
electromotors (not shown in fig. 3); as they are revolving in nearly
vacuous spaces they consume but little power, and even if the current were
stopped they would continue running at sufficiently high speeds to be
effective for a length of time. Still it must not be forgotten that energy
is wasted in friction, and work has to be done in bringing the car to a new
position of equilibrium, and this energy is supplied by the electromotors.
Should the gyrostats really stop, or fall to a certain low speed, two
supports are automatically dropped, one on either side of the car; each of
them drops till it reaches the ground; one of them dropping, perhaps, much
farther than the other.
The real full-size car, which he is now constructing, may be pulled with
other cars by any kind of locomotive using electricity or petrol or steam,
or each of the wheels may be a driving wheel. He would prefer to generate
electropower on his train, and to drive every wheel with an electric motor.
His wheels are so independent of one another that they can take very quick
curves and vertical inequalities of the rail. The rail is fastened to
sleepers lying on ground that may have sidelong slope. The model car is
supported by a mono-rail bogie at each end; each bogie has two wheels
pivoted both vertically and horizontally; it runs on a round iron gas pipe,
and sometimes on steel wire rope; the ground is nowhere levelled or cut,
and at one place the rail is a steel wire rope spanning a gorge, as shown
in fig. 2. It is interesting to stop the car in the middle of this rope and
to swing the rope sideways to see the automatic balancing of the car. The
car may be left here or elsewhere balancing itself with nobody in charge of
it. If the load on the car--great lead weights--be dumped about into new
positions, the car adjusts itself to the new conditions with great {144}
quickness. When the car is stopped, if a person standing on the ground
pushes the car sidewise, the car of course pushes in opposition, like an
indignant animal, and by judicious pushing and yielding it is possible to
cause a considerable tilt. Left now to itself the car rights itself very
quickly.
[Illustration: FIG. 3.]
{145}
[Illustration: FIG. _3^b_ (showing the ground-plan of Fig. 3).]
{146} Fig. 3 is a diagrammatic representation of Mr. Brennan's pair of
gyrostats in sectional elevation and plan. The cases G and G', inside which
the wheels F and F' are rotating _in vacuo_ at the same speed and in
opposite directions (driven by electromotors not shown in the figure), are
pivoted about vertical axes E J and E' J'. They are connected by
spur-toothed segments J J and J' J', so that their precessional motions are
equal and opposite. The whole system is pivoted about C, a longitudinal
axis. Thus when precessing so that H comes out of the paper, so will H',
and when H goes into the paper, so does H'. When the car is in equilibrium
the axes K H and K' H' are in line N O O' N' across the car in the plane of
the paper. They are also in a line which is at right angles to the total
resultant (vertical or nearly vertical) force on the car. I will call
N O O' N' the mid position. Let 1/2m be the moment of momentum of either
wheel. Let us suppose that suddenly the car finds that it is not in
equilibrium because of a gust of wind, or centrifugal force, or an
alteration of loading, so that the shelf D comes up against H, the spinning
axis (or a roller revolving with the spinning axis) of the gyrostat. H
begins to roll away from me, and if no slipping occurred (but there always
is slipping, and, indeed, slipping is a necessary condition) it would roll,
that is, the gyrostats would precess with a constant angular velocity
[alpha], and exert the moment m[alpha] upon the shelf D, and therefore on
the car. It is to be observed that this is greater as the diameter of the
rolling part is greater. This precession continues until the roller and the
shelf cease to touch. At first H lifts with the shelf, and afterwards the
shelf moving downwards is followed for some distance by the roller. If the
tilt had been in the opposite direction the shelf D' would have acted
upwards upon the roller H', and caused just the opposite kind of
precession, and a moment of the opposite kind.
We now have the spindles out of their mid position; how are they brought
back from O Q and O' Q' to O N and O' N', {147} but with H permanently
lowered just the right amount? It is the essence of Mr. Brennan's invention
that after a restoring moment has been applied to the car the spindles
shall go back to the position N O O' N' (with H permanently lowered), so as
to be ready to act again. He effects this object in various ways. Some ways
described in his patents are quite different from what is used on the
model, and the method to be used on the full-size wagon will again be quite
different. I will describe one of the methods. Mr. Brennan tells me that he
considers this old method to be crude, but he is naturally unwilling to
allow me to publish his latest method.
