Part 21

Let us suppose that the ellipse A C B D, Fig. 93, is the path described by a particle when suspended by a string from a point vertically above O, the centre of the ellipse. To produce this motion I withdraw the particle from its position of rest at O to A. If merely released, the particle would swing over to B, and back again to A; but I do not simply release it, I impart a velocity impelling it in the direction A T. Through O draw C D parallel to A T. If I had taken the particle at O, and, without withdrawing it from its position of rest, had started it off in the direction O D, the particle would continue for ever to vibrate backwards and forwards from C to D. Hence, when I release the particle at A, and give it a velocity in the direction A T, the particle commences to move under the action of two distinct vibrations, one parallel to A B, the other parallel to C D, and we have to find the effect of these two vibrations impressed simultaneously upon the same particle. They are performed in the same time, since all vibrations are isochronous. We must conceive one motion starting from A A towards O at the same moment that the other commences to start from O towards D. After the lapse of a short time, the body has moved through A Y in its oscillation towards O, and in the same time through O Z in its oscillation towards D; it is therefore found at X. Now, when the particle has moved through a distance equal and parallel to A O, it must be found at the point D, because the motion from O to D takes the same time as from A to O. Similarly the body must pass through B, because the time occupied by going from A to B, would have been sufficient for the journey from O to D, and back again. The particle is found at P, because, after the vibration returning from B has arrived at Q, the movement from D to O has travelled on to R. In this way the particle may be traced completely round its path by the composition of the two motions. It can be proved that for small motions the path is an ellipse, by reasoning founded upon the fact that the vibrations are isochronous.

653. Close examination reveals a very interesting circumstance connected with this experiment. It may be observed that the ellipse described by the body is not quite fixed in position, but that it gradually moves round in its plane. Thus, in Fig. 92, the ellipse which is being traced out by the brush will gradually change its position to the dotted line shown on the board. The axis of the ellipse revolves in the same direction as that in which the ball is moving. This phenomenon is more marked with an ellipse whose dimensions are considerable in proportion to the length of the string. In fact, if the ellipse be very small, the change of position is imperceptible. The cause of this change is to be found in the fact already mentioned (Art. 598), that though the vibrations of a pendulum are very nearly isochronous, yet they are not absolutely so; the vibrations through a long arc taking a minute portion of time longer than those through a short arc.

This difference only becomes appreciable when the larger arc is of considerable magnitude with reference to the length of the pendulum.

654. How this causes displacement of the ellipse may be explained by Fig. 94. The particle is describing the figure A D C B in the direction shown by the arrows. This motion may be conceived to be compounded of vibrations A C and B D, if we imagine the particle to have been started from A with the right velocity perpendicular to O A. At the point A, the entire motion is for the instant perpendicular to O A; in fact, the motion is then exclusively due to the vibration B D, and there is no movement parallel to O A. We may define the extremity of the major axis of the ellipse to be the position of the particle, when the motion parallel to that axis vanishes. Of course this applies equally to the other extremity of the axis C, and similarly at the points B or D there is no motion of the particle parallel to B D.

655. Let us follow the particle, starting from A until it returns there again. The movement is compounded of two vibrations, one from A to C and back again, the other along B D; from O to D, then from D to B, then from B to O, taking exactly double the time of one vibration from D to B. If the time of vibration along A C were exactly equal to that along B D, these two vibrations would bring the particle back to A precisely under the original circumstances. But they do not take place in the same time; the motion along A C takes a shade longer, so that, when the motion parallel to A C has ceased, the motion along D B has gone past O to a point Q, very near O. Let A P = O Q, and when the motion parallel to A C has vanished, the particle will be found at P; hence P must be the extremity of the major axis of the ellipse. In the next revolution, the extremity of the axis will advance a little more, and thus the ellipse moves round gradually.

THE COMPOSITION OF VIBRATIONS.

656. We have learned to regard the elliptic motion in the conical pendulum as compounded of two vibrations. The importance of the composition of vibrations justifies us in examining this subject experimentally in another way. The apparatus which we shall employ is represented in Fig. 95.

A is a ball of cast iron weighing 25 lbs., suspended from the tripod by a cord: this ball itself forms the support of another pendulum, B. The second pendulum is very light, being merely a globe of glass filled with sand. Through a hole at the bottom of the glass the sand runs out upon a drawing-board placed underneath to receive it.

Thus the little stream of sand depicts its own journey upon the drawing-board, and the curves traced out thus indicate the path in which the bob of the second pendulum has moved.

