Part 19

A B is a vertical spindle which is turned by the engine. P P is a piece firmly attached to the spindle and turning with it. P W, P W are arms terminating in weights W W; these are balls of iron, generally very massive: the arms are free to turn round pins at P P. At Q Q links are placed, attached to another piece R R, which is free to slide up and down the spindle. When A B rotates, W and W are carried round, and therefore fly outwards from the spindle; to do this they must evidently pull the piece R R up the shaft. We can easily imagine an arrangement by which R R shall be made to shut or open the steam-valve according as it ascends or descends. The problem is then solved, for if the engine begin to go too rapidly, the balls fly out further just as they did in Fig. 74: this movement raises the piece R R, which diminishes the supply of steam, and consequently checks the speed. On the other hand, when the engine works too slowly, the balls fall in towards the spindle, the piece R R descends, the valve is opened, and a greater supply of steam is admitted. The objection to this governor is that though it moderates, it does not completely check irregularity. There are other governors occasionally employed which depend also on circular motion; some of these are more sensitive than the governor-balls; but they are elaborate machines, only to be employed under exceptional circumstances.

578. The application of circular motion to sugar refining is a very beautiful invention. To explain it I must briefly describe the process of refining.

The raw sugar is dissolved in water, and the solution is purified by straining and by filtration through animal charcoal. The syrup is then boiled. In order to preserve the colour of the sugar, and to prevent loss, this boiling is conducted _in vacuo_, as by this means the temperature required is much less than would be necessary with the ordinary atmospheric pressure.

The evaporation having been completed, crystals of sugar form throughout the mass of syrup. To separate these crystals from the liquor which surrounds them, the aid of circular motion force is called in. A mass of the mixture is placed in a large iron tub, the sides of which are perforated with small holes. The tub is then made to rotate with prodigious velocity; its contents instantly fly off to the circumference, the liquid portions find an exit through the perforations in the sides, but the crystals are left behind. A little clear syrup is then sprinkled over the sugar while still rotating: this washes from the crystals the last traces of the coloured liquid, and passes out through the holes; when the motion ceases, the inside of the tub contains a layer of perfectly pure white sugar, some inches in thickness, ready for the market.

579. Circular motion is peculiarly fitted for this purpose; each particle of liquid strives to get as far away from the axis as possible. The action on the sugar is very different from what it would have been had the mass been subjected to pressure by a screw-press or similar contrivance; the particles immediately acted on would then have to transmit the pressure to those within; and the consequence would be that while the crystals of sugar on the outside would be crushed and destroyed, the water would only be very imperfectly driven from the interior: for it could lurk in the interstices of the sugar, which remain notwithstanding the pressure.

580. But with circular motion the water must go, not because it is pushed by the crystals, but because of its own inertia; and it can be perfectly expelled by a velocity of rotation less than that which would be necessary to produce sufficient pressure to make the crystals injure each other.

THE PERMANENT AXES.

581. There are some curious properties of circular motion which remain to be considered. These we shall investigate by means of the apparatus of Fig. 80. This consists of a pair of wheels B C, by which a considerable velocity can be given to a horizontal shaft. This shaft is connected by a pair of bevelled wheels D with a vertical spindle F. The machine is worked by a handle A, and the object to be experimented upon is suspended from the spindle.

582. I first take a disk of wood 18" in diameter; a hole is bored in the margin of this disk; through this hole a rope is fastened, by means of which the disk is suspended from the spindle. The disk hangs of course in a vertical plane.

583. I now begin to turn the handle round gently, and you see the disk begins to rotate about the vertical diameter; but, as the speed increases, the motion becomes a little unsteady; and finally, when I turn the handle very rapidly, the disk springs up into a horizontal plane, and you see it like the surface of a small table: the rope swings round and round in a cone, so rapidly that it is hardly seen.

584. We may repeat the experiment in a different manner. I take a piece of iron chain about 2' long, G; I pass the rope through the two last links of its extremities, and suspend the rope from the spindle. When I commence to turn the handle, you see the chain gradually opens out into a loop H; and as the speed increases, the loop becomes a complete ring. Still increasing the speed, I find the ring becomes unsteady, till finally it rises into a horizontal plane. The ring of chain in the horizontal plane is shown at I. When the motion is further increased, the ring swings about violently, and so I cease turning the handle.

