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Attwood's Machine.

47

slides move, which can be fixed by a screw in any position; one of these has a disc, A, and the other a ring, B (fig. 44). At the top of the column is a brass pulley whose axis, instead of resting in pivots, turns on four other wheels, r, r, r', r', called friction wheels, since they serve to diminish friction (fig. 39). Two exactly equal weights, K and K', are attached to the end of a fine silk thread, which passes round the pulley.

At the top of the column is a plate, , on which is placed the falling body (fig. 41). This plate is fixed to a horizontal axis which carries a small catch i, supported, when the plate is horizontal, by a lever, ab, movable in the middle. A spring placed behind the dial tends to keep this lever in the position represented in fig. 41, while an excentric, e, moved by the clockwork, tends to incline towards the right the upper arm of the lever ab. The parts are so arranged, that when the needle is at zero of the graduation, the lever ab is moved by the excentric; the plate n then lets fall the body which it sustained (fig. 42).

These details being premised we may add that the slackening which it produces in the fall of a body depends on the mechanical principle, that when a moving body meets another at rest it imparts to this latter a part of its velocity, which is greater the greater the mass of the second body compared with the first. For instance if a body with the mass 1, strikes against another at rest with the mass 19, the total mass being now 20, the common velocity after the impact is only a twentieth of the original velocity of the first.

First experiment. To demonstrate the second law, that the spaces traversed are proportional to the squares of the times, a weight K is placed upon the ledge » (fig. 41), and it is loaded with an overweight, which consists of a brass disc, m (fig. 47), open at the side, so as to let pass a rod fixed to the weight K. Then below the ledge n the slider A is placed at such a distance that Km requires a second to traverse the space nA, which is easily obtained after a few trials. If the mass m fell alone it would traverse in a second about 32 feet; but from the principle stated above it can only fall by imparting to the masses K and K' what it carries with it, and hence its fall is the more diminished the smaller the mass m, as compared with the sum of the masses K and K'.

The experiment being prepared as indicated in fig. 41, the pendulum is made to oscillate; the clockwork then begins to move, and as soon as the needle arrives at zero the plate ʼn drops (fig. 42), the weights K. and m fall too, and the space nA is traversed in a second

by a uniformly accelerated motion. The experiment is recommenced, the slider A being placed at four times its original distance, that is, that if the distance An were 8 inches (fig. 41) it is now 32 inches (fig. 43). But here when the plate n drops it is found that the weight Km requires exactly two seconds to traverse the space An. Increasing the space traversed to 72 inches, the time required for the purpose is found to be three seconds. That is, that when the times are twice or thrice as great, the spaces traversed are four, or nine times as great.

Second experiment.—To prove the law that the velocities are proportional to the times, the experiment is arranged as shown in figs. 44, 45, and 46, that is, the weights K and m being arranged as in the first experiment on the ledge n, at a distance of 8 inches below this the sliding ring B is placed, and at 16 inches below the disc A. When the ledge ʼn has dropped, the weights K and m still require a second to fall from n to B. But then the over-weight m being arrested by the ring B (fig. 45), the weight K only falls in virtue of its acquired velocity. The motion which was uniformly accelerated from o to B (19) is kept uniform from B to A; for the weight m was the cause of the acceleration, and this having ceased to act, the acceleration ceases. It is then found that the space oB, equal to 8, having been traversed in one second, the space BA, equal to 16, is also traversed in a second. That is, 16 represents the velocity of the uniform motion, which, starting from the point B, has succeeded to the uniformly accelerated motion.

The experiment is finally recommenced by placing the slidingring B at the distance 32 (fig. 46), and sliding-disc below B, also at the distance 32. The space oB being then four times as great as in fig. 44 the weights K and m require, in accordance with the second law, twice the time. But the mass m being again arrested by the slider B, it is found that the weight K falls alone and uniformly from B to A in one second. The number 32 from B to A represents then the velocity acquired, starting from the point B after two seconds of fall. In the first part of the experiment it was ascertained that the velocity acquired after one second was 16; hence, in double the time, the velocity acquired is double. It may be shown, in like manner, that, after three times the time, the velocity is trebled, and so on; thus proving the third law.

