THE SIMPLE PENDULUM

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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.

Fig. 82.

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 [Pg 286]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 [Pg 287]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.

Fig. 83.

[Pg 288]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 [Pg 289]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.

Fig. 84.

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 [Pg 291]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.[Pg 292]

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—

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 [Pg 293]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 [Pg 294]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, [Pg 295]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.

617. But there is a curve in which a weight may be made to move where the time of vibration is precisely the same, whatever be the amplitude. This curve is called a cycloid. It is the path described by a nail in the circumference of a wheel, as the wheel rolls along the ground. Thus, if a circle be rolled underneath the line a b (Fig. 85), a point on its circumference describes the cycloid a d c p b. The lower part of this curve does not differ very much from a circle whose centre is a certain point o above the curve.

618. Suppose we had a piece of wire carefully shaped to the cycloidal curve a d c p b, and that a ring could slide along it without friction, it would be found that, whether the ring be allowed to drop from c, p or b, it would fall to d precisely in the same time, and would then run up [Pg 296]the wire to a distance from d on the other side equal to that from which it had originally started. In the oscillations on the cycloid, the amplitude is absolutely without effect upon the time.

619. As a frictionless wire is impossible, we cannot adopt this method, but we can nevertheless construct a cycloidal pendulum in another way, by utilizing a property of the curve, o a (Fig. 85) as a half cycloid; in fact, o a is just the same curve as b d, but placed in a different position, so also is o b. If a string of length o d be suspended from the point o, and have a weight attached to it, the weight will describe the cycloid, provided that the string wrap itself along the arcs o a and o b; thus when the weight has moved from d to p, the string is wrapped along the curve through the space o t, the part t p only being free. This arrangement will always force the point p to move in the cycloidal arc.

Fig. 85.

620. We are now in a condition to ascertain experimentally, whether the time of oscillation in the cycloid is independent of the amplitude. We use for this purpose the apparatus shown in Fig. 86. d c e is the arc of the cycloid; two strings are attached at o, and equal weights a, b are suspended from them; [Pg 297]c is the lowest point of the curve. The time a will take to fall through the arc a c is of course half the time of its oscillation. If, therefore, I can show that a and b both take the same time to fall down to c, I shall have proved that the vibrations are isochronous.

Fig. 86.

621. Holding, as shown in the figure, a in one hand and b in the other, I release them simultaneously, and you see the result,—they both meet at c: even if I bring a [Pg 298]up to e, and bring b down close to c, the result is the same. The motion of a is so rapid that it arrives at c just at the same instant as b. When I bring the two balls on the same side of c, and release them simultaneously, a overtakes b just at the moment when it is passing c. Hence, under all circumstances, the times of descent are equal.

622. It will be noticed that the string attached to the ball b, in the position shown in the figure, is almost as free as if it were merely suspended from o, for it is only when the ball is some distance from the lowest point that the side arcs produce any appreciable effect in curving the string. The ball swings from b to c nearly in a circle of which the centre is at o. Hence, in the circular pendulum, the vibrations when small are isochronous, for in that case the cycloid and the circle become indistinguishable.

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