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CLOCK . The measurement of See also:time has always been based on the revolution of the See also:celestial bodies, and the See also:period of the apparent revolution of the See also:sun, i.e. the See also:interval between two consecutive crossings of a See also:meridian, has been the usual See also:standard for a See also:day . By the Egyptians the day was divided into 24 See also:hours of equal length . The Greeks adopted a different See also:system, dividing the day, i.e. the period from sunrise to sunset, into 12 hours, and also the See also:night . Whence it followed that it was only at two periods in the See also:year that the length of the hours during the day and night were See also:uniform (see See also:CALENDAR) . In consequence, those who adopted the See also:Greek system were obliged to furnish their See also:water-clocks (see See also:CLEPSYDRA) with a compensating See also:device so that the equal hours measured by those clocks should be rendered unequal, according to the exigencies of the See also:season . The hours were divided into minutes and seconds, a system derived from the sexagesimal notation which prevailed before the decimal system was finally adopted . Our mode of computing time, and our angular measure, are the only See also:relics of this obsolete system . The simplest measure of time is the revolution of the See also:earth See also:round its -See also:axis, which so far as we know is uniform, perfectly See also:regular, and has not varied in See also:speed during any period of human observation . The time of such a revolution is called a sidereal day, and is divided into hours, minutes and seconds . The period of rotation of the earth is practially measured by observations of the fixed stars (see TIME), the period between two successive transits of the same See also:star across a meridian constituting the sidereal day . But as the axis of the earth slowly revolves round in a See also:cone, whereby the phenomenon known as the precession of the equinoxes is produced, it follows that the astronomical sidereal day is not the true period of the earth's rotation on its axis, but varies from it by less than a twenty millionth See also:part, a fraction so small as to be inappreciable . But the See also:civil day depends not on the revolution of the earth with regard to the stars, but on its revolution as compared with the position of the sun . Therefore each civil day is on the See also:average longer than a sidereal one by nearly four minutes, or, to be exact, each sidereal day is to an average civil day as •99727 to 1, and the sidereal See also:hour, See also:minute and second are also shorter in like proportion . Hence a sidereal clock has a shorter, quicker-moving pendulum than an See also:ordinary clock . Ordinary civil time thus depends on the apparent revolution of the sun round the earth . As, however, this is not uniform, it is needful for See also:practical convenience to give it an artificial uniformity . For this purpose an imaginary sun, moving round the earth with the average velocity of the real sun, and called the " mean " sun, is taken as the measure of civil time . The day is divided into 24 hours, each hour into 6o minutes, and each minute into 6o seconds . After that the sexagesimal See also:division system is abandoned, and fractions of seconds are estimated in decimals . A clock consists of a See also:train of wheels, actuated by a See also:spring or See also:weight, and provided with a governing device which so regulates the speed as to render it uniform . It also has a mechanism by which it strikes the hours on a See also:bell or See also:gong (cp . Fr. clothe, Ger . Glocke, a bell; Dutch klok, bell, clock), whereas, strictly, a timepiece does not strike, but simply shows the time .
The earliest clocks seem to have come into use in See also:Europe during the 13th See also:century
.
For although there is See also:evidence that they may have been invented some centuries sooner, yet until that date they were probably only curiosities
.
The first See also:form they took was that of the See also:balance clock, the invention of which .is ascribed, but on very insufficient grounds, to See also:Pope See also:Silvester II. in A.D
.
996
.
A clock was put up in a former clock See also:tower at See also:Westminster with some See also:great bells in 1288, out of a See also:fine imposed on a See also:chief-See also:justice who had offended the See also:government, and the See also:motto Discite justitiam, moniti, inscribed upon it
.
