PENDULUM

, in Mechanics, any heavy body, so suspended as that it may swing backwards and forwards, about some fixed point, by the force of gravity.

These alternate ascents and descents of the Pendulum, are called its Oscillations, or Vibrations; each complete oscillation being the descent from the highest point on one side, down to the lowest point of the arch, and so on up to the highest point on the other side. The point round which the Pendulum moves, or vibrates, is called its Centre of Motion, or Point of Suspension; and a right line drawn through the centre of motion, parallel to the horizon, and perpendicular to the plane in which the Pendulum moves, is called the Axis of Oscillation. There is also a certain point within every Pendulum, into which, is all the matter that composes the Pendulum were collected, or condensed as into a point, the times in which the vibrations would be performed, would not be altered by such condensation; and this point is called Centre of Oscillation. The length of the Pendulum is always estimated by the distance of this point below the centre of motion; being usually near the bottom of the Pendulum; but in a cylinder, or any other uniform prism or rod, it is at the distance of one third from the bottom, or twothirds from and below the centre of motion.

The length of a Pendulum, so measured to its centre of oscillation, that it will perform each vibration in a second of time, thence called the second's Pendulum, has, in the latitude of London, been generally taken at 39 2/10 or 39 1/5 inches; but by some very ingenious and accurate experiments, the late celebrated Mr. George Graham found the true length to be 39 128/1000, inches, or 39 1/8 inches very nearly.

The length of the Pendulum vibrating seconds at Paris, was found by Varin, Des Hays, De Glos, and Godin, to be 440 5/<*> lines; by Picard 440 1/2 lines; and by Mairan 440 1<*>/30 lines.

Galileo was the first who made use of a heavy body annexed to a thread, and suspended by it, for measuring time, in his experiments and observations. But according to Sturmius, it was Riccioli who first observed the isochronism of Pendulums, and made use of them in measuring time. After him, Tycho, Langrene, Wendeline, Mersenne, Kircher, and others, observed the same thing; though, it is said, without any intimation of what had been done by Riccioli. But it was the celebrated Huygens who first demonstrated the principles and properties of Pendulums, and probably the first who applied them to clocks. He demon- strated, that if the centre of motion were perfectly fixed and immoveable, and all manner of friction, and resistance of the air, &c, removed, then a Pendulum, once set in motion, would for ever continue to vibrate without any decrease of motion, and that all its vibrations would be perfectly isochronal, or performed in the same time. Hence the Pendulum has universally been considered as the best chronometer or measurer of time. And as all Pendulums of the same length perform their vibrations in the same time, without regard to their different weights, it has been suggested, by means of them, to establish an universal standard for all countries. On this principle Mouton, canon of Lyons, has a treatise, De Mensura posteris transmittenda; and several others since, as Whitehurst, &c. See Universal Measure.

Pendulums are either simple or compound, and each of these may be considered either in theory, or as in practical mechanics among artisans.

A Simple Pendulum, in Theory, consists of a single weight, as A, considered as a point, and an inflexible right line AC, supposed void of gravity or weight, and suspended from a fixed point or centre C, about which it moves.

A Compound Pendulum, in Theory, is a Pendulum consisting of several weights moveable about one common centre of motion, but connected together so as to retain the same distance both from one another, and from the centre about which they vibrate.

The Doctrine and Laws of Pendulums.—1. A Pendulum raised to B, through the arc of the circle AB, will fall, and rise again, through an equal arc, to a point equally high, as D; and thence will fall to A, and again rise to B; and thus continue rising and falling perpetually. For it is the same thing, whether the body fall down the inside of the curve BAD, by the force of gravity, or be retained in it by the action of the string; for they will both have the same effect; and it is otherwise known, from the oblique descents of bodies, that the body will descend and ascend along the curve in the manner above described.

Experience also consirms this theory, in any finite number of oscillations. But if they be supposed infinitely continued, a difference will arise. For the resistance of the air, and the friction and rigidity of the string about the centre C, will take off part of the force acquired in falling; whence it happens that it will not rise precisely to the same point from whence it fell.

