Time and Clocks: A Description of Ancient and Modern Methods of Measuring Time

CHAPTER IV.

It is now time to give a description of the various parts of an ordinary pendulum clock. We will take the “Grandfather” clock as an example. We shall want an hour hand and a minute hand in the centre of the face, and a seconds hand to show seconds a little above them. There will be a seconds pendulum 39·14 inches long, and the centre of the face of the clock will be about seven feet above the ground, so as to give practically about five feet of fall for the weight.

Fig. 45.

In the first place, we have to consider the axle which carries the minute hand, and which turns round once in each hour. This is usually made of a piece of steel about one-sixth of an inch in diameter. Clockmakers usually call an axle an “arbor,” or “tree,” whence our word axletree.

This “arbor” is turned in the lathe, so as to have pivots on each end, fitted into holes in the clock plates, that is to say, the flat pieces of brass that serve as the body of the clock. The adjoining diagram shows S T the clock faces, and C, the arbor of the minute hand.

Inasmuch as the seconds hand is to turn round sixty times while the minute hand turns round once, it is obvious that the arbor of the minute hand must be connected to the arbor of the seconds hand by a train of cogwheels so arranged as to multiply by sixty. This of course involves us in having large and small cogwheels.

Fig. 46.

The small cogwheels usually have eight teeth, and are for convenience of manufacture, as also to stand prolonged wear, cut out of the solid steel of the arbor. They are nicely polished.

The easiest pair of wheels to use will be two pinions of eight teeth, or “leaves,” as they are called, and two cogwheels, one of sixty-four teeth, the other of sixty teeth.

It is then clear that if the arbor A turns round once in an hour, the arbor B will turn round eight times in an hour, and C will turn round (60 × 64)/(8 × 8) = 60 times in an hour, or once in each minute.

By having 480 teeth on the cogwheel on A, you could, of course, make C go round once in a minute without the use of any intermediate arbor such as B.

Fig. 47.

But this would not be a very convenient plan. For as the wheel on A is usually about two and a quarter inches in diameter, to cut 480 teeth on so small a wheel would involve us in cutting about sixty teeth to the inch. The teeth would thus be microscopically small, and would have to be set so fine that the least dirt would clog them. Moreover, the pinion of eight leaves would have to be microscopic. For these reasons, therefore, it is usual in clocks not to use wheels with teeth more than sixty or sixty-four in number, and to diminish the motion gradually by means, where needful, of intermediate arbors. We have next to consider how the weight is to be arranged so as to turn the arbor A once round in an hour. We know that we have five feet of space for the weights to fall in. If we arrange to have what is called a double fall, as shown in the sketch, then, allowing room for pulley wheels, we shall find that our string may be practically about nine feet in length.

Fig. 48.

The clock will be wanted to go for a week without winding, and as people may forget to wind it at the proper hour of the day, we will give it a day extra, and make an “eight-day” clock of it. Hence then, while nine feet of cord is being pulled out by a weight which falls four and a half feet, the minute hand is to be turned round as many times as there are hours in eight days, viz., 192 times. This could be accomplished, of course, by winding the cord round the arbor of the minute hand. But this would require 192 turns. If our cord is to be ordinary whipcord, or catgut, say one-twelfth of an inch in diameter, in order that the cord could be wound upon it, the arbor would have to be 192/12 inches long = 14? inches long. This would make the clock case unnecessarily deep. We must therefore again have recourse to an intermediate wheel.

Fig. 49.

If we put a pinion of eight leaves on the minute hand arbor c, and engage it with a wheel of sixty-four teeth on another arbor b, then b will obviously turn round once in eight hours, that is to say, twenty-four times in the period of eight days. And, if we fix on b a “drum” or cylinder two inches long, the twenty-four turns of our cord will just fit upon it, since, as has been said, our cord is to be one-twelfth of an inch in diameter. The diameter of the drum must be such that a cord nine feet long can be wound twenty-four times round it. That is to say, each lap must take (9 × 12)/24 = 4½ inches of cord. From this it is easy to calculate that the diameter of the drum must be rather less than one and a half inches. From this then it results that we want for a “Grandfather’s” clock a drum two inches long and one and a half inches diameter, on this a cogwheel of sixty-four teeth working into a minute hand arbor, with a pinion wheel with eight leaves, and a cogwheel of sixty-four teeth, an intermediate or idle wheel with an eight-leaved pinion, and a cogwheel of sixty teeth, engaging with a seconds hand arbor with a pinion of eight leaves. This is called the “train of wheels.” With it a weight such as can be arranged in an ordinary “Grandfather’s” clock case will cause by its fall during eight days the second hand arbor to turn round once in each minute during the whole time, and the minute hand arbor to turn round once in each hour.