D' is a circular shelf extending from the mid position in my direction; D
is a similar shelf extending from the mid position into the paper, or away
from me. It is on these shelves that H' and H roll, causing precession away
from N O O' N', as I have just described. When H' is inside the paper, or
when H is outside the paper, they find no shelf to roll upon. There are,
however, two other shelves L and L', for two other rollers M and M', which
are attached to the frames concentric with the spindles; they are free to
rotate, but are not rotated by the spindles. When they are pressed by their
shelves L or L' this causes negative precession, and they roll towards the
N O O' N' position. There is, of course, friction at their supports,
retarding their rotation, and therefore the precession. The important thing
to remember is that H and H', when they touch their shelves (when one is
touching the other is not touching) cause a precession away from the mid
position N O O' N' at a rate [alpha], which produces a restoring moment
m[alpha] of nearly constant amount (except for slipping), whereas where M
or M' touches its shelf L or L' (when one is touching the other is not
touching) the pressure on the shelf and friction determine the rate of the
precession towards the mid position N O O' N', {148} as well as the small
vertical motion. The friction at the supports of M and M' is necessary.
Suppose that the tilt from the equilibrium position to be corrected is R,
when D presses H upward. The moment m[alpha], and its time of action (the
total momental impulse) are too great, and R is over-corrected; this causes
the roller M' to act on L', and the spindles return to the mid position;
they go beyond the mid position, and now the roller H' acts on D', and
there is a return to the mid position, and beyond it a little, and so it
goes on, the swings of the gyrostats out of and into the mid position, and
the vibrations of the car about its position of equilibrium getting rapidly
less and less until again neither H nor H', nor M nor M' is touching a
shelf. It is indeed marvellous to see how rapidly the swings decay.
Friction accelerates the precession away from N O O' N'. Friction retards
the precession towards the middle position.
It will be seen that by using the two gyrostats instead of one when there
is a curve on the line, although the plane N O O' N' rotates, and we may
say that the gyrostats precess, the tilting couples which they might
exercise are equal and opposite. I do not know if Mr. Brennan has tried a
single gyrostat, the mid position of the axis of the wheel being vertical,
but even in this case a change of slope, or inequalities in the line, might
make it necessary to have a pair.
It is evident that this method of Mr. Brennan is altogether different in
character from that of Mr. Schlick. Work is here actually done which must
be supplied by the electromotors.
One of the most important things to know is this: the Brennan model is
wonderfully successful; the weight of the apparatus is not a large fraction
of the weight of the wagon; will this also be the case with a car weighing
1,000 times as {149} much? The calculation is not difficult, but I may not
give it here. If we assume that suddenly the wagon finds itself at the
angle R from its position of equilibrium, it may be taken that if the size
of each dimension of the wagon be multiplied by n, and the size of each
dimension of the apparatus be multiplied by p, then for a sudden gust of
wind, or suddenly coming on a curve, or a sudden shift of position of part
of the cargo, R may be taken as inversely proportional to n. I need not
state the reasonable assumption which underlies this calculation, but the
result is that if n is 10, p is 7.5. That is, if the weight of the wagon is
multiplied by 1,000, the weight of the apparatus is only multiplied by 420.
In fact, if, in the model, the weight of the apparatus is 10 per cent. of
that of the wagon, in the large wagon the weight of the apparatus is only
about 4 per cent. of that of the wagon. This is a very satisfactory
result.[15]
My calculations seem to show that Mr. Schlick's apparatus will form a
larger fraction of the whole weight of a ship, as the ship is larger, but
in the present experimental stage of the subject it is unfair to say more
than that this seems probable. My own opinion is that large ships are
sufficiently steady already.
In both cases it has to be remembered that if the _diameter_ of the wheel
can be increased in greater proportion than the dimensions of ship or
wagon, the proportional weight of the apparatus may be diminished. A wheel
of twice the diameter, but of the same weight, may have twice the moment of
momentum, and may therefore be twice as effective. I assume the stresses in
the material to be the same.
* * * * *
{150}
APPENDIX II.
Page 23; note at line 3. Prof. Osborne Reynolds made the interesting remark
(_Collected Papers_, Vol. ii., p. 154), "That if solid matter had certain
kinds of internal motions, such as the box has, pears differing, say, from
apples, the laws of motion would not have been discovered; if discovered
for pears they would not have applied to apples."
Page 38; note at line 8. The motion of a rifle bullet is therefore one of
precession about the tangent to the path. The mathematical solution is
difficult, but Prof. Greenhill has satisfied himself mathematically that
air friction damps the precession, and causes the axis of the shot to get
nearer the tangential direction, so that fig. 10 illustrates what would
occur in a vacuum, but not in air. It is probable that this statement
applies only to certain proportions of length to diameter.