The bells were sold,
or rather, it is said, gambled away, by See also: The See also:works of one of these old clocks still exist in a going See also:condition at the See also:Victoria and See also:Albert Museum . It came from See also:Wells cathedral, having previously been at See also:Glastonbury See also:abbey . These old clocks had what is called a See also:verge escapement, and a balance . The train of wheels ended with a See also:crown See also:wheel, that is, a wheel serrated with See also:teeth like those of a saw, placed parallel with its axis (fig . 1), These teeth, D, engaged with pallets CB, CA, mounted on a verge or See also:staff placed parallel to the See also:face of the crown wheel . As the crown wheel was turned round the teeth pushed the pallets alternately until one or the other slid past a tooth, and thus let the crown wheel rotate . When one pallet had slipped over a tooth, the other pallet caught a corre- sponding tooth on the opposite See also:side of the wheel . The verge was terminated by a balance See also:rod placed at right angles to it with a See also:ball at each end . It is evident that when the force of any tooth on the crown wheel began to See also:act on a pallet, it communicated See also:motion to the balance and thus caused it to rotate . This motion would of course be accelerated, not uniformly, but according to some See also:law dependent on the shape of the teeth and pallets . When the motion had reached its maximum, the tooth slipped past the pallet . The other pallet now engaged another tooth on the op- posite side of the wheel . The motion of the balls, however, went on and they continued to See also:swing round, but the pressure of the tooth . For a time they overcame that pressure, and drove the tooth back, causing a recoil . As, however, every motion if subjected to an adverse See also:acceleration (i.e. a retardation) must come to See also:rest, the balls stopped, and then the tooth, which had been forced to recoil, advanced in its turn, and the swing was repeated . The arrangement was thus very like a huge See also:watch balance wheel in which the See also:driving weight acted in a very irregular manner, not only as a driving force, but also as a regulating spring . The going of such clocks was influenced greatly by See also:friction and by the oil on the parts, and never could be satisfactory, for the time varied with every variation in the swing of the balls, and this again with every variation of the effective driving force . The first great step in the improvement of the balance clock was a very See also:simple one . In the 17th century Galileo had discovered the isochronism of the pendulum, but he made no practical use of it, except by the invention of a little See also:instrument for enabling doctors to See also:count their patients' See also:pulse-beats . His son, however, is supposed to have applied the pendulum to clocks . There is at the Victoria and Albert Museum a copy of an See also:early clock, said to be Galileo's, in which the pins on a rotating wheel kick a pendulum outwards, remaining locked after having done so till the pendulum returns and unlocks the next See also:pin, which then administers another kick to the pendulum (fig . 2) . The See also:interest of the specimen is that it contains the germ of the chronometer escapement and See also:free pendulum, which is possibly destined to be the escapement of the future . The essential component parts of a clock are: 1 . The pendulum or time-governing device; 2 . The escapement, whereby the pendulum controls the speed of going; 3 . The train of wheels, urged round by the weight or See also:main-spring, together with the recording parts, i.e. the See also:dial, hands and hour motion wheels; 4 . The striking mechanism . The See also:general construction of the going part of all clocks, except large or See also:turret clocks, is substantially the same, and fig . 3 is a See also:section of any or- dinary See also:house clock . B is the See also:barrel with the See also:cord coiled round it, generally 16 times for the 8 days; the barrel is fixed to its arbor K, which is prolonged into the winding square coming up to the face or dial of the clock; the dial is here shown as fixed either by small screws x, or by a socket and pin z, to the See also:pro-longed pillars p, p, which (4 or 5 in number) connect the plates or See also:frame of the clock together, though the dial is commonly set on to the front See also:plate by another set of pillars of its own . The great wheel G rides on the arbor, and is connected with the barrel by the ratchet R, the See also:action of which is shown more fully in fig . 25 . The FIG . 3.