The claim to the invention of the pendulum, like the claim to most invitations, is disputed; and we have no intention of trying to settle it. It was, like many other discoveries and inventions, probably made by various persons independently, and almost simultaneously, when the state of science had become ripe for it. The discovery of that peculiarly valuable property of the pendulum called isochronism, or the disposition to vibrate different arcs in very nearly the same time (provided the arc are none of them large), commonly attributed to Galileo, in the well-known story of his being struck with the isochronism of a chandelier hung by a long chain from the roof of the church at Florence. And Galileos son appears as a rival of Avicenna, Huyghens, Dr Hooke, and a London clockmaster named Harris, for the honour of having first applied the pendulum ro regulate the motion of a clock train, all in the early part of the 17th century. Be this as it may, there seems little doubt that Huyghens was the first who mathematically investigated, and therefore really knew, the true nature of those properties of the pendulum which may now be found explained in any mathematical book on mechanics. He discovered that if a simple pendulum (i.e., a weight or bob consisting of a single point, and hung by a rod or string of no weight) can be made to describe, not a circle, but a cycloid of which the string would be the radius of curvature at the lowest point, all its vibrations, however large, will be performed in the same time. For a little distance near the bottom, the circle very nearly coincides with the cycloid; and hence it is that, for small arcs, a pendulum vibrating as usual in a circle is nearly enough isochronous for the purposes of horology; more especially when contrivances are introduced either to compensate for the variations of the arc, or, better still, to destroy them altogether, by making the force on the pendulum so constant that its arc may never sensibly vary.
The difference between the time of any small arc of the circle and any arc of the cycloid varies nearly as the square of the circular arc; and again, the difference between the times of any two small and nearly equal circular arcs of the same pendulum, varies nearly as the arc itself. If a, the arc, is increased by a small amount da, the pendulum will lose 10800 ada seconds a day, which is rather more than 1 second, if a is 2º (from zero) and da is 10', since the numerical value of 2º is ·035. If the increase of arc is considerable, it will not do to reckon thus by differentials, but we must take the difference of time for the day as 5400 (a,2a2 ),which will be just 8 seconds if a=2º and a,=3º. For many years it was thought of great importance to obtain cycloidal vibrations of clock pendulums, and it was done by making the suspension string or spring vibrate between cycloidal cheeks, as they were called. But it was in time discovered that all this is a delusion,first, because there is and can be no such thing in reality as a simple pendulum, and cycloidal cheeks will only make a simple pendulum vibrate isochronously; secondly, because a very slight error in the form of the cheeks (as Huyghens himself discovered) would do more harm than he circular error uncorrected, even for an arc of 10º, which is much larger than the common pendulum arc; thirdly, because there was always some friction or adhesion between the cheeks and the string; and fourthly (a reason which applies equally to all the isochronous contrivances since invented), because a common clock escapement itself generally tends to produce an error exactly opposite to the circular error or to make the pendulum vibrate quicker the farther it swings; and therefore the circular error is actually useful for the purpose of helping to counteract the error due to the escapement, and the clock goes better than it would with a simple pendulum, describing the most perfect cycloid. At the same time, the thin spring by which pendulum are always suspended, except in some French clocks where a silk string is used is very inferior plan), causes the pendulum to deviate a little from circular and to approximate to cycloidal motion, becaue the bend does not take place at one point, but is spread over some length of the spring.
The accurate performance of a clock depends so essentially on the pendulum, that we shall go somewhat into detail respecting it. First then, the time of vibration depends entirely on the length of the pendulum, the effect of the spring being too small for consideration until we come to difference of a higher order. But the time does not vary as the length, but only as the square root of the length; i.e. a pendulum to vibrate two seconds must be four times as long as a seconds pendulum. The relation between the time of vibration and the length of a pendulum is expressed thust=&Mac185;&Mac195; l/g, where t is the time in seconds, &Mac185; the well-known symbol for 3·14159, the ratio of the circumference of a circle to its diameter, l the length of the pendulum, and g the force of gravity at the latitude where it is intended to vibrate. This letter g, in the latitude of London, is the symbol for 32·2 feet, that being the velocity (or number of feet per second) at which a body is founded by experiment to be moving at the end of the first second of its fall, being necessarily equal to twice the actual number of feet it has fallen in that second. Consequently, the length of a pendulum to beat seconds in London is 39·4 inches. But the same pendulum carried to the equator, where the force of gravity is less, would lose 2 _ minutes a day.