Fig. 50.

We must next provide an arrangement for winding the clock up. It is obvious that we cannot do so by twisting the hands back. It is true that this could be done, but it would take about five minutes to do each time and be wearisome. In order to save this trouble, an arrangement called a ratchet wheel and pall must be provided. A ratchet wheel consists of a wheel with a series of notches cut in it, as shown in the figure A. A pall is a piece of metal, mounted on a pin, and kept pressed up against the ratchet wheel by a spring C. It is obvious that if I turn the wheel A round, and thus wind up a weight, fastened to a cord wound round the drum D, that the pall B will go click-click-click as the ratchet wheel goes round, but that the pall will hold it from slipping back again. When, however, I take my hands away, and let the ratchet wheel alone, then the weight E will pull on the drum D, and try and turn the ratchet wheel back the opposite way to that in which I twisted it at first. If the pall B is held fast, it is impossible to move it, but if the pall is fixed to a cogwheel F, which rides loose on the arbor of the drum D, then the pull of the weight E will tend to twist the cogwheel F round, and this, if engaged with a pinion wheel on the minute hand arbor, will therefore drive the clock. As the clock arbors move, of course the weight E gradually runs down, and, at last all the string is unwound from the drum D. The clock is said then to have “run down,” but if I take a clock key, and by means of it wind the string up upon the drum D, then the pall lets the drum and ratchet slip; the clock hands are not affected. When I have given twenty-four turns to the arbor, the nine feet of cord will then be wound upon the drum again, and the clock will be ready to go for eight more days, and will begin to move as soon as I cease to press upon the clock key.

Fig. 51.

I have thus described the winding mechanism. It now remains to describe the escapement.

It is of course obvious that, if the weight and train of wheels were simply let go, the weight would rush down, and the seconds-hand wheel would fly round at a tremendous pace; but we want it to be so restrained as only to be allowed to go one-sixtieth part of its journey round in each second. In fact, we need an “escapement” and a pendulum.

The escapement usually employed in “Grandfather” clocks is the anchor escapement above described. It is not by any means the best sort of escapement, but it is the easiest to make; and hence its popularity in the days sometimes called the “dear, good old days,” when people had to file everything out by hand, and had to take a day to do badly what can now be done well in five minutes.

The escape wheel of an anchor escapement has thirty sharp angular teeth on its rim. The wheel is made as light as possible, so that the shock of stoppage at each tick of the clock may be as slight as possible, for a heavy blow of course wastes power and gradually wears out the clock. The anchor consists of two arms of the shape shown in the illustration (Fig.44). As the escape wheel goes round in the direction of the arrow, the anchor, mounted on its arbor, rocks to and fro. The wheel cannot run away, because the act of pushing one arm or “pallet,” as it is called, outwards, and thus freeing the tooth pulls the other pallet in, and this stops the motion of the tooth opposite to it, but when the anchor rocks back again, so as to disengage the pallet from the tooth that holds it, then the opposite tooth is free to fly forward against the other pallet. This tends to rock the anchor the other way, but at that instant the pallet just engages the next tooth of the wheel, and so the action goes on. The anchor rocks from side to side; the pallets alternately engage the teeth of the wheel, making at each rock of the anchor the tick-tock sound with which we are so familiar. If the anchor were free to rock at any speed it could, the ticking of the clock would be very quick; so, to restrain the vivacity of the anchor, we have a pendulum. The pendulum might be simply hung on to the anchor. But the disadvantage of doing this would be that the heavy bob of the pendulum would cause such a pressure on the arbor of the anchor that there would be great friction, and the arbor would soon be worn out, and the accurate going of the clock disturbed. The pendulum therefore is hung on a piece of steel spring on a separate hook, which lets it go backwards and forwards and carries the weight easily, while a rod projecting from the anchor has a pin, which works in a slot on the pendulum. The pendulum is therefore able to control and regulate the movements of the escapement, and thus the time of the clock.

Of course it is clear that the heavier the driving weight put on the drum of the clock, and the better the cut and finish of the wheels, and the greater the cleanliness and oil, the more will be the pressure tending to drive round the escape wheel, and the harder the pressure on the pallets, and hence the bigger the impulses on the pendulum, and therefore the larger the amplitude of its swing.