—Section of House Clock . intermediate wheel r in this See also:drawing is for a purpose which will be described hereafter, and for the See also:present it may be considered as omitted, and the click of the ratchet R as fixed to the great wheel . The great wheel drives the pinion c which is called the centre pinion, on the arbor of the centre wheel C, which goes through to the dial, and carries the See also:long, or minute-See also:hand; this wheel always turns in an hour, and the great wheel generally in 12 hours, by having 12 times as many teeth as the centre pinion . The centre wheel drives the " second wheel " D by its pinion d, and that again drives the scape-wheel E by its pinion e . If the pinions d and e have each 8 teeth or leaves (as the teeth of pinions are usually called), C will have 64 teeth and D 6o, in a clock of which the scapewheel turns in a minute, so that the seconds hand may be set on its arbor prolonged to the dial* A represents the pallets of the escapement, which will be described presently, and their arbor a goes through a large hole in the back plate near F, and its back See also:pivot turns in a See also:cock OFQ screwed on to the back plate . From. the pallet arbor at F descends the crutch Ff, ending in the See also:fork f, which embraces the pendulum P, so that as the pendulum vibrates, the crutch and the pallets necessarily vibrate with it . The pendulum is hung by a thin spring S from the cock Q, so that the bending point of the spring may be just opposite the end of the pallet arbor, and the edge of the spring as See also:close to the end of that arbor as possible . We may now go to the front (or See also:left hand) of the clock, and Escapement . describe the dial or " motion-See also:work." The minute hand fits on to a squared end of a See also:brass socket, which is fixed to the wheel M, and fits close, but not tight, on the prolonged arbor of the centre wheel . Behind this wheel is a See also:bent spring which is (or ought to be) set on the same arbor with a square hole (not a round one as it sometimes is) in the See also:middle, so that it must turn with the arbor; the wheel is pressed up against this spring, and kept there, by a cap and a small pin through the end of the arbor . The consequence is, that there is friction enough between the spring and the wheel to carry the hand round, but not enough to resist a moderate push with the See also:finger for the purpose of altering the time indicated . This wheel M, which is sometimes called the minute-wheel, but is better called the hour-wheel as it turns in an hour, drives another wheel N, of the same number of teeth, which has a pinion attached to it; and that pinion drives the twelve-hour wheel H, which is also attached to a large socket or See also:pipe carrying the hour hand, and See also:riding on the former socket, or rather (in See also:order to relieve the centre arbor of that extra weight) on an intermediate socket fixed to the See also:bridge L, which is screwed to the front plate over the hour-wheel M . The weight W, which drives the train and gives the impulse to the pendulum through the escapement, is generally hung by a See also:catgut See also:line passing through a palley attached to the weight, the other end of the cord being tied to some convenient See also:place in the clock frame or seat-See also:board, to which it is fixed by screws through the See also:lower pillars . Pendulum.—Suppose that we hive a See also:body P (fig . 4) at rest, and that it is material, that is to say, has " See also:mass." And for simplicity let us consider it a ball of some heavy See also:matter . Let it be free P to move horizontally, but attached A pendulum is isochronous for similar reasons . If the bob be See also:drawn aside from D to C (fig . 5), then the restitutional force tending to bring it back to rest is approximately the force which See also:gravitation would exert along the tangent CA, i.e . ACW= BC displacement BC g See also:cos g OC = glength of pendulum' Since g is See also:constant, and the length of the pendulum does not vary, it follows that when a pendulum is drawn aside through a small arc the force tending to bring it back to rest is proportional to the displacement (approximately) . Thus the pendulum bob under the See also:influence of gravity, if the arc of swing is small, acts as though instead of being acted on by gravity it was acted on by a spring tending to See also:drag. it towards D, and therefore is isochronous . The qualification " If the arc of swing is small " is introduced because, as was discovered by Christiaan See also:Huygens, the arc of vibration of a truly isochronous pendulum should not be a circle with centre 0, but a See also:cycloid DM, generated by the See also:rolling of a circle with See also:diameter DQ=OD, upon a straight line QM . However, for a See also:short distance near the bottom, the circle so nearly coincides with the cycloid that a pendulum swinging in the usual circular path is, for small arcs, isochronous for practical purposes . The See also:formula representing the time of oscillation of a pendulum, in a circular arc, is thus found:—Let OB (fig . 6) be the pendulum, B be the position from which the bob is let go, and P be its position at A some period during its swing . Put FC =h, and MC=x, and OB=I . Now when a C body is allowed to move under the force FIG . 