The seconds we are here speaking of are the seconds of common clock indiacting mean solar time. But as clocks are also required for sidereal time, it may be as well to mention the proportions between a mean and a sidereal pendulum. A sidereal day is the interval between two successive transits over the meridian of a place by that imaginary point in the heavens called _, the first point of Aries, at the intersection of the equator and the ecliptic; and there is one more sidereal day than there are solar days in a year, since the earth has to turn more than once round in space before the sun can come a second time to the meridian, on account of the earths own motion in its orbit during the day. A sidereal day or hour is shorter than a mean solar one in the ratio of ·99727, and consequently a sidereal pendulum must be shorter than a mean time pendulum in the square of that ratio, or in the latitude of London the sidereal seconds pendulum is 38·87 inches. As we have mentioned what is 9 to 24 oclock by sidereal time, we may as well add, that the mean day is also reckoned in astronomy by 24 hours, and not from midnight as in civil reckoning, but from the following noon; thus, what we call 11 A.M. May 1 in common life is 23 h. April 30 with astronomers.
It must be remembered that the pendulums whose lengths we have been speaking of are simple pendulums; and as that is a thing which can only exist in theory, the reader may ask how the length of a real pendulum to vibrate in any required time is ascertained. In every pendulum, that is to say, in every body hung so as to be capable of vibrating freely, there is a certain point, always somewhere below the centre of gravity, which possesses these remarkable propertiesthat if the pendulum were turned upside down, and set vibrating about this point, it would vibrate in the same time as before, and moreover, that that the distance of this point form the point of suspension is exactly the length of that imaginary simple pendulum which would vibrate in the same time. This point is therefore called the centre of oscillation. The rules for finding it by calculation are too complicated for ordinary use, except in bodies of certain simple and regular forms; but they are fortunately not requisite in practice, because in all clock pendulum the centre of oscillation is only a short distance below the centre of gravity of the whole pendulum, and generally so near to the centre of gravity of the whole pendulum, and generally so near to the centre of gravity of the bobin fact a little above itthat there is no difficulty in making a pendulum for any given time of vibratiuon near enough to the proper length at once, and then adjusting it by screwing the ob up or down until it is found to vibrate in the proper time.
Revolving of Conical Pendulum
Thus far we have been speaking of vibrating pendulums; but the notice of pendulums would be incomplete without some allusion to revolving or conical pendulums, as they called, because they describe a cone in revolving. Such pendulums are used where a continuous instead of an intermittent motion of the clock train is required as in the clocks for keeping an equatorial telescope directed to a star, by driving it the opposite way to the motion of the earth, to whose axis the axis on which the telescope turns is made parallel. Clocks with such pendulums may also be used in bedrooms by persons who cannot bear the ticking of a common clock. The pendulum, instead of being hung by a flat spring, is hung by a thin piece of piano-forte wire; and it should be understood that it has no tendency to twist on its own axis, and so to twist off the wire, as may be apprehended; in fact, it would require some extra force to make it twist, if it were wanted to do so. The time of revolution of such a pendulum may be easily ascertained as follows:Let l be its length; a the angle which it makes with the vertical axis of the cone which it describes; _2 sin a; and as this is the force which keeps the pendulum away from the vertical, it must balance the force which draws it to the vertical, which is g tan. a: and therefore &Mac195; g/l cos. _=the angular velocity, or the angle described in a second of time; and the time of complete revolution through the angle 360º or 2&Mac185; is 2&Mac185;/w=2&Mac185;&Mac195; l cos.a/g; that is to say, the time of revolution of a pendulum of any given length is less than the time of a doubvle oscillation of the same pendulum, in the proportion of the cosine of the angle which it makes with the axis of revolution to unity.
A rotary pendulum is kept in motion by the train of the clock ending in a horizontal wheel with a vertical axis, from which projects an arm pressing against a spike at the bottom of the pendulum; and it has this advantage that any inequality in the force of the train, arising from variations of friction or any other cause, is immediately transmitted to the pendulum; whereas it will be seen that in several kinds of escapements which can be applied to a vibrating pendulum, the variations of force can be rendered nearly or quite insensible. And it is a mistake to imagine that there is any self-correcting power in a conical pendulum analogous to that of the governor of a stream-engine; for that apparatus, though it is a couple of conical pendulums, has also a communication by a system of levers with the valve which supplies the steam. The governor apparatus has itself been applied to telescope-driving clocks, with a lever ending in a spring which acts by friction on some revolving plate in the clock, increasing the friction, and so diminishing the force as the balls of the governor fly out farther under any increase in the force. And with the addition of some connection with the hand of the observer, by which the action can be father moderated, the motion can be made sufficiently uniform for that purpose.
