Part 15

_Pendulum._--Suppose that we have a body P (fig. 4) at rest, and that it is material, that is to say, has "mass." And for simplicity let us consider it a ball of some heavy matter. Let it be free to move horizontally, but attached to a fixed point A by means of a spring. As it can only move horizontally and not fall, the earth's gravity will be unable to impart any motion to it. Now it is a law first discovered by Robert Hooke (1635-1703) that if any elastic spring be pulled by a force, then, within its elastic limits, the amount by which it will be extended is proportional to the force. Hence then, if a body is pulled out against a spring, the restitutional force is proportional to the displacement. If the body be released it will tend to move back to its initial position with an acceleration proportioned to its mass and to its distance from rest. A body thus circumstanced moves with harmonic motion, vibrating like a stretched piano string, and the peculiarity of its motion is that it is isochronous. That is to say, the time of returning to its initial position is the same, whether it makes a large movement at a high velocity under a strong restitutional force, or a small movement at a lower velocity under a smaller restitutional force (see MECHANICS). In consequence of this fact the balance wheel of a watch is isochronous or nearly so, notwithstanding variations in the amplitude of its vibrations. It is like a piano string which sounds the same note, although the sound dies away as the amplitude of its vibrations diminishes.

A pendulum is isochronous for similar reasons. If the bob be drawn aside from D to C (fig. 5), then the restitutional force tending to bring it back to rest is approximately the force which gravitation would exert along the tangent CA, i.e.

BC displacement BC g cos ACW = g -- = g ------------------. OC length of pendulum

Since g is 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 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 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 Huygens, the arc of vibration of a truly isochronous pendulum should not be a circle with centre O, but a cycloid DM, generated by the rolling of a circle with diameter DQ = ½OD, upon a straight line QM. However, for a 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 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 some period during its swing. Put FC = h, and MC = x, and OB = l. Now when a body is allowed to move under the force 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 = sqrt(2gFM) = sqrt(2g(h - x)). Let Pp = ds, and the vertical distance of p below P = dx, then Pp = velocity at P × dt; that is, dt = ds/v.

ds l l Also -- = -- = ---------------, dx MP sqrt(x(2l - x))

ds ldx 1 whence dt = -- = --------------- · --------------- v sqrt(x(2l - x)) sqrt(2g(h - x))

1 / l dx 1 = --- / --- · -------------- · ---------------- 2 \/ g sqrt(x(p - x)) sqrt(1 - (x/2l))

Expanding the second part we have

1 / l dx / x \ dt = --- / --- · -------------- · ( 1 + --- + ... ). 2 \/ g sqrt(x(h - x)) \ 4l /

If this is integrated between the limits of 0 and h, we have

/ l / h \ t = [pi] / --- · ( 1 + --- + ... ), \/ g \ 8l /

where t is the time of swing from B to A. The terms after the second may be neglected. The first term, [pi] sqrt(l/g), is the time of swing in a cycloid. The second part represents the addition necessary if the swing is circular and not cycloidal, and therefore expresses the "circular error." Now h = BC²/l = 2[pi]²[theta]²l / 360², where [theta] is half the angle of swing expressed in degrees; hence h/(8l) = [theta]²/52520, and the formula becomes

/ l / [theta]² \ t = [pi] / --- ( 1 + -------- ). \/ g \ 52520 /

Hence the ratio of the time of swing of an ordinary pendulum of any length, with a semiarc of swing = [theta] degrees is to the time of swing of a corresponding cycloidal pendulum as 1 + [theta]²/52520 : 1. Also the difference of time of swing caused by a small increase [theta]' in the semiarc of swing = 2[theta][theta]' / 52520 second per second, or 3.3[theta][theta]' seconds per day. Hence in the case of a seconds pendulum whose semiarc of swing is 2° an increase of .1° in this semiarc of 2° would cause the clock to lose 3.3 × 2 × 0.1 = .66 second a day.

Huygens proposed to apply his 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 = [pi] sqrt(L/g). From this formula it follows that the times of vibration of pendulums are directly proportional to the square 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 London is 32.2 ft. per second per second, whence it results that the length of a pendulum for London to beat seconds of mean 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 radius of oscillation, from the point of suspension of the pendulum is 39.14 in. From this it might be imagined that if a sphere, say of iron, were suspended from a 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 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 = [Sigma](ma²). If k be defined by the relation [Sigma](ma²) = [Sigma](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 l be the length of a simple pendulum that oscillates in the same time, then lh = k'² = h² + k². Now k can be calculated if we know the form of the bob, and l 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, l = 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 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.

Regulation.