6. of gravity in any path from a height h, the velocity it attains is the same as a body would attain falling freely vertically through the distance h . Whence if v be the velocity of the bob at P, v='I 2gFM = s/ 2g(h—x) . Let Pp=ds, and the See also:vertical distance of p below P=dx, then Pp=velocity at PXdt; that is, dt=ds/v . Also dx=MP=x(21—x)' whence dl =— = s ldx sl x(2l—x) ',12g(h—x) =IA . l de 1 2 g x(p—x) ,/ i—x/2l Expanding the second part we have dt=2,~g . fi x . (1+41~-...) . If this is integrated between the limits of o and h, we have t=Ir/Vg (1+81+...) where 6 is the time of swing from B to A . The terms after the second may be neglected . The first See also:term, x sl l/g, is the time of swing in a cycloid . The second part represents the addition necessary if the swing is circular and not eycloidal, and therefore expresses the " circular See also:error." Now- h = BC2/l =27f2021/36o2, where 0 is See also:half the See also:angle of swing expressed in degrees; hence h/81=82/52520, and the formula becomes t =,rV - (1+ 52520) . . Hence the ratio of the time of swing of an ordinary pendulum of any length, with a semiarc of swing =8 degrees is to the time of swing of a corresponding cycloidal pendulum as 1+82/52520:1 . Also the difference of time of swing caused by a small increase 0' in the semiarc of swing=288'/52520 second per second, or 3.388' seconds per day . Hence in the See also:case of a seconds pendulum whose semiarc of swing is 2° an increase of .10 in this semiarc of 20 would cause the clock to lose 3.3 X2 Xo• 1= •66 second a day . Huygens proposed to apply his See also:discovery to clocks, and since the evolute of a cycloid is an equal cycloid, he suggested the use of a flexible pendulum swinging between cycloidal cheeks . But this was only an example of theory pushed too far, because the friction on the cycloidal cheeks involves more error than they correct, and other disturbances of a higher degree of importance are left uncorrected . In fact the application of pendulums to clocks, though governed in the abstract by theory, has to be modified by experiment . Neglecting the circular error, if L be the length of a pendulum and g the acceleration of gravity at the place where the pendulum is, then T, the time of a single vibration = 7r\i (L/g) . From this formula it follows that the times of vibration of pendulums are directly 'proportional to the square See also:root of their lengths, and inversely proportional to the square root of the acceleration of gravity at the place where the pendulum is swinging . The value of g for See also:London is 32.2 ft. per second per second, whence it results that the length of a pendulum for London to See also:beat seconds of mean See also:solar time=39.14 in. nearly, the length of an astronomical pendulum to beat seconds of sidereal time being 38.87 in . This length is calculated on the supposition that the arc of swing is cycloidal and that the whole mass of the pendulum is concentrated at a point whose distance, called the See also:radius of oscillation, from the point of suspension of the pendulum is 39.14 in . From this it might be imagined that if a See also:sphere, say of See also:iron, were suspended from a See also:light rod, so that its centre were 39.14 in. below its point of support, it would vibrate once per second . This, however, is not the case . For as the pendulum swings, the ball also tends to turn in space to and fro round a See also:horizontal axis perpendicular to the direction of its motion . Hence the force stored up in the pendulum is expended, not only in making it swing, but also in causing the ball to oscillate to and fro through a small angle about a horizontal axis . We have therefore to consider not merely the vibrations of the rod, but the oscillations of the bob . The moment of the momentum of the system round the point of suspension, called its moment of inertia, is composed of the sum of the mass of each particle multiplied into the square of its distance from the axis of rotation . Hence the moment of inertia of the body I=E(ma2) . If k be defined by the relation E(ma2) = (m) X k2, then k is called the radius of gyration . If k be the radius of gyration of a bob round a horizontal axis through its centre of gravity, h the distance of its centre of gravity below its point of suspension, and k' the radius of gyration of the bob round the centre of suspension, then k'2=h2+k2 . If I be the length of a simple pendulum that oscillates in the same time, then lh=k'2=112+k2 . Now k can be calculated if we know the form of the bob, and 1 is the length of the simple pendulum =39.14 in.; hence h, the distance of the centre of gravity of the bob below the point of suspension, can be found . In an ordinary pendulum, with a thin rod and a bob, this distance h is not very different from the theoretical length, 1=39.14 in., of a simple theoretical pendulum in which the rod has no weight and the bob is only a single heavy point . For the effect of the weight of the rod is to throw the centre of oscillation a little above the centre of gravity of the bob, while the effect of the See also:size of the bob is to throw the centre of oscillation a little down . In ordinary practice it is usual to make the pendulum so that the centre of gravity is about 39 in. below the upper free end of the suspension spring and leave the exact length to be determined by trial . Since T =ref E]g, we have, by differentiating, dL/L = 2dT/T, that is, any small percentage of increase in L. will correspond to See also:double the percentage of increase in T . Therefore with a seconds See also:pen- dulum, in order to make a second's difference in 2 a day, See also:equivalent to 1/86,400 of the pendulum's See also:rate of vibration, since there are 86,400 seconds in 24 hours, we must have a difference of length amounting to 2/86,400 =143,200 of the length of the rod . This is 39.138/43,200=•000906 in . Hence if under the pendulum bob be put a See also:nut working a See also:screw of 32 threads to the See also:inch and having its See also:head divided into 30 parts, a turn of this nut through one division will alter the length of the pendulum by .0009 in. and See also:change the rate of the clock by about a second a day . To accelerate the clock the nut has always to be turned to the right, or as you would drive in a corkscrew and See also: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 See also:collar, as shown in fig . 