A pendulous body vibrates when it is suspended so that the centre of
its mass is not placed directly under the point of suspension, because
then the alternating influences of weight and velocity are constantly
impressing it with motion. Weight carries it down as far as it can go
towards the earth's attraction; acquired velocity then carries it
onwards; but as the onward movement is constrained to be upward
against the direction of the earth's attraction, that force
antagonises, and at last arrests it, for velocity flags when it has to
drag its load up-hill, and soon gives over the effort. The body swings
down-hill with increasing rapidity, because weight and velocity are
then both driving it; it swings up-hill with diminishing rapidity,
because then weight is pulling it back in opposition to the force of
velocity. Weight pulls first this way, then that way; velocity carries
first this way, then that way: but the two powers do not act evenly
and steadily together; they now combine with, and now oppose each
other; now increase their influence together, and now augment and
diminish it inversely and alternately; and so the suspended body is
tossed backwards and forwards between them, and made to perform its
endless dance.
It is related of Galileo, that he once stood watching a swinging lamp,
hung from the roof of the cathedral at Pisa, until he convinced
himself that it performed its vibratory movement in the same time,
whether the vibration was one of wide or of narrow span. This
traditionary tale is most probably correct in its main features, for
the Newtons and Galileos of all ages do perceive great truths in
occurrences that are as commonplace as the fall of an apple, or the
disturbance of a hanging lamp. Trifles are full of meaning to them,
because their minds are already prepared to arrive at certain
conclusions by means of antecedent reflections. Simple and familiar
incidents, thus accidentally associated with the history of grand
discoveries, are the channels through which the accumulating waters at
length descend, rather than the rills which feed the swelling of their
floods. The orchard at Woolsthorpe, and the cathedral at Pisa, were
outlets of this kind, through which the pent-up tide of gathering
knowledge burst. If they had never offered themselves, the laws of
universal gravitation and isochronous vibration would still have
reached the world.
If the reader will hang up two equal weights upon nearly the same
point of suspension, and by means of two strings of exactly the same
length, he will have an apparatus at his command that will enable him
to see, under even more favourable conditions, what Galileo saw in the
cathedral at Pisa. Upon drawing one of them aside one foot from the
position of rest, and the other one yard, and then starting them off
both together to vibrate backwards and forwards, he will observe, that
although the second has a journey of two yards to accomplish, while
the first has but a journey of two feet, the two will, nevertheless,
come to the end at precisely the same instant. As the weights swing
from side to side in successive oscillations, they will always present
themselves together at the point which is the middle of their
respective arcs. This is what is called isochronous vibration--the
passing through unequal arcs in equal periods of time.
At the first glance, this seems a very singular result. The careless
observer naturally expects that a weight hung upon a string ought to
take longer to move through a long arc than through a short one, if
impelled by the same force; but the subject appears in a different
light upon more mature reflection, for it is then seen, that the
weight which performs the longer journey starts down the steeper
declivity, and therefore acquires a greater velocity. A ball does not
run down a steep hill and a more gently inclined one at the same pace;
neither, therefore, will the suspended weight move down the steeper
curve, and the less raised one, at equal rates. The weight which moves
the fastest, of necessity gets through more space in a given period
than its more leisurely companion does. The equality of the periods in
which two weights vibrate, is perfect so long as both the unequal arcs
of motion are short ones, when compared with the length of the
suspending strings; but even when one of the arcs is five times longer
than the other, ten thousand vibrations will be completed before one
weight is an entire stride in advance of the other; and even this
small amount of difference is destroyed when the arc in which the
weights swing is a little flattened from the circular curve.
But there is yet another surprise to be encountered. Hang a weight of
a pound upon one of the strings, and a weight of two pounds upon the
other, and set them vibrating in arcs of unequal length as before, and
still their motions will be found to be isochronous. Unequal weights,
as well as equal ones, when hung on equal strings, will swing through
arcs of unequal length in equal periods of time. This seeming
inconsistency also admits of a satisfactory explanation. It has been
stated, that the motion of swinging bodies is caused by the earth's
attraction. But what are the facts that are more particularly implied
in this statement? What discoveries does the philosophic inquirer make
when he looks more narrowly into it? For the sake of familiar
illustration, let it be imagined that a man stands at the top of the
Monument of London, with two leaden bullets in his hand, each weighing
an ounce, and that he drops these together. They go to the earth,
because the earth's mass draws them thither; and since the two bodies
exactly resemble each other, and start at the same instant upon their
descent, they must of course both strike the pavement beneath
simultaneously. There can be no reason why one should get down before
the other, for the same influence causes the fall of each. The entire
mass of the huge earth attracts each bullet alike, and the bullets,
therefore, yield like obedience to the influence, and fall together to
the ground.