If the amplitude of the pendulum’s swing affected the time of its swing, then the time kept by the clock would vary with the weight, and the dirt and friction, and the drying up of the oil. But here precisely is where the value of the beautiful law governing the harmonic motion of the pendulum comes in. The time of the pendulum is (for small arcs) independent of the length of swing, and therefore of the driving force of the clock, and hence within limits the clock, even though roughly made and foul with the dirt of years, continues to keep good time. But the anchor escapement has imperfections. The only way in which a pendulum can be relied on to keep accurate time is by leaving it unimpeded. But the pressure of the teeth on the pallets in an anchor escapement constantly interferes with this.

Fig. 52.

A little consideration will easily show that there are some times during the swing of a pendulum at which interference is far more fatal to its time-keeping than at others. Thus the bob of a pendulum may be regarded as a weight shot outwards from its position of rest against the influence of a retarding force varying as its distance from rest—in fact, shot out against a spring. The time of going out and coming in again will be quite independent of the force exerted to throw it out, quite independent of its original velocity. Therefore a variation in the impulse given to the bob is of no consequence, provided that impulse is given when the bob is near the position of rest. This follows from the nature of the motion. If a ball be attached to a piece of elastic thread, and thrown from the hand, so that it flies out, and then stops and is brought back by the elastic force of the thread, the time of the outward motion and the return is the same whatever be the force of the throw. And so if a pendulum be impelled outwards from a position of rest, the time of the swing out and back is the same, however big (within limits) is the impelling force and the consequent length of the swing. The use of a pendulum as a measure of time is to impel it outwards, and then let it fly freely out and back. But if its motion is not free, if forces other than gravity act upon it while on its path, then its time of swing will be disturbed. It does not matter with what force you originally impel it, but what does matter is, that when it once starts it should be allowed to travel unimpeded and uninfluenced. Now that is what an anchor escapement does not do. The impulse is given the whole way out on one of the pallets, and then when it is at its extreme of swing, and ought to be left tranquil, the other pallet fastens on it. But a perfect escapement ought to give its impulse at the middle point of the swing, when the pendulum is at the lowest, and then cease, and allow the pendulum to adapt its swing to the impulse it has received, and thus therefore to keep its time constant. This is done by an escapement called the dead beat escapement, which, though in an imperfect way, realises these conditions.

The alteration is made in the shape of the pallets of the anchor. The wheel is much the same. Each pallet consists of two faces: a driving face a b and a sliding face b c.

When the tooth b has done its work by pressing on the driving face, and thus driving the anchor over, say, to the left, then the tooth on the opposite side falls on the sliding face of the other pallet. This being an arc of a circle, has no effect in driving the anchor one way or the other; hence the pendulum is free to swing to the left as far as it likes and return when it feels inclined, always with the exception of a little friction of the tooth on the faces of the pallets, but when it returns and begins to move towards the right, the tooth slides back along the face of the pallet till the pendulum is almost at the middle of its swing; then an impulse is given by the pressure of the tooth upon the inclined plane a´ b´. As soon, however, as the tooth leaves b´, another tooth on the other side at once engages the sliding face b c of the other pallet, and so the motion goes on.

This beautiful escapement is at present used for astronomical clocks; the pallets are made of agate or sapphire, and therefore do not grind away the teeth of the wheel perceptibly, and the loss by friction on the sliding surfaces is exceedingly small.

There are several other ways even better than this for securing a free pendulum movement. We have now to return to our clock.

The centre arbor moves round once in an hour, and carries the minute hand. In order to provide an hour hand, which shall turn round once in twelve hours, we fasten a cogwheel and tube N on to the minute hand arbor by means of a small spring, which keeps it rather tight, but allows it to slip if turned round hard (see Fig.45). This spring is a little bent plate slipped in behind the cogwheel on which its ends rest; its centre presses on a shoulder on the minute hand arbor; it is a sort of small carriage spring. The cogwheel n has thirty teeth. This cogwheel engages another cogwheel o with thirty teeth, on a separate arbor, which carries a third cogwheel, p, with six teeth, and this again engages a fourth cogwheel, q, with seventy-two teeth, mounted on a tube which slips over the tube to which the cogwheel a is attached. It is now easy to see that for each turn of the minute hand arbor the arbor p makes one turn, and for each turn of the arbor p the cogwheel d, makes one-twelfth of a turn. From which it follows that for each turn of the minute hand arbor the cogwheel d with its tube, or, as it is sometimes called, its “slieve,” makes one-twelfth of a turn, and thus makes a hand fastened to it show one hour for every complete turn of the minute hand.