585. The principles already enunciated will explain these remarkable results; we shall only describe that of the chain, as the same explanation will include that of the disk of wood. We shall begin with the chain hanging vertically from the spindle: the moment rotation commences, the chain begins to spin about a vertical axis; the parts of the chain fly outwards from this axis just as the ball flies outwards in Fig. 74; this is the cause of the looped form H which the chain assumes. As the speed is increased the loop gradually opens more and more, just as the diameter of the circle Fig. 74 increases with the velocity. But we have also to inquire into the cause of the remarkable change of position which the ring undergoes; instead of continuing to rotate about a vertical diameter, it comes into a horizontal plane. This will be easily understood with the help of Fig. 81. Let O P represent the rope attached to the ring, and O C be the vertical axis. Suppose the ring to be spinning about the axis O C, when O C was a diameter; if then, from any cause, the ring be slightly displaced, we can show that the circular motion will tend to drive the ring further from the vertical plane, and force it into the horizontal plane. Let the ring be in the position represented in the figure; then, since it revolves about the vertical line O C, the tendency of P P and Q Q is to move outwards in the directions of the arrows, thus evidently tending to bring the ring into the horizontal plane.

586. In Art. 103 we have explained what is meant by stable and unstable equilibrium; we have here found a precisely analogous phenomenon in motion. The rotation of the ring about its diameter is unstable, for the minutest deviation of the ring from this position is fatal; circumstances immediately combine to augment the deviation more and more, until finally the ring is raised into the horizontal plane. Once in the horizontal plane, the motion there is stable, for if the ring be displaced the tendency is to restore it to the horizontal.

587. The ring, when in a horizontal plane, rotates permanently about the vertical axis through its centre; this axis is called permanent, to distinguish it from all other directions, as being the only axis about which the motion is stable.

588. We may show another experiment with the chain: if instead of passing the rope through the links at its ends, I pass the rope through the centre of the chain, and allow the ends of the chain to hang downwards. I now turn the handle; instantly the parts of the chain fly outwards in a curved form; and by increasing the velocity, the parts of the chain at length come to lie almost in a straight line.

LECTURE XVIII. _THE SIMPLE PENDULUM._

Introduction.—The Circular Pendulum.—Law connecting the Time of Vibration with the Length.—The Force of Gravity determined by the Pendulum.—The Cycloid.

INTRODUCTION.

589. If a weight be attached to a piece of string, the other end of which hangs from a fixed point, we have what is called a simple pendulum. The pendulum is of the utmost importance in science, as well as for its practical applications as a time-keeper. In this lecture and the next we shall treat of its general properties; and the last will be devoted to the practical applications. We shall commence with the simple pendulum, as already defined, and prove, by experiment, the remarkable property which was discovered by Galileo. The simple pendulum is often called the circular pendulum.

THE CIRCULAR PENDULUM.

590. We first experiment with a pendulum on a large scale. Our lecture theatre is 32 feet high, and there is a wire suspended from the ceiling 27' long; to the end of this a ball of cast iron weighing 25 lbs. is attached. This wire when at rest hangs vertically in the direction O C (Fig. 82).

I draw the ball from its position of rest to A; when released, it slowly returns to C, its original position; it then moves on the other side to B, and back again to my hand at A. The ball—or to speak more precisely, the centre of the ball—moves in a circle, the centre being the point O in the ceiling from which the wire is suspended.

591. What causes the motion of the pendulum when the weight is released? It is the force of gravity; for by moving the ball to A I raise it a little, and therefore, when I release it gravity compels it to return to C it being the only manner in which the mode of suspension will allow it to fall. But when it has reached its original position at C, why does it continue its motion?—for gravity must be acting against the ball during the journey from C to B. The first law of motion explains this. (Art. 485). In travelling from A to C the ball has acquired a certain velocity, hence it has a tendency to go on, and only by the time it has arrived at B will gravity have arrested the velocity, and begin to make it descend.

592. You see, the ball continues moving to and fro—oscillating, as it is called—for a long time. The fact is that it would oscillate for ever, were it not for the resistance of the air, and for some loss of energy at the point of suspension.

593. By the time of an oscillation is meant the time of going from A to B, but not back again. The time of our long pendulum is nearly three seconds.

594. With reference to the time of oscillation Galileo made a great discovery. He found that whether the pendulum were swinging through the arc A B, or whether it had been brought to the point A´, and was thus describing the arc A´ B´, the time of oscillation remained nearly the same. The arc through which the pendulum oscillates is called its amplitude, so that we may enunciate this truth by saying that _the time of oscillation is nearly independent of the amplitude_. The means by which Galileo proved this would hardly be adopted in modern days. He allowed a pendulum to perform a certain number of vibrations, say 100, through the arc A B, and he counted his pulse during the time; he then counted the number of pulsations while the pendulum vibrated 100 times in the arc A´ B´, and he found the number of pulsations in the two cases to be equal. Assuming, what is probably true, that Galileo’s pulse remained uniform throughout the experiment, this result showed that the pendulum took the same time to perform 100 vibrations, whether it swung through the arc A B, or through the arc A´ B´. This discovery it was which first suggested the employment of the pendulum as a means of keeping time.