7 at C, about midway down the rod, capable of having very small weights laid upon it, 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 1 its length (down to the centre of oscillation), and m a small weight added at the distance n 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 -dt m n n2l of Westminster Clock Pendulum. from which it is evident that if n=l/2, then =dt/t=m/8M . But as there are 86400 seconds in a day, —dT, the daily acceleration, =86400 dt, or io800 m/M, or if m is the io800th 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 15 lb weight (700o gr. being =1 lb.), or an See also:ounce on a pendulum of 6 cwt . In like manner if n=1/3 from either See also:top or bottom, m must =M/7200 to accelerate the clock a second a day . The higher up the collar the less is the See also:risk of disturbing the pendulum in putting on or taking off the regulating weights, but the bigger the weight required to produce the effect . The weights should be made in a See also:series, and marked 4, 4, 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 . The length of pendulum rods is also affected by temperature and also, if they are made of See also:wood, by See also:damp .
Hence, to ensure See also:good gym_ time-keeping qualities in a clock, it is necessary (I) to make pensatlan. the rods of materials that are as little affected by such
influences as possible, and (2) to provide means of See also:compensation by which the effective length of the rod is kept constant in spite of expansion or contraction in the material of which it is composed
.
Fairly good pendulums for ordinary use may be made out of very well dried wood, soaked in a thin See also:solution of shellac in See also:spirits of See also:wine, or in melted See also:paraffin See also:wax; but wood shrinks inso uncertain a manner that such pendulums are not admissible for clocks of high exactitude
.
See also:Steel is an excellent material for pendulum rods, for the See also:metal is strong, is not stretched by the weight of the bob, and does not suffer great changes in molecular structure in the course of time
.
But a steel rod expands on the average lineally by •o000064 of its length for each degree F. by which its temperature rises; hence an expansion of .00009 in. on a pendulum rod of 39.14 in., that is .000023 of its length, will be caused by an increase of temperature of about 4° F., and that is sufficient to make the clock lose a second a day
.
Since the summer and See also:winter temperatures of a See also:room may differ by as much as 5o° F., the going of a clock may thus be affected by an error of 12 seconds a day
.
With a pendulum rod of brass, which has a coefficient of expansion of •0000i, a clock might gain one-third of a minute daily in winter as compared with its rate in summer
.
The coefficients of linear expansion per degree F. of some other materials used in making pendulums are as follows: See also: The glass having a small coefficient of expansion, the lateral expansion of the mercury will be checked by it, and this will help to raise the See also:column . For the linear coefficient of expansion of glass is •0000048 per degree F., whence the sectional See also:area of a glass vessel increases by •0000096 per degree F., and therefore the coefficient of vertical expansion of a column of mercury whose volumetric expansion coefficient is •000i per degree F. is (000i —.0000096) = •0000904 . Let x be the height of the vessel necessary to compensate a steel rod upon the bottom of which it rests . Then, the coefficient of expansion of steel being •000oo66 per degree F., we have -(•0000904—•0000066)=•0000066X39.14, whence x=64 in . It must, however, be remembered that the glass See also:jar has some weight and that it does not rise by anything like the amount of the mercury . This tends to keep the centre of gravity down . So that the height of mercury of 64 in. will not be sufficient to effect the compensation, and about 61 to 7 in. will be required . Some authors specify 7 in.; this is when the diameter of the jar is small . A certain amount of negative compensation must also be deducted to allow for the changes of temperature in the See also:air, as will presently be seen; this amounts in the case of mercury to about § in . In consequence of the complication of all these calculations it is usual to allow about 64 to 7 in. of mercury in the glass vessel and to adjust the exact amount of mercury by trial . Another very good form of mercurial pendulum was proposed by E . J .