657. If the lengths of the two pendulums be equal, and their vibrations be in different planes, the curve described is an ellipse, passing at one extreme into a circle, and at the other into a straight line. This is what we might have expected, for the two vibrations are each performed in the same time, and therefore the case is analogous to that of the conical pendulum of Art. 648.

658. But the curve is of a very different character when the cords are unequal. Let us study in particular the case in which the second pendulum is only one-fourth the length of the cord supporting the iron ball. This is the experiment represented in Fig. 95. The form of the path delineated by the sand is shown in Fig. 96. The arrowheads placed upon the curve show the direction in which it is traced. Let us suppose that the formation of the figure commences at A; it then goes on to B, to O, to C, to D, and back to A: this shows us that the bob of the lower pendulum must have performed two vibrations up and down, while that of the upper has made one right and left. The motion is thus compounded of two vibrations at right angles, and the time of one is half that of the other.

The time of vibration is proportional to the square root of the length; and, since the lower pendulum is one-fourth the length of the upper, its time of vibration is one-half that of the upper. In this experiment, therefore, we have a confirmation of the law of Art 605.

LECTURE XX. _THE MECHANICAL PRINCIPLES OF A CLOCK._ Introduction.—The Compensating Pendulum.—The Escapement.—The Train of Wheels.—The Hands.—The Striking Parts.

INTRODUCTION.

659. We come now to the most important practical application of the pendulum. The vibrations being always isochronous, it follows that, if we count the number of vibrations in a certain time, we shall ascertain the duration of that time. It is simply the product of the number of vibrations with the period of a single one. Let us take a pendulum 39·139 inches long; which will vibrate exactly once a second in London, and is therefore called a seconds pendulum (See Art. 607). If I set one of these pendulums vibrating, and contrive mechanism by which the number of its vibrations shall be recorded, I have a means of measuring time. This is of course the principle of the common clock: the pendulum vibrates once a second and the number of vibrations made from one epoch to another epoch is shown by the hands of the clock. For example, when the clock tells me that 15 minutes have elapsed, what it really shows is that the pendulum has made 60 × 15 = 900 vibrations, each of which has occupied one second.

660. One duty of the clock is therefore to count and record the number of vibrations, but the wheels and works have another part to discharge, and that is to sustain the motion of the pendulum. The friction of the air and the resistance experienced at the point of suspension are forces tending to bring the pendulum to rest; and to counteract the effect of these forces, the machine must be continually invigorated with fresh energy. This supply is communicated by the works of the clock, which will be sufficiently described presently.

661. When the weight driving the clock is wound up, a store of energy is communicated which is doled out to the pendulum in a very small impulse, at every vibration. The clock-weight is just large enough to be able to counterbalance the retarding forces when the pendulum has a proper amplitude of vibration. In all machines there is some energy lost in maintaining the parts in motion in opposition to friction and other resistances; in clocks this represents the whole amount of the force, as there is no external work to be performed.

THE COMPENSATING PENDULUM.

662. The actual length of the pendulum used, depends upon the purposes for which the clock is intended, but it is essential for correct performance that the pendulum should vibrate at a constant rate; a small irregularity in this respect may appreciably affect the indications of the clock. If the pendulum vibrates in 1·001 seconds instead of in one second, the clock loses one thousandth of a second at each beat; and, since there are 86,400 seconds in a day, it follows that the pendulum will make only 86,400 - 86·3 vibrations in a day, and therefore the clock will lose 86·3 seconds, or nearly a minute and a half daily.

663. For accurate time-keeping it is therefore essential that the time of vibration shall remain constant. Now the time of vibration depends upon the length, and therefore it is necessary that the length of the pendulum be absolutely unalterable. If the length of the pendulum be changed even by one-tenth of an inch, the clock will lose or gain nearly two minutes daily, according to whether the pendulum has been made longer or shorter. In general we may say that, if the alteration in the length amount to _k_ thousandths of an inch, the number of seconds gained or lost per day is 1·103 × _k_ with a seconds pendulum.

664. This explains the practice of raising the bob of the pendulum when the clock is going too slow or lowering it when going too fast. If the thread of the screw used in doing this have twenty threads to the inch; then one complete revolution of the screw will raise the bob through 50 thousandths of an inch, and therefore the effect on the rate will be 1·103 × 50 = 55 nearly. Thus, the rate of the clock will be altered by about 55 seconds daily. Whatever be the screw, its effect can be calculated by the simple rule expressed as follows. Divide 1103 by the number of threads to the inch; the quotient is the number of seconds that the clock can be made to gain or lose daily by one revolution of the screw on the bob of the pendulum.