Various other contrivances have been invented for producing a continuous clock-motion. The great equatorial telescope at Greenwich is kept in motion by a kind of water clock called in books on hydrostatics Barkers Mill, in which two horizontal pipes branching out from a vertical tubular axis have each a hole near their ends on opposite sides, from which water flows, being poured constantly into the tubular axis, which revolves on a pivot. The resistance of the air to the water issuing from the holes drives the mill round, and there are means of regulating it. Another plan is to connect a clock train having a vibrating pendulum with another clock having a conical pendulum by one of the lower wheels in the train, with a spring connection; the telescope is driven by the revolving clock train, and the other pendulum keeps it sufficiently in order, though allowing it to expatiate enough for each beat of the pendulum. The more complicated plan of Wagner of Paris described in Sir E. Becketts Rudimentary Treatise on Clocks and Watches and Bells does not appear to have ever come into use, and therefore it is now omitted.
The suspension of the pendulum on what are called knife-edges, like those of a scale-beam, has often been advocated. But though it may do well enough for short experiments, in which the effects of the elasticity of the spring are wanted to be eliminated, if fails altogether in use, even if the knife-edges and the plates which carry them are made of the hardest stones. The suspension which is now used universally, in all but some inferior foreign clocks, which have spring instead, is a thin and short spring, with one end let into the top of the pendulum, and the other screwed between two chops of metal with a pin through them, which rests firmly in a nick in the cock which carries the pendulum as shown in Fig. 2 a little farther on; and the steadiness of this cock, and its firm fixing to a wall, are essential to the accurate performance of the clock. The thinner the spring the better; provided, or coarse, it is strong enough to carry the pendulum without being bent beyond its elasticity, or bent short; not that there is much risk of that in practice. Pendulum springs are much oftener too thick than too thin; and it is worth notice that, independently of their greater effect on the natural time of vibration of the pendulum, thick and narrow springs are more liable to break than thin and broad ones of the same strength. It is of great importance that the spring should be of uniform thickness throughout its breadth; and the bottom of the chops which carry it should be exactly horizontal; otherwise the pendulum will swing with a twist, as they may be often seen to do in ill-made clocks. If the bottom of the chops is left sharp, where they clip the spring, it is very likely to break there; and therefore the sharp edges should be taken off.
The bob of the pendulum used to be generally made in the shape of a lens, with a view to its passing through the air with the least resistance. But after the importance of making the bob heavy was discovered, it became almost necessary to adopt a form of more solid content in proportion to its surface. A sphere has been occasionally used, but it is not a good shape, because a slight error in the place of the hole for the rod may make a serious difference in the amount of weight on each side, and give the pendulum a tendency to twist in motion. The mercurial jar pendulum suggested the cylindrical form, which is now generally adopted for astronomical clocks, and in the best turret clocks, with a round top to prevent any bits of mortar or dirt falling and resting upon it, which would alter the time; it also looks better than a flat-topped cylinder. There is no rule to be given for the weight of pendulums. It will be shown hereafter that, whatever escapement may be used, the errors due to any variation of force are expressed in fractions which invariably have the weight and the length of the pendulum in the denominator, though some kind of escapements require a heavy pendulum to correct their errors much less than others. And as a heavy pendulum requires very little more force to keep it in motion than a light one, being less affected by the resistance of the air, we may almost say that the heavier and longer a pendulum can be made the better; at any rate, the only limit is one of convenience; for instance, it would obviously be inconvenient to put a large pendulum of 100 _ weight in the case of an astronomical or common house clock. It may perhaps be laid down as a rule, that no astronomical clock or regulator (as they are also called) will go as well as is now expected of such clocks with a pendulum of less than 28 _ weight, and no turret clock with less than 1 cwt. Long pendulums are generally made with heavier bobs that short ones; and such a clock as that of the Houses of Parliament, with a two-seconds pendulum of 6 cwt., ought to go 44 times as well as a small turret clock with a one-second pendulum of 60_. Pendulums longer than 14 feet (2 seconds) are inconvenient, liable to be disturbed by wind, and expensive to compensate, and they are now quite disused, and most or all of the old ones removed, with their clocks, for better ones.