But now, suppose that the two bullets were to be all at once fused
into one, and that this combined mass were then dropped from the top
of the Monument as a single bullet, would there then be any reason why
the two ounces of lead should make a more rapid descent than they
would have made while in separate halves? Clearly not. There is but
the same earth to attract, and the same number of particles to be
drawn in each case, and therefore the same result must ensue. Each
particle still renders its own individual obedience, and makes its own
independent fall, although joined cohesively to its neighbours. It is
the mass of the attracting body, and not the mass of the attracted
body, that determines the velocity with which the latter moves. The
greater mass of an attracted body expends its superior power, not in
increasing its own rate of motion, but in pulling more energetically
against the attracting mass. Every particle of matter when at rest
resists any attempt to impress it with motion. The amount of this
resistance is called its inertia. When many particles are united
together into one body, they not only, therefore, take to that body
many points upon which the earth's attraction can tell, but they also
carry to it a like quantity of resistance or inertia, which must be
overcome before any given extent of motion can be produced. If the
earth's force be but just able to make particle 1 of any body go
through 200 inches in a second, it will also be but just able to make
particles 2, 3, and 4 do the same; consequently, whether those
particles be separate or combined together, their rate of travelling
will be the same. Hence all bodies descend to the earth with exactly
the same velocities, however different their natures may be in the
matter of weight, always provided there be no retarding influence to
act unequally upon their different bulks and surfaces. It is well
known that even a guinea and feather will fall together when the
atmospheric resistance is removed from their path.
The reader will now, of course, see that what is true of the motion of
free bodies, must also be true of the motion of suspended ones, since
the same terrestrial attraction causes both. There is no reason why
the two-pound weight in the experiment should vibrate quicker than the
one-pound weight, just as there is no reason why a two-ounce bullet
should fall quicker than a one-ounce bullet. Here, also, there are
only the same number of terrestrial particles to act upon each
separate particle of the two unequal weights. Hence it is that the
vibrations of unequal weights are isochronous when hung on strings of
equal lengths.
Thus far our dealings have been with what has seemed to be a very
single-purposed and determined agent. We have hung a weight upon a
piece of string and set it swinging, and have then seen it persisting
in making the same number of beats in the same period of time, whether
we have given it a long journey or a short one to perform; and also
whether we have added to or taken from its mass. But now we enter upon
altogether new relations with our little neophyte, and find that we
have reached the limits of its patience.
Take three pieces of string of unequal lengths--one being one foot
long; the second, four feet; and the third, nine feet. Hang them up by
one extremity, and attach to each of the other ends a weight. Then
start the three weights all off together vibrating, and observe what
happens. The several bodies do not now all vibrate in the same times
as in the previous experiments. By making the lengths of the strings
unequal, we have introduced elements of discord into the company. The
weight on the shortest string makes three journeys, and the weight on
the next longest string makes two journeys, while the other is
loitering through one.
This discrepancy, again, is only what the behaviour of the vibrating
masses in the previous experiments should have taught the observer to
anticipate. Each of the weights in this new arrangement of the
strings, has to swing in the portion of a circle, which, if completed,
would have a different dimension from the circles in which the other
weights swing. The one on the shortest string swings in the segment of
a circle that would be two feet across; the one on the longest string
swings in the segment of a circle that would be eighteen feet across.
Now, if these two weights be made to vibrate in arcs that shall
measure exactly the twelfth part of the entire circumference of their
respective circles, then one will go backwards and forwards in a
curved line only half a foot long, while the other will move in a line
four feet and a half long.
But both these weights, the one going upon the short journey, and the
other upon the long, will start down exactly the same inclination or
declivity. The reader will see that this must be the case if he will
draw two circles on paper round a common centre, the one at the
distance of one inch, and the other at the distance of nine inches.
Having done this, let him cut a notch out of the paper, extending
through both the circles to the centre, and including a twelfth part,
or thirty degrees, of each between its converging sides. He will then
observe, that the two arcs cut out by the notch are everywhere
concentric with each other; therefore, their beginnings and endings
are concentric or inclined in exactly the same degree to a
perpendicular crossing their centres. These concentric beginnings and
endings represent correctly the concentric directions in which the
swinging weights commence their downward movements.