Since T = [pi]sqrt(L/g), we have, by differentiating, dL/L = 2dT/T, that is, any small percentage of increase in L will correspond to double the percentage of increase in T. Therefore with a seconds pendulum, in order to make a second's difference in a day, equivalent to 1/86,400 of the pendulum's rate of vibration, since there are 86,400 seconds in 24 hours, we must have a difference of length amounting to 2/86,400 = 1/43,200 of the length of the rod. This is 39.138/43,200 = .000906 in. Hence if under the pendulum bob be put a nut working a screw of 32 threads to the inch and having its head divided into 30 parts, a turn of this nut through one division will alter the length of the pendulum by .0009 in. and 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 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. 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 l 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 n² \ --- = --- ( --- - ---- ): t 2M \ l l² /

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 10800 m/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 15 lb weight (7000 gr. being = 1 lb.), or an ounce on a pendulum of 6 cwt. In like manner if n = l/3 from either top or bottom, m must = M/7200 to accelerate the clock a second a day. The higher up the collar the less is the 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 series, and marked ¼, ½, 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.

The length of pendulum rods is also affected by temperature and also, if they are made of wood, by damp. Hence, to ensure good time-keeping qualities in a clock, it is necessary (1) to make the rods of materials that are as little affected by such influences as possible, and (2) to provide means of 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 solution of shellac in spirits of wine, or in melted paraffin wax; but wood shrinks in so uncertain a manner that such pendulums are not admissible for clocks of high exactitude. Steel is an excellent material for pendulum rods, for the 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 .0000064 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 winter temperatures of a room may differ by as much as 50° 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 .00001, 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: white deal, .0000024; flint glass, .0000048; iron, .000007; lead, .000016; zinc, .000016; and mercury, .000033. The solid or cubical expansions of these bodies are three times the above quantities respectively.

The first method of compensating a pendulum was invented in 1722 by George Graham, who proposed to use a bob of mercury, taking advantage of the high coefficient of expansion of that metal. As now employed, the mercurial pendulum consists of a rod of steel terminating in a stirrup of the same metal on which rests a glass vessel full of mercury, having its centre of gravity about 39 in. below the point of suspension of the pendulum. For each Fahrenheit degree of temperature the centre of gravity of the bob is lowered by the expansion of the rod about 1/4000 of an inch. The glass vessel and the mercury in it have therefore to be so contrived, that their centre of gravity will rise 1/4000 in. per degree F. 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 column. For the linear coefficient of expansion of glass is .0000048 per degree F., whence the sectional 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 .0001 per degree F. is (.0001 - .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 .0000066 per degree F., we have

x --- (.0000904 - .0000066) = .0000066 X 39.14, whence x = 6¼ in. 2

It must, however, be remembered that the glass 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 6¼ in. will not be sufficient to effect the compensation, and about 6¾ 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 air, as will presently be seen; this amounts in the case of mercury to about 1/5 in.

In consequence of the complication of all these calculations it is usual to allow about 6¾ 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 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. The depth of mercury required in an iron jar is stated by Lord Grimthorpe to be 8½ to 9 in. The 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 compromise. A small amount of compensation of a purely experimental character is also allowed to compensate the changes which temperature effects on the suspension spring. This is sometimes made as much as 1/6 of the length correction.

As an alternative to the mercurial pendulum other systems have been employed. The "gridiron" pendulum consists of a 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 John Harrison. Assuming that 9 rods are used--5 of steel and 4 of brass--their lengths may be as follows from pin to pin:--Centre steel rod 31.5 in.; 2 steel rods next the centre 24.5 in.; 2 steel rods farthest from centre 29.5 in.; from the lower end of outside steel rods to centre of bob 3 in.; total 89.5 in. 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 88½ in. of steel is counteracted by the expansion of 49 1/8 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 are for a very soft brass and steel. Thos. Reid, with more ordinary steel and brass, prescribed a ratio of 112 to 71, Lord Grimthorpe a ratio of 100 to 61. It is absolutely necessary to put the actual rods to be used for making the pendulum in a hot water bath, and measure their expansions with a microscope.

John Smeaton, taking advantage of a far greater expansion coefficient 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 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 2/3 (39.14) = 26¼ 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 mere theory to determine what will be its behaviour. The zinc tube must be of rolled zinc, hard drawn through a 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 10½ to 13 in. in height, resting on a nut at the bottom of the rod.

Invar.

The old methods of pendulum compensation for heat may now be considered as superseded by the invention of "invar," a combination of nickel and steel, due to Charles E. Guillaume, of the International Office of Weights and Measures at Sèvres near Paris. This alloy has a linear coefficient of expansion on the average of .000001 per degree centigrade, that is to say, only about 1/11 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 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 computations are made of the moments of inertia of every separate individual part. Thus, for instance, since an addition of 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 1/10 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 elasticity with change of temperature. Moreover, since rods of invar steel subjected to 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.