The minute hand is attached to the tube or slieve which carries the cogwheel N. The hour hand is attached to the tube or slieve which carries the cogwheel Q, and one goes twelve times as slowly as the other.

But if you want to set the clock it is easy to do so by reason of the fact that the minute hand is not fixed to the arbor, but only to the slieve on the cogwheel that fits on the arbor, and is held somewhat tight to the arbor by means of the spring. The hands can thus be turned, but they are a little stiff. A washer on the minute hand arbor keeps the slieve on the cogwheel pressed tight against the spring, being secured in its turn by a very small lynch-pin driven through a hole in the minute hand arbor.

It remains to explain a few subsidiary arrangements, not always found upon all clocks, but which are useful.

In order to prevent the overwinding of the clock (see Fig.43), which would cause the cord to overrun the drum, an arm is provided, fitted with a spring. As the weight is wound up the free part of the cord travels along the drum or the fusee; and the cord, when it is near the end of the winding, comes up against the arm and pushes it a little aside. This causes the end of the arm to be pushed against a stop on the axis of the fusee, and thus prevents the clock being further wound up. The stop, being ratchet-shaped, does not prevent the weight from pulling the ratchet wheel round the other way, and thus driving the clock; it only prevents the rotation of that wheel when the string is near it, and the winding is finished.

Another arrangement is the “maintaining spring.”

It will be remembered that during the process of winding the clock the hand twisting the key takes the pressure of the ratchet wheel off the pall, so that during that operation no force is at work to drive the clock. In consequence the pendulum receives no impulse, but swings simply by virtue of its former motion. If the process of winding were done slowly enough the clock might even stop. To avoid this, a very ingenious arrangement is made to keep the cogwheel mounted on the winding shaft going during the winding-up process. This is called a maintaining spring.

The arrangement shown in Fig.53 will explain it.

Fig. 53.

The cogwheel a and the ratchet wheel are both mounted loosely on the arbor carrying the drum. a is linked to b by a spring c. The ratchet wheel b is engaged by a pall fixed to some convenient place on the body of the clock frame. When the weight pulls on the drum the pull is communicated to the ratchet wheel b, and this acts on the spring c and pulls it out a little. As soon as the spring c is pulled out as far as its elasticity permits, a pull is communicated to the cogwheel a, and the clock is driven round. When the clock is wound the pressure of the weight is removed, and therefore the ratchet wheel e no longer presses on the pall, and thus no pressure is communicated to the ratchet wheel b, or through it to the clock. But here the spring c comes into play. For since the ratchet wheel b is held fast by the pall d, the spring c pulls at the wheel a, and thus for a minute or so will continue to drive the clock. This driving force, it is true, is less than that caused by the weight, but it is just enough to keep the pendulum going for a short time, so that the going of the clock is not interfered with.

If the reader can get possession of a clock, preferably one that does not strike, and, with the aid of a small pair of pincers and one or two screwdrivers, will take it to pieces and put it together again, the mechanism above described will soon become familiar to him. Not every clock is provided with maintaining spring and overwinding preventer.

The cause of stoppage of a clock generally is dirt. Where possible, clocks should always be put under glass cases. “Grandfather” clocks will go much better if brown paper covers are fitted over the works under the cases. In this way a quantity of dust may be avoided. To get a good oil is very important. It will be noticed that pivot-holes in clocks are usually provided with little cup-like depressions. This is to aid in keeping in the oil. The best clock oil is that which does not easily solidify or evaporate. Ordinary machine oil, such as used for sewing machines, is good as a lubricant, but rapidly evaporates. Olive oil corrodes the brass.

It is best to procure a little clock oil, or else the oil used for gun locks, sold by the gunsmiths. The holes should be cleaned out with the end of a wooden lucifer match, cut to a tapering point. The pivots should be well rubbed with a rag dipped in spirits of wine. If the pivots are worn they should be repolished in the lathe. If the cogs of the wheels are worn, there is no remedy but to get new ones. Old clocks sometimes want a little addition to the driving weight to make them go.