Dent; it consists of a See also:cast-iron jar into the top of which the steel pendulum rod is screwed, having its end plunged into the mercury contained in the jar
.
By this means the mercury, jar and rod rapidly acquire the same temperature
.
This pendulum is less likely to break than the form just described
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The See also:depth of mercury required in an iron jar is stated by See also:Lord See also:Grimthorpe to be 84 to 9 in
.
The See also:reason why it is greater than it is when a glass jar is employed is that iron has a larger coefficient of expansion than glass, and that it is also heavier,
.
In all cases, however, of mercury pendulums experiment seems to be the only ultimate test of the quantity of mercury required, for the results are so complicated by the behaviour of the oil and the barometric errors that at its best the regulation of a clock can only be ultimately a matter of scientifically guided See also:compromise
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A small amount of compensation of a purely experimental See also:character is also allowed to compensate the changes which temperature effects on the suspension spring
.
This is sometimes made as much as 4 of the length correction
.
As an alternative to the mercurial pendulum other systems have been employed
.
The " gridiron " pendulum consists of a See also:group of alternate rods of steel and brass, so arranged that the expansion of the brass acts upwards and counteracts that of the steel downwards
.
It was invented in 1726 by See also: Of the 4 brass rods the 2 outside ones are 26.87 in.; and the two inside ones 22.25 in.; total 49.12 in . Thus the expansion of 88a in. of steel is counteracted by the expansion of 494 in. of brass . Everything depends, however, on the expansion coefficient of the steel and brass employed; the requirement in every case being that of total lengths of the brass and iron should be in proportion to the linear coefficients of expansion of those metals . The above figures Regsda- tbn . are for a very soft brass and steel . Thos . See also:Reid, with more ordinary steel and brass, prescribed a ratio of 112 to 71, Lord Grimthorpe a ratio of Too to 61: It is absolutely necessary to put the actual rods to be used for making the pendulum in a hot water See also:bath, and measure their expansions with a See also:microscope . John See also:Smeaton, taking advantage of a far greater expansion co-efficient of zinc as compared with brass, proposed to use a steel rod with a collar at the bottom, on which rested a hard drawn zinc rod . From this rod hung a steel See also:tube to which the bob was attached . The total length of the steel rod and of the steel tube down to the centre of the bob was made to the total length of the zinc tube, in the ratio of 5 to 2 (being the ratio of the expansions of zinc and steel) ; for a 39.14 in.pendulum we should therefore want a zinc tube equal in length to 1 (39.14) =264 in . In practice the zinc tube is made about 27 in. long, and then gradually cut down by trial . In fact the weight of a heavy pendulum squeezes the zinc, and it is impossible by See also:mere theory to determine what will be its behaviour . The zinc tube must be of rolled zinc, hard drawn through a See also:die, and must not be cast . Ventilating holes must be made in suitable places in the steel tube and the collar on which it rests, to ensure that changes of temperature are rapidly communicated throughout the system . A pendulum with a rod of dry varnished deal is tolerably compensated by a bob of lead or of zinc See also:lot to 13 in. in height, resting on a nut at the bottom of the rod . The old methods of pendulum compensation for See also:heat may now be considered as superseded by the invention of " See also:invar," a comlnvar. bination of See also:nickel and steel, due to Charles E . See also:Guillaume, of the See also:International See also:Office of Weights and See also:Measures at Sevres near See also:Paris . This alloy has a linear coefficient of expansion on the average of •00000i per degree centigrade, that is to say, only about ; i that of ordinary steel . Hence it can be easily compensated by means of brass, lead or any other suitable metal . Brass is usually employed . In the invar pendulum introduced into Great See also:Britain by Mr Agar Baugh a departure is made from the previous practice of merely calculating the length of the compensator, fastening it to the lower part of the pendulum, and attaching it to the centre of the bob . In the case of these pendulums, accurate cornputations are made of the moments of inertia of every See also:separate individual part . Thus, for instance, since an addition of See also:volume due to the effect of heat to the upper part of the bob has a different effect upon the moment of inertia from that of an equal quantity added to the lower part of the bob, the bob is suspended not from its centre, but from a point about ii.o in. below it, the distance varying according to the shape of the bob, so that the heat expansion of the bob may cause its centre of gravity to rise and compensate the effect