595. We shall adopt a different method to show that the time does not depend upon the amplitude. I have here an arrangement which is represented in Fig. 83. It consists of two pendulums A D and B C, each 12' long, and suspended from two points A B, about 1' apart, in the same horizontal line. Each of these pendulums carries a weight of the same size: they are in fact identical.

596. I take one of the balls in each hand. If I withdraw each of them from its position of rest through equal distances and then release them, both balls return to my hands at the same instant. This might have been expected from the identity of the circumstances.

597. I next withdraw the weight C in my right hand to a distance of 1', and the weight D in my left hand to a distance of 2', and release them simultaneously. What happens? I keep my hands steadily in the same position, and I find that the two weights return to them at the same instant. Hence, though one of the weights moved through an amplitude of 2' (C E) while the other moved through an amplitude of 4' (D F), the times occupied by each in making two oscillations are identical. If I draw the right-hand ball away 3', while I draw the left hand only 1' from their respective positions of rest, I still observe the same result.

598. In two oscillations we can see no effect on the time produced by the amplitude, and we are correct in saying that, when the amplitude is only a small fraction of the length of the pendulum, its effect is inappreciable. But if the amplitude of one pendulum were very large, we should find that its time of oscillation is slightly greater than that of the other, though to detect the difference would require a delicate test. One consequence of what is here remarked will be noticed at a later page. (Art. 655.)

599. We next inquire whether the weight which is attached to the pendulum has any influence upon the time of vibration. Using the 12' pendulums of Fig. 83, I place a weight of 12 lbs. on one hook and one of 6 lbs. on the other. I withdraw one in each hand; I release them; they return to my hand at the same moment. Whether I withdraw the weights through long arcs or short arcs, equal or unequal, they invariably return together, and both therefore have the same time of vibration. With other iron weights the same law is confirmed, and hence we learn that, besides being independent of the amplitude, the time of vibration is also independent of the weight.

600. Finally, let us see if the _material_ of the pendulum can influence its time of vibration. I place a ball of wood on one wire and a ball of iron on the other; I swing them as before: the vibrations are still performed in equal times. A ball of lead is found to swing in the same time as a ball of brass, and both in the same time as a ball of iron or of wood.

601. In this we may be reminded of the experiments on gravity (Art. 491), where we showed that all bodies fall to the ground in equal times, whatever be their sizes or their materials. From both cases the inference is drawn that the force of gravity upon different bodies is proportional to their masses, though the bodies be made of various substances. It was indeed by means of experiments with the pendulum that Newton proved that gravity had this property, which is one of the most remarkable truths in nature.

LAW CONNECTING THE TIME OF VIBRATION WITH THE LENGTH.

602. We have seen that the time of vibration of a pendulum depends neither upon its amplitude, material, nor weight; we have now to learn on what the time _does_ depend. It depends upon the _length_ of the pendulum. The shorter a pendulum the less is its time of vibration. We shall find by experiment the relation between the time and the length of the cord by which the weight is suspended.

603. I have here (Fig. 84) two pendulums A D, B C, one of which is 12' long and the other 3'; they are mounted side by side, and the weights are at the same distance from the floor. I take one of the weights in each hand, and withdraw them to the same distance from the position of rest. I release the balls simultaneously; C moves off rapidly, arrives at the end C´ while D has only reached D´, and returns to my hand just as D has completed one oscillation. I do not seize C: it goes off again, only to return at the same moment when D reaches my hand. Thus C has performed four oscillations while D has made no more than two. This proves that when one of two pendulums is a quarter the length of the other, the time of vibration of the shorter one is half that of the other.

604. We shall repeat the experiment with another pendulum 27' long, suspended from the ceiling, and compare its vibrations with those of a pendulum 3' long. I draw the weights to one side and release them as before; and you see that the small pendulum returns twice to my hand while the long pendulum is still absent; but that, keeping my hands steadily in the same place throughout the experiment, the long pendulum at last returns simultaneously with the third arrival of the short one. Hence we learn that a pendulum 27' long takes three times as much time for a single vibration as a 3' pendulum.

605. The lengths of the three pendulums on which we have experimented (27', 12', 3'), are in the proportions of the numbers 9, 4, 1; and the times of the oscillations are proportional to 3, 2, 1: hence we learn that _the period of oscillation of a pendulum is proportional to the square root of its length_.