The weight necessary to drive the clock depends on its goodness of construction, and on the weight of the pendulum. If the clock is driven for eight days with a cord of nine feet in length with a double fall, then during each beat of the pendulum that weight will descend by an amount =

9/(2 × 24 × 60 × 60 × 8) feet or 1/12800th inch.

Whence, if the clock weight is 10 lbs., the impulse received by the clock at each beat is equivalent to a weight of 10 lbs. falling through 1/12800th of an inch, or to the fall of six grains through an inch. The power thus expended goes in friction of the wheels and hands, and in maintaining the pendulum in spite of the friction of the air.

The work therefore that is put into the clock by the operation of winding is gradually expended during the week in movement against friction. The work is indestructible. The friction of the parts of the clock develops heat, which is dissipated over the room and gradually absorbed in nature. But this heat is only another form of work. Amounts of work are estimated in pressures acting through distances. Thus, if I draw up a weight of 1 lb. against the accelerative force of gravity through a distance of one foot, I am said to do a foot-pound of work.

One pound of coal consumed in a perfect engine would do eight millions foot-pounds of work. Hence, if the energy in a pound of coal could be utilized, it would keep about 100,000 grandfather’s clocks going for a week. As it is consumed in an ordinary steam engine it will do about half a million foot-pounds of work. One pound of bread contains about three million foot-pounds of energy. A man can eat about three pounds of bread in a day, and, as he is a very good engine, he can turn this into about three-quarters of a million foot-pounds of work. The rest of the work contained in the bread goes off in the form of heat.

Fig. 54.

As has been previously said, the power of the action of gravity in drawing back a pendulum that has been pushed aside from its position of rest becomes less in proportion as the pendulum is longer, and hence as the pendulum is longer the time of vibrations increases. In the appendix to this chapter a short proof will be given showing that the length of a pendulum varies as the square of the time of its vibration. A pendulum which is 39·14 inches in length vibrates at London once in each second. Of course at other parts of the earth, where the force of gravity is slightly different, the time of vibration will be different, but, since the earth is nearly a globe in shape, the force of gravity at different parts of it does not vary much, and therefore the time of vibration of the same pendulum in different parts of the earth does not vary very much. The length of a pendulum is measured from its point of suspension down to a point in the bob or weight. At first sight one would be inclined to think that the centre of gravity of the pendulum would be the point to which you must measure in order to get its length. So that if B were a circular bob, and the rod of the pendulum were very light, the distance A B to the centre of the bob would be the length of the pendulum. But if we were to fly to this conclusion, we should be making the same error that Galileo made when he allowed a ball to roll down an inclined plane. He forgot that the motion was not a simple one of a body down a plane, but was also a rolling motion. The pendulum does not vibrate so as always to keep the bob immovable with the top side C always uppermost. On the contrary, at each beat the bob rotates on its centre and makes, as it were, some swings of its own. Therefore in the total motions of the pendulum this rotation of the bob has to be taken into account. Of course, if the pendulum were so arranged that the bob did not rotate, and the point C were always uppermost, as, for instance, if the pendulum consisted of two parallel rods, A B and C D, suspended from A and C, then we might consider the bob as that of a pendulum suspended from E, and the pendulum would swing once in a second if A B = C D = E F were equal to 39·14 inches, for by this arrangement there would be no rotation of the bob. But as pendulums are generally made with the bob rigidly fixed to the rod E F, the rotation must be taken into account.

Fig. 55.

It wants some rather advanced mathematical knowledge to do this. But in practice clockmakers take no account of it. The correction is not a large one, so they make the rod as nearly true as they can, arrange a screw on the bob to allow of adjustment, and then screw the bob up and down until in practice the time of oscillation is found to be correct.

Fig. 56.

The mode of suspension of a pendulum of the best class is that shown in Fig.56, which allows the pendulum to fall into its true position without strain. A is a tempered steel spring, which bends to and fro at each oscillation. It is wonderful how long these springs can be bent to and fro without breaking. Inasmuch as lengthening the pendulum increases the time, so that the time of vibration t varies as the square of the length of the pendulum, a very small lengthening of the pendulum causes a difference in the time. In practice, for each thousandth of an inch that we lengthen the pendulum we make a difference of about one second a day in the going of the clock. If we cut a screw with eighteen threads to the inch on the bottom of the pendulum rod, and put a circular nut on it, with the rim divided into sixty parts, then each turn through one division will raise or lower the bob by 1/1080th of an inch, and this first causes an alteration of time of the clock by one second in the day. This is a convenient arrangement in practice, for it affords an easy means of adjusting the pendulum. We need only observe how many seconds the clock loses or gains in the day, and then turn the nut through a corresponding number of divisions in order to rectify the pendulum.