Thus, the ascent continually diminishing the oscillation, this will be at last stopped, and the Pendulum will hang at rest in its natural direction, which is perpendicular to the horizon.

Now as to the real time of oscillation in a circular arc BAD: it is demonstrated by mathematicians, that if , denote the circumference of a circle whose diameter is 1; feet or 193 inches, the space a heavy body falls in the first second of time; and the length of the Pendulum; also the height of the arch of vibration; then the| time of each oscillation in the arc BAD will be equal to into the infinite series , where is the diameter of the arc described, or twice the length of the Pendulum.

And here, when the arc is a small one, as in the case of the vibrating Pendulum of a clock, all the terms of this series after the 2d may be omitted, on account of their smallness; and then the time of a whole vibration will be nearly equal to . So that the times of vibration of a Pendulum in different small arcs of the same circle, are as , or 8 times the radius, added to the versed sine of the semiarc.

And farther, if D denote the number of degrees in the semiarc AB, whose versed fine is a, then the quantity last mentioned, for the time of a whole vibration, is changed to . And therefore the times of vibration in different small arcs, are as , or as the number 52524 added to the square of the number of degrees in the semiarc AB. See my Conic Sections and Select Exercises, p. 190.

2. Let CB be a semicycloid, having its base EC parallel to the hori<*>on, and its vertex B downwards; and let CD be the other half of the cycloid, in a similar position to the former. Suppose a Pendulum string, of the same length with the curve of each semicycloid BC, or CD, having its end sixed in C, and the thread applied all the way close to the cycloidal curve BC, and consequently the body or Pendulum weight coinciding with the point B. If now the body be let go from B, it will descend by its own gravity, and in descending it will unwind the string from off the arch BC, as at the position CGH; and the ball G will describe a semicycloid BHA, equal and similar to BGC, when it has arrived at the lowest point A; after which, it will continue its motion, and ascend, by another equal and similar semicycloid AKD, to the same height D, as it fell from at B, the string now wrapping itself upon the other arch CID. From D it will descend again, and pass along the whole cycloid DAB, to the point B; and thus perform continual successive oscillations between B and D, in the curve of a cycloid; as it before oscillated in the curve of a circle, in the former case.

This contrivance to make the Pendulum oscillate in the curve of a cycloid, is the invention of the celebrated Huygens, to make the Pendulum perform all its vibrations in equal times, whether the arch, or extent of the vibration be great or small; which is not the case in a circle, where the larger arcs take a longer time to run through them, than the smaller ones do, as is well known both from theory and practice.

The chief properties of the cycloidal Pendulum then, as demonstrated by Huygens, are the following. 1st, That the time of an oscillation in all arcs, whether larger or smaller, is always the same quantity, viz, whether the body begin to descend from the point B, and describe the semiarch BA; or that it begins at H, and describes the arch HA; or that it sets out from any other point; as it will still descend to the lowest point A in exactly the same time. And it is farther proved, that the time of a whole vibration through any double arc BAD, or HAK. &c, is in proportion to the time in which a heavy body will freely fall, by the force of gravity, through a space equal to (1/2)AC, half the length of the Pendulum, as the circumference of a circle is to its diameter. So that, if feet denote the space a heavy body falls in the first second of time, the circumference of a circle whose diameter is 1, and the length of the Pendulum; then, because, by the nature of descents by gravity, that is the time in which a body will fall through (1/2)r, or half the length of the Pendulum; therefore, by the above proportion, as , which is the time of an entire oscillation in the cycloid.

And this conclusion is abundantly confirmed by experience. For example, if we consider the time of a vibration as 1 second, to find the length of the Pendulum that will so oscillate in 1 second; this will give the equation ; which reduced, gives inches = 39.11 or 39 1/9 inches, for the length of the second's Pendulum; which the best experiments shew to be about 39 1/8 inches.