The regulation of pendulums, or their exact adjustment to the proper length, is primarily effected by a nut on the end of the rod, by which the bob can be screwed up or down. In the best clocks the rim of this nut is divided, with an index over it; so the exact quantity of rise or fall, or the exact acceleration or retardation, may be known, the amount due to one turn of the nut being previously ascertained. By the calculation used below for compensation of pendulums, it may be seen that if the length of the pendulum rod is l, and the breadth of one thread of the screw is called dl, then one turn of the nut will alter the rate of the clock by 43200 dl/l seconds a day; which would be just 30 seconds, if the pendulum rod is 45 incres long, and the screw had 32 thread in the inch. To accelerate the clock the nut has always to be turned to the right, as it is called, and vice versa. But in astronomical and in large turret clocks, it is desirable to avoid stopping, or in any way disturbing the pendulum; and for the finer adjustments other methods of regulation are adopted. The best is that of fixing a collar, as shown in fig. 2, capable of having very small weights laid upon it, half-way down the pendulum, this being the place where the addition of any small weight produces the greatest effect, and where, it may be added, any moving of that weight up or down on the rod produces the least effect. If M is the weight of the pendulum and l its length (down to the centre of oscillation), and m a small weight added at the distance d below the centre of suspension or above the c.o. (since they are reciprocal), t the time of vibration, and dt the acceleration due to adding m; then
-dl/l=m/2m (d/l d2 / l2 ):
from which it is evident that if d= l/2, then d T the daily acceleration = 10800m/ M; or if m is the 10800th of the weight of the pendulum it will accelerate the clock a second a day, or 10 grains will do that on a pendulum of 15lb. weight (7000 gr. being = 1lb.), or an ounce on a pendulum of 6 cwt. In like manner if d=l/3 from either top or bottom, m must+ M/7200 to accelerate the clock a second a day. The higher up the collar is the less risk there is of disturbing the pendulum in putting on or taking off the regulating weights. The weights should be made in a series, and marked _. _, 1, 2, according to the number of seconds a day by which they will accelerate; and the pendulum adjusted at first to lose a little, perhaps a second a day, when there are no weights on the collar, so that it may always have some weight on, which can be diminished or increased from time to time with certainty, as the rate may vary.
Compensation of Pendulums
Soon after pendulums began to be generally used in clocks, it was discovered tha they contained within themselves a source of error independent of the action of the clock upon them, and that they lost time in the hot weather and gained in cold, in consequence of all the substances of which they could be made expanding as the temperature increases. If l is the length of a pendulum, and dl the small increase of it from increased head, t time of the pendulum k, and t+d+ that of the pendulum l+dl; then
since (dl/l)2 may be neglected as very small; or dt=tdl/2l; and the daily loss of the clock will be 43200 dl/l seconds. The following is a table of the values of dl/l for 1000º Fahr. of heat in different substances, and also the weight of a cubic inch of each:
..·0024 ·036 lb
...·0048 ·116 lb
·0064 ·28 lb
·007 ·26 lb
·010 ·30 lb
·016 ·41 lb
. ·017 ·25 lb
Mercury (in bulk, not in length).. ·100 ·49 lb
Thus a common pendulum with an iron wire red would lose 43200 x ·00007=3 seconds a day for 10º of heat; and if adjusted for the winter temperature it would lose about a minute a week in summer, unless something in the clock happened to produce a counteracting effect, as we shall see may be the case when we come to escapements. We want therefore some contrivance which will always keep that point of the pendulum on which its time depends, viz., the centre of oscillation, at the same distance from the point of suspension. A vast number of such contrivances have been made, but there are only three which can be said to be at all in common use; and the old gridiron pendulum, made of 9 alternate bars of brass and steel is not one of them, having been superseded by one of zinc and iron, exactly on the same principle, but requiring much fewer bars on account of the greater expansion of zinc than brass. The centre of oscillation so nearly coincides in most clock pendulums with the centre of the bob that we may practically say that the object of compensation is to keep the bob always at the same height. For this purpose we must hang the bob from the top of a column of some metal which has so much more expansion than the rod that its expansion upwards will neutralize that of the rod, and of the wires or tube by which the bob is hung, downwards. The complete calculation, taking into account the weight of all the rods and tubes is too long and complicated to be worth going through, especially as it must always be finally adjusted by trial ether of that every pendulum or of one exactly similar. For practical purposes it is found sufficient to treat the expansion of zinc as being ·016 to steel ·0064m instead of ·017 as it is really; and fro large pendulums with very heavy tubes even the ·016 is a little too much. Moreover the c.o is higher above the c.g. of the bob in such large pendulums than in small ones with light rods and tubes.