665. Let us suppose that the length of the pendulum has been properly adjusted so that the clock keeps accurate time. It is necessary that the pendulum should not alter in length. But there is an ever-present cause tending to change it. That cause is the variation of temperature. We shall first illustrate by actual experiment the well known law that bodies expand under the action of heat; then we shall consider the irregularities thus introduced into the motion of the pendulum; and, finally, we shall point out means by which these irregularities may be effectually counteracted.

666. We have here a brass bar a yard long; it is at present at the temperature of the room. If we heat the bar over a lamp, it becomes longer; but upon cooling, it returns to its original dimensions. These alterations of length are very small, indeed too small to be perceived except by careful measurement; but we shall be able to demonstrate in a simple way that elongation is the consequence of increased temperature. I place the bar A D in the supports shown in Fig. 97. It is firmly secured at B by means of a binding screw, and passes quite freely through C; if the bar elongate when it is heated by the lamp, the point D must approach nearer to E. At H is an electric battery, and G is a bell rung by an electric current. One wire of the battery connects H and G, another connects G with E, and a third connects H with the end of the brass rod B. Until the electric current becomes completed, the bell remains dumb, the current is not closed until the point touches E: when this is the case, the current rushes from the battery along the bar, then from D to E, from that through the bell, and so back to the battery. At present the point is not touching E, though extremely close thereto. Indeed if I press E towards the point, you hear the bell, showing that the circuit is complete; removing my finger, the bell again becomes silent, because E springs back, and the current is interrupted.

667. I place the lamp under the bar: which begins to heat and to elongate; and as it is firmly held at B, the point gradually approaches E: it has now touched E; the circuit is complete, and the bell rings. If I withdraw the lamp, the bar cools. I can accelerate the cooling by touching the bar with a damp sponge; the bar contracts, breaks the circuit, and the bell stops: heating the bar again with the lamp, the bell again rings, to be again stopped by an application of the sponge. Though you have not been able to see the process, your ears have informed you that heat must have elongated the bar, and that cold has produced contraction.

668. What we have proved with respect to a bar of brass, is true for a bar of any material; and thus, whatever be the substance of which a pendulum is made, a simple uncompensated rod must be longer in hot weather than in cold weather: hence a clock will generally have a tendency to go faster in winter than in summer.

669. The amount of change thus produced is, it is true, very small. For a pendulum with a steel rod, the difference of temperature between summer and winter would cause a variation in the rate of five seconds daily, or about half a minute in the week. The amount of error thus introduced is of no great consequence in clocks which are only intended for ordinary use; but in astronomical clocks, where seconds or even portions of a second are of importance, inaccuracies of this magnitude would be quite inadmissible.

670. There are, it is true, some substances—for example, ordinary timber—in which the rate of expansion is less than that of steel; consequently, the irregularities introduced by employing a pendulum with a wooden rod are less than those of the steel pendulum we have mentioned; but no substance is known which would not originate greater variations than are admissible in the performance of an astronomical clock.

We must, therefore, devise some means by which the effect of temperature on the length of a pendulum can be avoided. Various means have been proposed, and we shall describe one of the best and simplest.

671. The mercurial pendulum (Fig. 98) is frequently used in clocks intended to serve as standard time-keepers. The rod by which the pendulum is suspended is made of steel; and the bob consists of a glass jar of mercury. The distance of the centre of gravity of the mercury from the point of suspension may practically be considered as the length of the pendulum. The rate of expansion of mercury is about sixteen times that of steel: hence, if the bob be formed of a column of mercury one-eighth part of the length of the steel rod, the compensation would be complete. For, suppose the temperature of the pendulum be raised, the steel rod would be lengthened, and therefore the vase of mercury would be lowered; on the other hand, the column of mercury would expand by an amount double that of the steel rod: thus the centre of the column of mercury would be elevated as much as the steel was elongated; hence the centre of the mercury is raised by its own expansion as much as it is lowered by the expansion of the steel, and therefore the effective length of the pendulum remains unaltered. By this contrivance the time of oscillation of the pendulum is rendered independent of the temperature. The bob of the mercurial pendulum is shown in Fig. 98. The screw is for the purpose of raising or lowering the entire vessel of mercury in order to make the rate correct in the first instance.

THE ESCAPEMENT.

672. Practical skill as well as some theoretical investigation has been expended upon that part of a clock which is called the _escapement_, the excellence of which is essential to the correct performance of a timepiece. The pendulum must have its motion sustained by receiving an impulse at every vibration: at the same time it is desirable that the vibration should be hampered as little as possible by mechanical connection. The isochronism on which the time-keeping depends is in strictness only a characteristic of oscillations performed with a total freedom from constraint of every description; hence we must endeavour to approximate the clock pendulum as nearly as possible to one which is swinging quite freely. To effect this, and at the same time to maintain the arc of vibration tolerably constant, is the property of a good escapement.