3. Hence also, we have a method of determining, from the experimented length of a Pendulum, the space a heavy body will fall perpendicularly through in a given time: for, since , therefore, by reduction, is the space a body will fall through in the first second of time, when r denotes the length of the second's Pendulum; and as constant experience shews that this length is nearly 39 1/8 inches, in the latitude of London, in this case g or becomes inches = 16 1/12 feet, very nearly, for the space a body will fall in the first second of time, in the latitude of London: a fact which has been abundantly confirmed by experiments made there. And in the same manner, Mr. Huygens found the same space fallen through at Paris, to be 15 French feet.

The whole doctrine of Pendulums, oscillating between two semicycloids, both in theory and practice,| was delivered by that author, in his Horologium Oscillatorium, sive Demonstrationes de Motu Pendulorum. And every thing that regards the motion of Pendulums has since been demonstrated in different ways, and particularly by Newton, who has given an admirable theory on the subject, in his Principia, where he has extended to epicycloids the properties demonstrated by Huygens of the cycloids.

4. As the cycloid may be considered as coinciding, in A, with any small are of a circle described from the centre C, passing through A, where it is known the two curves have the same radius and curvature; therefore the time in the small arc of such a circle, will be nearly equal to the time in the cycloid; so that the times in very small circular arcs are equal, because these small arcs may be considered as portions of the cycloid, as well as of the circle. And this is one great reason why the Pendulums of clocks are made to oscillate in as small arcs as possible, viz, that their oscillations may be the nearer to a constant equality.

This may also be deduced from a comparison of the times of vibration in the circle, and in the cycloid, as laid down in the foregoing articles. It has there been shewn, that the times of vibration in the circle and cycloid are thus, viz, time in the circle nearly , time in the cycloidal arc ; where it is evident, that the former always exceeds the latter in the ratio of to 1; but this ratio always approaches nearer to an equality, as the arc, or as its versed sine a, is smaller; till at length, when it is very small, the term a/8r may be omitted, and then the times of vibration become both the same quantity, viz .

Farther, by the same comparison, it appears, that the time lost in each second, or in each vibration of the second's Pendulum, by vibrating in a circle, instead of a cycloid, is a/8r, or ; and consequently the time lost in a whole day of 24 hours, is (5/3)D² nearly. In like manner, the seconds lost per day by vibrating in the are of ▵ degrees, is (5/3)▵². Therefore if the Pendulum keep true time in one of these arcs, the seconds lost or gained per day, by vibrating in the other, will be (5/3)(D² - ▵²). So, for example, if a Pendulum measure true time in an arc of 3 degrees, on each side of the lowest point, it will lose 11 2/3 seconds a day by vibrating 4 degrees; and 26 2/3 seconds a day by vibrating 5 degrees; and so on.

5. The action of gravity is less in those parts of the earth where the oscillations of the same Pendulum are slower, and greater where these are swifter; for the time of oscillation is reciprocally proportional to √g. And it being found by experiment, that the oscillations of the same Pendulum are slower near the equator, than in places farther from it; it follows that the force of gravity is less there; and consequently the parts about the equator are higher or farther from the centre, than the other parts; and the shape of the earth is not a true sphere, but somewhat like an oblate spheroid, flatted at the poles, and raised gradually towards the equator. And hence also the times of the vibration of the same Pendulum, in different latitudes, afford a method of determining the true figure of the earth, and the proportion between its axis and the equatorial diameter.

Thus, M. Richer found by an experiment made in the island Cayenna, about 4 degrees from the equator, where a Pendulum 3 feet 8 2/5 lines long, which at Paris vibrated seconds, required to be shortened a line and a quarter to make it vibrate seconds. And many other observations have confirmed the same principle. See Newton's Principia, lib. 3, prop. 20. By comparing the different observations of the French astronomers, Newton apprehends that 2 lines may be considered as the length a seconds Pendulum ought to be decreased at the equator.

From some observations made by Mr. Campbell, in 1731, in Black<*>river, in Jamaica, 18° north latitude, it is collected, that if the length of a simple Pendulum that swings seconds in London, be 39.126 English inches, the length of one at the equator would be 39.00, and at the poles 39.206. Philos. Trans. numb. 432; or Abr. vol. 8, part 1, pa. 238.