606. But the time of vibration must also depend upon gravity; for it is only owing to gravity that the pendulum vibrates at all. It is evident that, if gravity were increased, all bodies would fall to the earth more than 16' in the first second. The effect on the pendulum would be to draw the ball more quickly from D to D´ (Fig. 84), and thus the time of vibration would be diminished.

It is found by calculation, and the result is confirmed by experiment, that the time of vibration is represented by the expression,

_____________________________ 3·1416 √ ( Length / Force of gravity).

607. The accurate value of the force of gravity in London is 32·1908, so ________ that the time of vibration of a pendulum there is 0·5537 √ length: the length of the seconds pendulum is 3'·2616.

THE FORCE OF GRAVITY DETERMINED BY THE PENDULUM.

608. The pendulum affords the proper means of measuring the force of gravity at any place on the earth. We have seen that the time of vibration can be expressed in terms of the length and the force of gravity; so conversely, when the length and the time of vibration are known, the force of gravity can be determined and the expression for it is—

Length × (3·1416 / Time)².

609. It is impossible to observe the time of a single vibration with the necessary degree of accuracy; but supposing we consider a large number of vibrations, say 100, and find the time taken to perform them, we shall then learn the time of one oscillation by dividing the entire period by 100. The amplitudes of the oscillations may diminish, but they are still performed in the same time; and hence, if we are sure that we have not made a mistake of more than one second in the whole time, there cannot be an error of more than 0·01 second, in the time of one oscillation. By taking a still larger number the time may be determined with the utmost precision, so that this part of the inquiry presents little difficulty.

610. But the length of the pendulum has also to be ascertained, and this is a rather baffling problem. The ideal pendulum whose length is required, is supposed to be composed of a very fine, perfectly flexible cord, at the end of which a particle without appreciable size is attached; but this is very different from the pendulum which we must employ. We are not sure of the exact position of the point of suspension, and, even if we had a perfect sphere for the weight, the distance between its centre and the point of suspension is not the same thing as the length of the simple pendulum that would vibrate in the same time. Owing to these circumstances, the measurement of the pendulum is embarrassed by considerable difficulties, which have however been overcome by ingenious contrivances to be described in the next chapter.

611. But we shall perform, in a very simple way, an experiment for determining the force of gravity. I have here a silken thread which is fastened by being clamped between two pieces of wood. A cast iron ball 2"·54 in diameter is suspended by this piece of silk. The distance from the point of suspension of the silk to the ball is 24"·07, as well as it can be measured.

The length of the ideal pendulum which would vibrate isochronously with this pendulum is 25"·37, being about 0"·03 greater than the distance from the point of suspension to the centre of the sphere.

612. The length having been ascertained, the next element to be determined is the time of vibration. For this purpose I use a stop-watch, which can be started or stopped instantaneously by touching a little stud: this watch will indicate time accurately to one-fifth of a second. It is necessary that the pendulum should swing in a small arc, as otherwise the oscillations are not strictly isochronous. Quite sufficient amplitude is obtained by allowing the ball to rotate to and fro through a few tenths of an inch.

613. In order to observe the movement easily, I have mounted a little telescope, through which I can view the top of the ball. In the eye-piece of the telescope a vertical wire is fastened, and I count each vibration just as the silken thread passes the vertical wire. Taking my seat with the stop-watch in my hand, I write down the position of the hands of the stop-watch, and then look through the telescope. I see the pendulum slowly moving to and fro, crossing the vertical wire at every vibration; on one occasion, just as it passes the wire, I touch the stud and start the watch. I allow the pendulum to make 300 vibrations, and as the silk arrives at the vertical wire for the 300th time, I promptly stop the watch; on reference I find that 241·6 seconds have elapsed since the time the watch was started. To avoid error, I repeat this experiment, with precisely the same result: 241·6 seconds are again required for the completion of 300 vibrations.

614. It is desirable to reckon the vibrations from the instant when the pendulum is at the middle of its stroke, rather than when it arrives at the end of the swing. In the former case the pendulum is moving with the greatest rapidity, and therefore the time of coincidence between the thread and the vertical wire can be observed with especial definiteness.

615. The time of a single vibration is found, by dividing 241·6 by 300, to be 0·805 second. This is certainly correct to within a thousandth part of a second. We conclude that a pendulum whose length is 25"·37 = 2·114, vibrates in 0·805 second; and from this we find that gravity at Dublin is 2'·114 × (3·1416 / 0.805)² = 32·196. This result agrees with one which has been determined by measurement made with every precaution.

Another method of measuring gravity by the pendulum will be described in the next lecture (Art. 637).

THE CYCLOID.

616. If the amplitude of the vibration of a circular pendulum bear a large proportion to the radius, the time of oscillation is slightly greater than if the amplitude be very small. The isochronism of the pendulum is only true for small arcs.