673. A common form of escapement is shown in Fig. 99. The arrangement is no doubt different from that actually found in a clock; but I have constructed the machine in this way in order to show clearly the action of the different parts. G is called the escapement-wheel: it is surrounded by thirty teeth, and turns round once when the pendulum has performed sixty vibrations,—that is, once a minute. I represents the escapement; it vibrates about an axis and carries a fork at K which projects behind, and the rod of the pendulum hangs between its prongs. The pendulum is itself suspended from a point O. At N, H are a pair of polished surfaces called the pallets: these fulfil a very important function.

674. The escapement-wheel is constantly urged to turn round by the action of the weight and train of wheels, of which we shall speak presently; but the action of the pallets regulates the rate at which the wheel can revolve. When a tooth of the wheel falls upon the pallet N, the latter is gently pressed away: this pressure is transmitted by the fork to the pendulum; as N moves away from the wheel, the other pallet H approaches the wheel; and by the time N has receded so far that the tooth slips from it, H has advanced sufficiently far to catch the tooth which immediately drops upon H. In fact, the moment the tooth is free from N, the wheel begins to revolve in consequence of the driving weight; but it is quickly stopped by another tooth falling on H: and the noise of this collision is the well known tick of the clock. The pendulum is still swinging to the left when the tooth falls on H. The pressure of the tooth then tends to push H outwards, but the inertia of the pendulum in forcing H inwards is at first sufficient to overcome the outward pressure arising from the wheel; the consequence is that, after the tooth has dropped, the escapement-wheel moves back a little, or “recoils,” as it is called. If you look at any ordinary clock, which has a second-hand, you will notice that after each second is completed the hand recoils before starting for the next second. The reason of this is, that the second-hand is turned directly by the escapement-wheel, and that the inertia of the pendulum causes the escapement-wheel to recoil. But the constant pressure of the tooth soon overcomes the inertia of the pendulum, and H is gradually pushed out until the tooth is able to “escape”; the moment it does so the wheel begins to turn round, but is quickly brought up by another tooth falling on N, which has moved sufficiently inwards.

The process we have just described then recurs over again. Each tooth escapes at each pallet, and the escapements take place once a second; hence the escapement-wheel with thirty teeth will turn round once in a minute.

675. When the tooth is pushing N, the pendulum is being urged to the left; the instant this tooth escapes, another tooth falls on H, and the pendulum, ere it has accomplished its swing to the left, has a force exerted upon it to bring it to the right. When this force and gravity combined have stopped the pendulum, and caused it to move to the right, the tooth soon escapes at H, and another tooth falls on N, then retarding the pendulum. Hence, except during the very minute portion of time that the wheel turns after one escapement, and before the next tick, the pendulum is never free; it is urged forwards when its velocity is great, but before it comes to the end of its vibration it is urged backwards; this escapement does not therefore possess the characteristics which we pointed out (Art. 672) as necessary for a really good instrument. But for ordinary purposes of time-keeping, the recoil escapement works sufficiently well, as the force which acts upon the pendulum is in reality extremely small. For the refined applications of the astronomical clock, the performance of a recoil escapement is inadequate.

The obvious defect in the recoil is that the pendulum is retarded during a portion of its vibration; the impulse forward is of course necessary, but the retarding force is useless and injurious.

676. The “dead-beat” escapement was devised by the celebrated clockmaker Graham, in order to avoid this difficulty. If you observe the second-hand of a clock, controlled by this escapement, you will understand why it is called the dead beat: there is no recoil; the second-hand moves quickly over each second, and remains there fixed until it starts for the next second.

The wheel and escapement by which this effect is produced is shown in Fig. 100. A and B are the pallets, by the action of the teeth on which the motion is given to the crutch, which turns about the centre O; from the axis through this centre the fork descends, so that as the crutch is made to vibrate to and fro by the wheel, the fork is also made to vibrate, and thus sustain the motion of the pendulum. The essential feature in which the dead-beat escapement differs from the recoil escapement is that when the tooth escapes from the pallet A, the wheel turns: but the tooth which in the recoil escapement would have fallen on the other pallet, now falls on a surface D, and not on the pallet B. D is part of a circle with its centre at O, the centre of motion; consequently, the tooth remains almost entirely inert so long as it remains on the circular arc D.