And hence Mr. Emerson has computed the following Table, shewing the length of a Pendulum that swings seconds at every 5th degree of latitude, as also the length of the degree of latitude there, in English miles.

6. If two Pendulums vibrate in similar arcs, the times of vibration are in the sub-duplicate ratio of their lengths. And the lengths of Pendulums vibrating in similar arcs, are in the duplicate ratio of the times| of a vibration directly; or in the reciprocal duplicate ratio of the number of oscillations made in any one and the same time. For, the time of vibration t being as , where p and g are constant or given, therefore t is as √r, and r as t². Hence therefore the length of a half-second Pendulum will be 1/4r or inches; and the length of the quarter-second Pendulum will be inches; and so of others.

7. The foregoing laws, &c, of the motion of Pendulums, cannot strictly hold good, unless the thread that sustains the ball be void of weight, and the gravity of the whole ball be collected into a point. In practice therefore, a very fine thread, and a small ball, but of a very heavy matter, are to be used. But a thick thread, and a bulky ball, disturb the motion very much; for in that case, the simple Pendulum becomes a compound one; it being much the same thing, as if s<*>ral weights were applied to the same inflexible rod in several places.

8. M. Krafft in the new Petersburgh Memoirs, vols 6 and 7, has given the result of many experiments upon Pendulums, made in different parts of Russia, with deductions from them, from whence he derives this theorem: If x be the length of a Pendulum that swings seconds in any given latitude l, and in a temperature of 10 degrees of Reaumur's thermometer, then will the length of that Pendulum, for that latitude, be thus expressed, viz, lines of a French foot. And this expression agrees very nearly, not only with all the experiments made on the Pendulum in Russia, but also, with those of Mr. Graham, and those of Mr. Lyons in 79° 50′ north latitude, where he found its length to be 441.38 lines. See Oblateness.

Simple Pendulum, in Mechanics, an expression commonly used among artists, to distinguish such Pendulums as have no provision for correcting the effects of heat and cold, from those that have such provision. Also Simple Pendulum, and Detached Pendulum, are terms sometimes used to denote such Pendulums as are not connected with any clock, or clock-work.

Compound Pendulum, in Mechanics, is a Pendulum whose rod is composed of two or more wires or bars of metal. These, by undergoing different degrees of expansion and contraction, when exposed to the same heat or cold, have the difference of expansion or contraction made to act in such manner as to preserve constantly the same distance between the point of suspension, and centre of oscillation, although exposed to very different and various degrees of heat or cold. There are a great variety of constructions for this purpose; but they may be all reduced to the Gridiron, the Mercurial, and the Lever Pendulum.

It may be just observed by the way, that the vulgar method of remedying the inconvenience arising from the extension and contraction of the rods of common Pendulums, is by applying the bob, or small ball, with a screw, at the lower end; by which means the Pendulum is at any time made longer or shorter, as the ball is screwed downwards or upwards, and thus the time of its vibration is kept continually the same.

The Gridiron Pendulum was the invention of Mr. John Harrison, a very ingenious artist, and celebrated for his invention of the watch for finding the difference of longitude at sea, about the year 1725; and of several other time keepers and watches since that time; for all which he received the parliamentary reward of between 20 and 30 thousand pounds. It consists of 5 rods of steel, and 4 of brass, placed in an alternate order, the middle rod being of steel, by which the Pendulum ball is suspended; these rods of brass and steel, thus placed in an alternate order, and so connected with each other at their ends, that while the expansion of the steel rods has a tendency to lengthen the Pendulum, the expansion of the brass rods, acting upwards, tends to shorten it. And thus, when the lengths of the brass and steel rods are duly proportioned, their expansions and contractions will exactly balance and correct each other, and so preserve the Pendulum invariably of the same length. The simplicity of this ingenious contrivance is much in its favour; and the difficulty of adjustment seems the only objection to it.

Mr. Harrison in his first machine for measuring time at sea, applied this combination of wires of brass and steel, to prevent any alterations by heat or cold; and in the machines or clocks he has made for this purpose, a like method of guarding against the irregularities arising from this cause is used.