Fig. 57.

Another needful correction of the pendulum is that due to changes in temperature. If the rod of the pendulum be made of thoroughly dried mahogany, soaked in a weak solution of shellac in spirits of wine, and then dried, there will not be much variation either from heat or moisture. But for clocks required to have great precision the pendulum rod is usually made of metal. A rod of iron expands about 1/160000th of its length for each degree Fahrenheit; and therefore for each degree Fahrenheit a pendulum rod of 39·14 inches will expand about 1/4000 thousandths of an inch, and thus make a difference in the going of the clock of about one-fourth of a second per day. The expansion will, of course, make the clock go slower. It would be possible to correct this expansion if some arrangement could be made, whenever it occurred, to lift up the bob of the pendulum by an amount corresponding to it, as, for instance, to make the bob of some material which expanded very much more by heat than the material of which the pendulum rod was made.

Fig. 58.

Thus if we hang on to the end of a pendulum of iron a bottle of iron about seven inches long, and almost fill it with mercury, then, as soon as the heat increases, the iron of the rod and of the bottle expands, and the centre of oscillation of the pendulum is lowered. But as the linear expansion of mercury contained in a bottle is about five times that of iron, the mercury rises in the bottle, and thus the expansion downwards of the pendulum rod is compensated by the expansion upwards of the mercury in the bottle. The rod may be fastened to the mouth of the bottle by a screw, so that as the bottle is turned round it may be raised or lowered on the rod, and thus the length of the pendulum may be adjusted. The bottle is made of steel tube, screwed into a thin turned iron top and bottom. Of course no solder must be used to unite the iron, for mercury dissolves solder. A little oil and white-lead will make the screwed joints tight. This is an excellent form of pendulum. Another plan is to use zinc as the metal which is to counteract the expansion of the iron. The expansion of zinc is about three times that of iron.

Fig. 59.

Hence a zinc tube, about twenty inches long (shown shaded in Fig.59), is made to rest upon a disc fastened to the lower part of the iron pendulum rod. On the top of the zinc rests a flat ring A, from which is suspended an iron tube A, which carries the bob B. The expansion of the zinc tube is large enough to compensate the expansion both of the rod and the tube, and the bob consequently remains at the same depth below the point of suspension, whatever be the temperature. There is, however, a new method which is far superior to all these, and this is due to the discovery by M. Guilliaume, of Paris, of a compound of nickel and steel which expands so little that it can be compensated by a bob of lead instead of by a bob of mercury. This material is sold in England under the name of “invar.” An invar rod with a properly proportioned lead bob makes an almost perfect pendulum, the expansion of the invar and the lead going on together. The exact expansion of the invar is given by the makers, who also supply information as to the size and suspension of the bob proper to use with it.

It has been already shown that the uniformity of time of swing of a pendulum is only true when the arc through which it swings is very small. If the total swing from one side to another is not more than about two inches very little difference in time-keeping is made by putting a little more driving weight on the clock, and thus increasing its arc of swing; but when the arc of swing becomes say three inches, or one and a half inches on each side of the pendulum, then the time of vibration is affected. At this distance each tenth of an inch increase of swing makes the pendulum go slower by about a second a day.

The resistance of the air, of course, has a great influence on a pendulum, and is one of the chief causes that bring it ultimately to rest. Even the variations of pressure of the atmosphere which the barometer shows as the weather varies have an effect on the going of a clock. Attempts have been made by fixing barometers on to pendulums with an ingenious system of counter balancing to counteract this, but these refinements are not in common use, and are too complicated to be susceptible of effective regulation.

Appendix to Chapter IV.

It may be useful to give a simple form of proof of the law which governs the time of oscillation of a pendulum whose length is given.

Unfortunately, it is impossible to give one so simple as to be comprehended by those who know nothing whatever of mathematics. It is, however, possible to give a proof that requires very little mathematical knowledge.