But neglecting these minutiae for the first approximation, and supposing the bob either to be of iron, in which case it may be considered fixed anywhere to the iron tube which hangs from the top of the zinc tube, or a lead bob attached at its own centre, which obviates the slowness of the transmission of a change of temperature through it, the following calculation will hold. Let r be the length of the steel rod and spring, z that of the zinc tube, b half the height of the bob; the length of the iron tube down the centre of the bob is z b. If the iron tube is of steel for simplicity of calculation, we must evidently have ·064 (r+z-b)= ·.z=2/3 (r-b). It is practically found that for a second pendulum with a lead cylindrical bob 9 in. x 3 hung by its middle r has to be about to be about 44 inches, and z nearly 27. At any rate it is safety to make it 27 at first, especially if the second tube is iron, which expands a little more than steel; and the tube can be shortened after trial but not lengthened. The rod of the standard sidereal pendulum at Greenwich (down to the bottom of the bob, which is such as has been described and weighs 26_), is 43 _ and z is the 26 inches, the descending wires being steel. A solar time pendulum is about _ inch longer, as stated above. If the bob were fixed at its bottom to the steel tube the zinc would have to be 4·88 longer. Fig. 2 is a section of the great Westminster pendulum. The iron rod which runs from top to bottom, ends in a screw, with a nut N, for adjusting the length of the pendulum after it was made by calculation as near the right length as possible. On this nut rests a collar M, which can slide up the rod a little, but is prevented from turning by a pin through the rod. On a groove or annular channel in the top of this collar stands a zinc tube 10 feet 6 inches long, and nearly half an inch think, made of three tubes all drawn together, so as to become like one (for it should be observed that cast zinc cannot be depended on; it must be drawn). On the top of this tube or hollow column fits another collar with an annular groove much like the bottom one M. The object of these grooves is to keep the zinc column in its place, not touching the rod within it, as contact might produce friction, which would interfere with their relative motion under expansion and contraction. Round the collar C is screwed a large iron tube, also not touching the zinc, and its lower end fits loosely on the collar M; and round its outside it has another collar D of its own fixed to it on which the bob rests. The iron tube has a number of large holes in its down each side, to let the air get to the zinc tube; before that was done, it was found that the compensation lagged a day or two behind the changes of temperature, in consequence of the iron rod tube being exposed, while the zinc tube was enclosed without touching the iron. The bottom of the bob is 14 feet 11 inches from the top of the spring A, and the bob itself is 18 inches high, with a dome-shaped top, and twelve inches in diameter. As it is a 2-seconds pendulum, its centre of oscillation is 13 feet from the top A, which is higher than usual above the centre of gravity of the bob, on account of the great weight of the compensation tubes, the whole weighs very nearly 700 lb, and is probably the heaviest pendulum in the world.
The second kind of compensation pendulum in use is still more simple, but not so effective or certain in its action; and that is merely a wooden rod with a long lead bob resting on a nut at the bottom. According to the above table, it would appear that this bob ought to be 14 inches high in a 1-second pendulum; but the found capable of being depended on, and a somewhat shorter bob is said to be generally more correct in point of compensation. All persons who have tried wooden pendulums severely have come to the same conclusion, that they are capricious in their action, and consequently unfit for the highest class of clocks.
The best of all the compensation was long thought to be the mercurial, which was invented by Graham, a London clock-maker, above a century ago, who also invented the well-known dead escapement for clocks, which will be hereafter explained, and the horizontal or cylinder escapement for watches. And the best form of the mercurial pendulum is that which was introduced by the late E.J. Dent, in which the mercury is enclosed in a cast iron jar or cylinder, into the top of which the steel rod is screwed, with its end plunged into the mercury itself. For by this means the mercury, the rod, and the jar all acquire the new temperature at any change more simultaneously than when the mercury is in a glass jar hung by a stirrup (as it is called) at the bottom of the rod; and moreover the pendulum is safe to carry about, and the jar can be made perfectly cylindrical by turning, and also air-tight, so as protect the mercury from oxidation; and, if necessary, it cane be heated in the jar so as to drive off any moisture, without the risk of breaking. The height of mercury required in a cast-iron jar, 2 inches in diameter, is about 6·8 inches; for it must be remembered, in calculating the rise of the mercury, that the jar itself expands laterally, and that expansion has to be deducted from that ofteh mercury in bulk.
The success of the Westminster clock pendulum, however, and of smaller zinc and steel pendulums at Greenwich and elsewhere, has established the conclusion that it is unnecessary to incur the expense of a heavy mercurial pendulum, which has become more serious from the great rise in the price of mercury and the admitted necessity for much heavier bobs than were once thought sufficient for astronomical clocks. The complete calculation for a compensated pendulum in which the rods and tubes form any considerable proportion of the whole weights, as they must in a zinc pendulum, is too complicated to be worth undertaking generally, especially as it is always necessary to adjust them finally by trial, and for that ought to be by calculation, except where one is exactly copying pendulums previously tried.
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