The Mercurial Pendulum was the invention of the ingenious Mr. Graham, in consequence of several experiments relating to the materials of which Pendulums might be formed, in 1715. Its rod is made of brass, and branched towards its lower end, so as as to embrace a cylindric glass vessel 13 or 14 inches long, and about 2 inches diameter; which being filled about 12 inches deep with mercury, forms the weight or ball of the Pendulum. If upon trial the expansion of the rod be found too great for that of the mercury, more mercury must be poured into the vessel: if the expansion of the mercury exceeds that of the rod, so as to occasion the clock to go fast with heat, some mercury must be taken out of the vessel, so as to shorten the column. And thus may the expansion and contraction of the quicksilver in the glass be made exactly to balance the expansion and contraction of the Pendulum rod, so as to preserve the distance of the centre of oscillation from the point of suspension invariably the same.

Mr. Graham made a clock of this sort, and compared it with one of the best of the common sort, for 3 years together; when he found the errors of his but about one-eighth part of those of the latter. Philos. Trans. numb. 392.

The Lever Pendulum. From all that appears concerning this construction of a Pendulum, we are inclined to believe that the idea of making the difference of the expansion of different metals operate by means of a lever, originated with Mr. Graham, who in the year 1737 constructed a Pendulum, having its rod composed of one bar of steel between two of brass, which acted upon the short end of a lever, to the other end of which, the ball or weight of the Pendulum was sus- pended.|

This Pendulum however was, upon trial, found to move by jerks; and therefore laid aside by the inventor, to make way for the mercurial Pendulum, just mentioned.

Mr. Short informs us in the Philos. Trans. vol. 47, art. 88, that a Mr. Frotheringham, a quaker in Lincolnshire, caused a Pendulum of this kind to be made: it consisted of two bars, one of brass, and the other of steel, fastened together by screws, with levers to raise or let down the bulb; above which these levers were placed. M. Cassini too, in the History of the Royal Academy of Sciences at Paris, for 1741, describes two sorts of Pendulums for clocks, compounded of bars of brass and steel, and in which he applies a lever to raise or let down the bulb of the Pendulum, by the expansion or contraction of the bar of brass.

Mr. John Ellicott also, in the year 1738, constructed a Pendulum on the same principle, but differing from Mr. Graham's in many particulars. The rod of Mr. Ellicott's Pendulum was composed of two bars only; the one of brass, and the other of steel. It had two levers, each sustaining its half of the ball or weight; with a spring under the lower part of the ball to relieve the levers from a considerable part of its weight, and so to render their motion more smooth and easy. The one lever in Mr. Graham's construction was above the ball: whereas both the levers in Mr. Ellicott's were within the ball; and each lever had an adjusting screw, to lengthen or shorten the lever, so as to render the adjustment the more perfect. See the Philos. Trans. vol. 47, p. 479; where Mr. Ellicott's methods of construction are described, and illustrated by figures.

Notwithstanding the great ingenuity displayed by these very eminent artists on this construction, it must farther be observed, in the history of improvements of this nature, that Mr. Cumming, another eminent artist, has given, in his Essays on the Principles of Clock and Watch-work, Lond. 1766, an ample description, with plates, of a construction of a Pendulum with levers, in which it seems he has united the properties of Mr. Graham's and Mr. Ellicott's, without being liable to any of the defects of either. The rod of this Pendulum is composed of one flat bar of brass, and two of steel; he uses three levers within the ball of the Pendulum; and, among many other ingenious contrivances, for the more accurate adjusting of this Pendulum to mean time, it is provided with a small ball and screw below the principal ball or weight, one entire revolution of which on its screw will only alter the rate of the clock's going one second per day; and its circumference is divided into 30, one of which divisions will therefore alter its rate of going one second in a month.

Pendulum Clock, is a clock having its motion regulated by the vibration of a Pendulum.

It is controverted between Galileo and Huygens, which of the two first applied the Pendulum to a clock. For the pretensions of each, see Clock.