We know that when a mass of matter is whirled round at the end of a string it tends to fly outwards and puts a strain on the string. The faster the speed at which the mass is whirled, the stronger will be the strain on the string. Suppose that the length of the string equals R, the velocity of the mass as it flies round equals V. Let a be the body whirled round by a string o a from a centre at O. The body always, of course, tends to fly on in a straight line from the point at which it is at any instant. But that tendency is frustrated by the pull of the string which constrains it to take a circular path. It is, of course, all one whether the force that tends to pull the body inwards towards O is a string or an attractive force of any kind acting through a distance without any string at all. Evidently if the body keeps its place in the circle it must be because the centrifugal force tending to whirl it out is equal to the centripetal or attractive force tending to pull it in.

Fig. 60.

The strain on the body, due to the force tending to pull it inwards, we shall designate by F, meaning by F the number of feet of velocity that would in one second be imparted to the body by the attractive force.

Suppose that at some given instant of time the body is at a point a. At that instant its direction will be along a b, tangential to the circle at a, and that is the path it would take if the centripetal or attractive force ceased to act just as the body got to a. In that case the body would be whirled off like a stone from a sling along the line a b, and would at the end of a given time, let us suppose a second, arrive at b. But it is not so whirled off; it is attracted towards O and pulled inwards, and comes to c. Hence, then, the attractive force acting during one second must have been sufficient to pull the mass in from b to c. But we know that if an accelerating force (F) acts on a body for a second it produces a final velocity equal to F at the end of the second, and an average velocity half F during the second.

Hence, then, the space b c, by which the body has been pulled in, is represented by half F, but a b, the space which the body would have travelled forwards, will be represented by V, the velocity of the body in a second; but if the motion be such that the distance b c travelled in a second is very small, then the triangles a b d and a b c are approximately similar, and the smaller a b is the more nearly similar they are. Whence then (a b)/(b c) = (a d)/(a b), that is to say (a b)² = a d × b c.

But a b represents the space which would have been traversed by the body in one second at the rate it was going, and hence is equal to V; a d is the diameter of the circle, and hence equals 2 R; b c is the space through which the body has been drawn in the second by the attractive force F, and therefore equals half F.

Whence then V² = 2 R × half F = R F. We took a second as the limit of time during which the motion was to be considered. Of course any other time could have been taken. Now what is true of the motion of a body during a very short time is also true of the body during the whole of its path, assuming that the path is a circle, and that F remains constant, as it obviously will if the path is a circle, and the velocity is uniform. Whence then we may generally say that if a body is being whirled round at the end of a string the strain F on the string is directly proportional to the square of the velocity, and is inversely proportional to the length of the string.

The time of rotation, is of course = length of the path ÷ velocity

= (2pR)/V = (2pR)/v(R F) = 2pv(R/F).

Whence then we see that for motion in a circle of a mass under the attraction of a centripetal force, or pull of a string, the time of rotation will be uniform, provided that the centripetal force always varies as the radius of the path. From this it is evident that a body fixed on to an elastic thread where the pull varies as the extension would make its rotations always in equal times. If your sling consists of elastic, whirl as you will, you can only whirl the body round so many times in a second, and no more. Any increase in your efforts only makes the string stretch, and the circle get bigger. The velocity of the body in its path of course increases, but the time it takes to go once round is invariable.

It also follows that if a body hung by a string of length l, under the action of gravity, be travelling in a circle round and round, then, if the circle is a small one compared with the length of the string, the inward acceleration f towards the centre will be approximately proportional to the radius r of the circle, and the time of rotation will be

t = 2pv(r/f).

But in this case f, the inward acceleration, is to g the acceleration downwards of gravity as A B:A P or

f/g = (A B)/(A P) = (A P)/(O P) = r/l.

Fig. 61.

Fig. 62.

Whence then the time of rotation of this body would be if the circle of rotation was small

= 2pv(l/g).

And if you try you will find that this is so. For instance, take a thread 39-1/7 inches long, that is 3·25 feet. Hang anything heavy from one end of it, and cause it to swing round and round in a small circle. Now g the acceleration of gravity = 32·2 feet per second. p the ratio of the circumference of a circle to its diameter = 3·14. From which it follows that the time of rotation = 2 × 3·14v(3·25/32·2) seconds = 2 seconds. But if we look at the rotating body sideways, it appears to act as a pendulum; it matters nothing whether we swing it round and round or to and fro. For in any case the accelerative force tending to bring it back to a position of rest is always proportional to the distance of displacement, and, therefore, its time of motion must always be 2pv(l/g) and its motion harmonic.

The length of a seconds pendulum, that is a pendulum that makes its double swing in two seconds, will therefore be

l = 4/((2p)²) × g feet

= (g × 12)/p² inches

= 39·14 inches.

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