of its increased moment of inertia . Again the suspension spring is measured for isochronism, and an alloy of steel prepared for it which does not alter its See also:elasticity with change of temperature . Moreover, since rods of invar steel subjected to See also:strain do not acquire their final coefficients of expansion and elasticity for some time, the invar is artificially " aged " by exposure to strain and heat . These considerations serve as a See also:guide in arranging for the compensation of the expansion of the rod and bob due to change of temperature . But they are not the only ones required; we See also:ave also to deal with changes due to the See also:density of the air in which the pendulum is moving . A body suspended in a fluid loses in weight by an amount equal to the weight of the fluid displaced, whence it follows that a pendulum suspended in air has not the weight which ought truly to correspond to its mass . M remains constant while Mg is less than in a vacuum . If the density of the air remained constant, this loss of weight, being constant, could be allowed for and would make no difference to the time-keeping . The period of swing would only be a little increased over what it would be in vacua . But the weight of a given volume of air varies both with the barometric pressure and also with temperature . If the bob be of type metal it weighs less in air than in a vacuum by about .000103 part, and for each 1 ° F. rise in temperature (the See also:barometer remaining constant and therefore the pressure remaining the same), the variation of density causes the bob to gain .00000024 of its weight . This, of course, makes the pendulum go quicker . Since the time of vibration varies as the inverse square root of g, it follows that a small increment of weight, the mass remaining constant, produces a diminution of one half that increment in time of swing . Hence, then, a rise of temperature of 1 ° F. will produce a diminution in the time of swing of -00000012th part or .c104 second in a day . But in making this calculation it has been assumed that the mass moved remains unaltered by the temperature . This is not so . A pendulum when swinging sets in motion a volume of air dependent on the size of the bob, but in a 10 lb bob nearly equal to its own volume . Hence while the rise of 1° of temperature increases the weight by . .00000012th part, it also decreases the mass by about the same proportion, and therefore the increase of period due to a rise of temperature of 1° F. will, instead of being •0104 second a day, be about •02 second . This must be compensated negatively by lengthening the pendulum by about 10020 in. for each degree of rise of temperature, which will require a piece of brass about 2 in. long . It follows, therefore, that with an invar rod having a linear expansion coefficient of -0000002 per degree F., which requires a piece of brass about •8 in. long to compensate it, the compensation which is to regulate both the expansion of the rod and also that of the air must be •8 in.-2 in., or -1.2 in.; so that the bob must be hung downwards from a piece of brass nearly xi in. in length . If the co-efficient of expansion of the invar were .00000053 per degree F., then the two corrections, one for the expansion of the rod and the other for the expansion of the air, would lust neutralize one another, and the pendulum rod would require no compensator at all . There are a number of other refinements which might be added, but which are too long for insertion here . By taking in all the See also:sources of error of higher orders, it has been possible to calculate a pendulum so accurately that, when the clock is loaded with the weight sufficient to give the pendulum the arc of swing for which it is designed, a rate of error has been produced of only half a minute in a year . These refinements, however, are only required for clocks of precision; for ordinary clocks an invar pendulum with a lead bob and brass compensator is quite sufficient . Invar pendulum rods are often made of steel with coefficients of expansion of about •00000l2 linear per C.; such a bob as this would require about 6.7 cm. of brass to compensate it, and, deducting 5 cm. of brass for the air compensation, this leaves about 1.7 cm. of See also:positive compensation for the pendulum . But as has been said, the exact See also:deduction depends on the shape and size of the bob, and the metal of which it is made . The diameters of the rods are 8 mm. for a 15 lb bob, 5 mm. for a 4 lb bob, and 12 to 15 mm. for a 6o lb bob . The bob is either a single See also:cylinder or two cylinders with the rod between them . Lenticular and spherical bobs are not used . The great See also:object is to allow the air ready See also:access to all parts of the rod and compensator, so that they are all heated or cooled simultaneously . The bobs are usually made of a See also:compound of lead,-See also:antimony, and See also:tin, which forms a hard metal, free from bubbles and with a specific gravity of about to . The usual weight of the bobs of the best pendulums for an ordinary astronomical clock is about 15 lb . A greater weight than this |