After Huygens had discovered, that the vibration made in arcs of a cycloid, however unequal they might be in extent, were all equal in time; he soon perceived, that a Pendulum applied to a clock, so as to make it describe arcs of a cycloid, would rectify the otherwise <*>oidable irregularities of the motion of the clock; since, though the several causes of those irrogularities should occasion the Pendulum to make greater or smaller vibrations, yet, by virtue of the cycloid, it would still make them perfectly equal in point of time; and the motion of the clock governed by it, would therefore be preserved perfectly equable. But the difficulty was, how to make the Pendulum describe arcs of a cycloid; for naturally the Pendulum, being tied to a fixed point, can only describe circular arcs about it.

Here M. Huygens contrived to fix the iron rod or wire, which bears the ball or weight, at the top to a silken thread, placed between two cycloidal cheeks, or two little arcs of a cycloid, made of metal. Hence the motion of vibration, applying successively from one of those arcs to the other, the thread, which is extremely flexible, easily assumes the figure of them, and by that means causes the ball or weight at the bottom to describe a just cycloidal arc.

This is doubtless one of the most ingenious and useful inventions many ages have produced: by means of which it has been asserted there have been clocks that would not vary a single second in several days: and the same invention also gave rise to the whole doctrine of involute and evolute curves, with the radius and degree of curvature, &c.

It is true, the Pendulum is still liable to its irregularities, how minute soever they may be. The silken thread by which it was suspended, shortens in moist weather, and lengthens in dry; by which means the length of the whole Pendulum, and consequently the times of the vibrations, are somewhat varied.

To obviate this inconvenience, M. De la Hire, instead of a silken thread, used a little fine spring; which was not indeed subject to shorten and lengthen, from those causes; yet he found it grew stiffer in cold weather, and then made its vibrations faster than in warm; to which also we may add its expansion and contraction by heat and cold. He therefore had recourse to a stiff wire or rod, firm from one end to the other. Indeed by this means he renounced the advantages of the cycloid; but he found, as he says, by experience, that the vibrations in circular arcs are performed in times as equal, provided they be not of too great extent, as those in cycloids. But the experiments of Sir Jonas Moore, and others, have demonstrated the contrary.

The ordinary causes of the irregularities of Pendulums Dr. Derham ascribes to the alterations in the gravity and temperature of the air, which increase and diminish the weight of the ball, and by that means make the vibrations greater and less; an accession of weight in the ball being found by experiment to accelerate the motion of the Pendulum; for a weight of 6 pounds added to the ball, Dr. Derham found made his clock gain 13 seconds every day.

A general remedy against the inconveniences of Pendulums, is to make them long, the ball heavy, and to vibrate but in small arcs. These are the usual means employed in England; the cycloidal checks being g<*>nerally neglected. See the foregoing article.

Pendulum clocks resting against the same rail have been found to influence each other's motion. See the Philos. Trans. numb. 453, sect. 5 and 6, where Mr. Ellicott has given a curious and exact account of this phenomenon.|

Pendulum Royal, a name used among us for a clock, whose Pendulum swings seconds, and goes 8 days without winding up; shewing the hour, minute, and second. The numbers in such a piece are thus calculated. First cast up the seconds in 12 hours, which are the beats in one turn of the great wheel; and they will be found to be 43200 = 12 X 60 X 60. The swing wheel must be 30, to swing 60 seconds in one of its revolutions; now let the half of 43200, viz 21600, be divided by 30, and the quotient will be 720, which must be separated into quotients. The first of these must be 12, for the great wheel, which moves round once in 12 hours. Now 720 divided by 12, gives 60, which may also be conveniently broken into two quotients, as 10 and 6, or 12 and 5, or 8 and 7 1/2; which last is most convenient: and if the pinions be all taken 8, the work will stand thus:

According to this computation, the great wheel will go round once in 12 hours, to shew the hour; the next wheel once in an hour, to shew the minutes; and the swing-wheel once in a minute, to shew the seconds. See Clock-work.

Ballistic Pendulum. See Ballistic Pendulum.

Level Pendulum. See Level.

Pendulum Watch. See Watch.