Now, since it has been shewn that bodies begin to run down equal
descents with equal velocities, it follows that the weight on the
short string and that on the long string must commence to move down
the concentric curves of their respective arcs at an equal rate. But
it has been also shewn that the one of these weights has a nine times
longer journey to perform than the other; it is clear, therefore, that
both cannot accomplish their respective distances in the same time.
The weight on the shortest string in reality makes three vibrations,
and the weight on the string that is next to this in length makes two
vibrations, while the weight on the longest string is occupied about
one; and the differences would be as 9, 4, and 1, instead of as 3, 2,
1, but that the weights moving in the longer arcs benefit most from
acceleration of velocity. Although all the vibrating bodies begin to
move at equal rates, they pass the central positions directly beneath
their points of suspension at unequal ones. Those that have been the
longest in getting down to these positions, have of necessity
increased their paces the most while upon their route.
Suspended weights, then, only vibrate in equal times when hung upon
equal strings; but they continue to make vibrations in equal times
notwithstanding the diminution of the arcs in which they swing. This
was the fact that caught the attention of Galileo; he observed that
the vibrations of the lamp slowly died away as the effect of the
disturbing force was destroyed bit by bit, but that, nevertheless, the
last faint vibration that caught his eye, took the same apparent time
for its performance as the fullest and longest one in the series.
The instrument that has been designated by the learned name of
pendulum, is simply a weight of this description placed on the end of
a metallic or wooden rod, and hung up in such a way that free sideways
motion is permitted. This freedom of motion is generally attained by
fixing the top of the rod to a piece of thin, highly elastic steel. A
pendulum fitted up after this fashion, will continue in motion, if
once started, for many hours. It only stops at last, because the air
opposes a slight resistance to its passage, and because the suspending
spring is imperfectly elastic. The effects of these two causes
combined arrest the vibration at last, but not until they have long
accumulated. The weight does not stand still at once, but its arc of
vibration grows imperceptibly less and less, until at last there comes
a time when the eye cannot tell whether the body is still moving or in
absolute repose.
Now, suppose that a careful and patient observer, aware of the exact
length of the suspending-rod of a vibrating pendulum, were to set
himself down to count how many beats it would make in a given period,
he would thenceforward be able to assign a fixed value to each beat,
and would consequently have acquired an invariable standard whereby he
might estimate short intervals. If he found that his instrument had
made exactly 86,400 beats at the end of a mean solar day, and knew
that the length of its rod was a trifle more than 39 inches, he would
be aware that each beat of such a pendulum might always be taken as
the measure of a second. The length of the rod of a pendulum which
beats exact seconds in London is 39.13 inches.
But there are few persons who would be willing to go through the
tedious operation of counting 86,400 successive vibrations. The
invention of a mechanical contrivance that was able to break the
monotony of such a task, would be hailed by any one who had to perform
it as an invaluable boon. Even a piece of brass with sixty notches
upon it, which he might slip through his fingers while noting the
swinging body, would enable him to keep his reckoning by sixties
instead of units, and so far would afford him considerable relief. But
if the notched brass could be turned into a ring, and the pendulum be
made to count the notches off for itself, round and round again
continuously, registering each revolution as it was completed for
future reference, the observer would attain the same result without
expending any personal trouble about it. It is this magical conversion
of brass and iron into almost intelligent counters of the pendulum's
vibrations, that the clock-maker effects by his beautiful mechanism.
In the pendulum clock, the top of the swinging-rod is connected with a
curved piece of steel, which dips its teeth-like ends on either hand
into notches deeply cut in the edges of a brass wheel. The notched
wheel is connected with a train of wheel-work kept moving by the
descent of a heavy weight; but it can only move onwards in its
revolution under the influence of the weight, as the two ends of the
piece of steel are alternately lifted out of the notches by the
swaying of the pendulum. The other wheels and pinions of the movement
are so arranged that they indicate the number of turns the wheel at
the top of the pendulum completes, by means of hands traversing round
a dial-plate inscribed with figures and dots.
It is found convenient in practice to make the direct descent of a
weight the moving power of the wheel-work, instead of the swinging of
the pendulum, for the simple reason, that the excess of its power
beyond what is required to overcome the friction of the wheel-work, is
then employed in giving a slight push to the pendulum; this push just
neutralises the retarding effects before named as inseparable from the
presence of air and imperfect means of suspension. The train of
wheel-work in a clock, therefore, serves two purposes--it records the
number of beats which the pendulum makes, and it keeps that body
moving when once started. As far as the activity of the pendulum is
concerned, the wheel-work is a recording power, and a preserving
power, but _not_ an originating power. If there were no air, and no
friction in the apparatus of suspension, the pendulum would continue
to go as well without the wheel-work as with it. With the wheel-work
it beats as permanently and steadily upon material supports and
plunged in a dense atmosphere, as it would if it were hung upon
nothing, and were swinging in nothing; and also performs its backward
and forward business in solitude and darkness, to the same practical
purpose that it would if the eyes of watchful and observant guardians
were turned incessantly towards it.
Galileo published his discovery of the isochronous property of the
pendulum in 1639. Richard Harris of London took the hint, and
connected the pendulum with clock-work movement in 1641. Huyghens
subsequently improved the connection, and succeeded in constructing
very trustworthy time-keepers, certainly before 1658.
But notwithstanding all that the knowledge and skill of Huyghens could
do, his most perfect instruments were still at the mercy of
atmospheric changes. It has been said, that the time of a pendulum's
vibration depends upon the length of its suspending-rod. This length
is measured, not down to the bottom of the weight, but to the centre
of its mass. For the weight itself is necessarily a body of
considerable dimensions, and in this body some particles must be
nearer to, and others further from the point of suspension. Those
which are nearest will, of course, in accordance with the principles
already explained, have a tendency to make their vibrations in shorter
periods; and those which are furthest, in longer periods. But all
these particles are bound together firmly by the power of cohesion,
and must move connectedly. They, therefore, come to an agreement to
move at a mean rate--that is, between the two extremes. The top
particles hurry on the middle ones; the bottom particles retard them
in a like degree. Consequently, the whole of the weight moves as if
its entire mass were concentered in the position of those middle
particles; and the exact place of this central position in relation to
the point of suspension, becomes the important condition which
determines the time in which the instrument swings.
In pendulums of ordinary construction, this relation is by no means an
unvarying one--changes of temperature alter the bulk of all kinds of
bodies. A metal rod runs up and down under increase and diminution of
heat, as certainly as the thread of mercury in the tube of the
thermometer does. A hot day, therefore, lengthens the metallic
suspending-rod of a pendulum, and carries the centre of its weight to
a greater distance from the point of suspension. By this means, the
period of each vibration is of necessity lengthened. An increase of
temperature to the extent of ten of Fahrenheit's degrees, will make a
second's pendulum with a brass rod lose five vibrations in a day. All
substances do not, however, suffer the same amount of expansion under
like increments of heat. If the rod of the pendulum be made of
varnished or black-leaded wood, an addition of ten degrees of heat
will not cause it to lose more than one vibration in a day. But even
this small irregularity is too vast for the purposes of precise
science, and accordingly ingenuity has been taxed to the utmost to
find some means of removing the source of inaccuracy, to invent some
plan whereby the pendulum may be made sensitive enough to discover and
correct its own varying dimensions as different temperatures are
brought to bear upon its material.
The first successful attempt to accomplish this useful purpose was
made by George Graham in 1715. He replaced the solid weight at the
bottom of the rod by a glass jar containing mercury. The rod he
formed of steel of the usual length; and because mercury expands five
times more than steel, he fixed the height of the column of mercury in
the jar at only 6-1/2 inches. In this arrangement he found that
additional heat carried up the mercury in the jar, as much as it
carried down the jar by the elongation of the rod. Consequently, the
motion of the one perfectly compensated the motion of the other, and
the effective centre of the weight always remained at the same precise
distance from the top of the rod. By the application of this
compensating pendulum, clocks are now constructed that do not vary to
the extent of a tenth of a second in a day.
Soon after the invention of Graham's mercurial pendulum, John
Harrison--the same clever mechanician who received L.20,000 from
government for making a chronometer that went to Jamaica in one year
and returned in another with an accumulated error of only 1 minute and
54 seconds--hit upon another means of gaining the same end. He brought
a steel rod down from the point of suspension, turned it up into a
copper rod of less length; and from the top of this hung the weight.
He fixed the lengths of the steel and copper rods, which expand
unequally, in such a way that the steel carried the copper down
exactly as much as the copper carried the weight up; and thus the
centre of the weight was still kept at the same distance from the real
point of suspension. Harrison's pendulum is generally seen in somewhat
the form of a gridiron, because many parallel bars of copper and steel
are used in its construction, for the sake of rendering it firm and
unyielding in all its parts.