55. Pendulum.—This is the name given to any heavy mass suspended by a thread to a fixed point, or to any metallic rod movable about a horizontal axis. The ball, m, suspended by the thread cm, which is fixed at the top at c (fig. 48), is a pendulum.

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Pendulum.

49

So long as the thread is vertical, which is the case when the centre of gravity of the ball is exactly below the point of suspension, c, the pendulum remains at rest, for the action of gravity is destroyed by the resistance at this point. This is no longer the case when the pendulum is removed from its vertical position; when it is placed, for instance, in the direction cn (fig. 49). The ball being raised, gravity tends to make it fall; it returns from n to m, and reaches the latter point with exactly the velocity it would have had by falling vertically through the height, om. The ball, accordingly, does not stop at m, but, in virtue of its inertia, and of its acquired velocity, it continues to move in the direction mp; as the ball rises, however, gravity, which had acted from n to m as an accelerating force, now exerts a retarding action, for it acts in a direction contrary to that of the motion; the motion, accordingly, becomes slower, and the ball stops at a distance, mp, which would

be exactly equal to mn, were it not for the resistance of the air, and also the rigidity of the thread, cm, which, as it is, offers a certain resistance to being bent about the point c, in passing from the position cn to cp, and vice versâ.

This being premised, the moment the ball stops at , gravity acting so as to make it fall again, brings it from p to m, when,

E

owing to its inherent velocity, it rises virtually as far as n, and so on; a backward and forward motion is thus produced from n towards, and from p towards n, which may last several hours.

This motion is described as an oscillating motion. The path of the ball from n to p, or from p to n, is known as a semi-oscillation, a complete oscillation being the motion from n to p, and from p to n. In France the former is known as a single oscillation, and the backward and forward motion as a double oscillation.

We distinguish in

The amplitude of the oscillation is the distance between the extreme positions, cn and cp, and is measured by the arc, pn. 56. Simple and compound pendulum. physics between the simple and the compound pendulum. The former would be that formed by a single material point, suspended by a thread without weight. Such a pendulum has only a theoretical existence; and it has only been assumed in order to arrive at the laws of the oscillations of the pendulum which we shall presently describe.

CR

Fig. 51.

air during each oscillation.

A compound or physical pendulum may be defined to be any body which can oscillate about a point or an axis. The pendulum described above (fig. 48) is of this kind. The form may be greatly varied, but the most ordinary one is a glass or steel rod (fig. 52) fixed at the top to a thin flexible steel plate, or to a knife edge like that of the balance (fig. 34). At the bottom of the rod is a heavy lens-shaped mass of metal, usually of brass, and known as the bob. The lenticular is preferred to the spherical form, for it presents less resistance to the

57. Laws of the pendulum. Galileo.-Whatever be the form of the pendulum, its oscillations always fall under the following laws. The first of these, that one and the same pendulum makes

-58] Verification of the Laws of the Pendulum.

51

its oscillations in equal times, was discovered by Galileo, the celebrated physicist and astronomer, at the end of the sixteenth century. It is related that he was led to this discovery, while still young, by observing the regular motion of a lamp suspended to the vault of the cathedral at Pisa. This property of the pendulum has received the name of isochronism, from two Greek words which mean equal times, and the oscillations are said to be isochronous.

First law; or, law of isochronism.— The oscillations of one and the same pendulum are isochronous, that is, are effected in equal times. This law is only perfectly exact for oscillations of small amplitude, four or five degrees at most; for a greater amplitude the oscillation is longer.

Second law; or, law of lengths. With pendulums of different lengths the duration of the oscillations is proportional to the square root of the length of the pendulum; that is to say, that if the lengths of the pendulums are as 1, 4, 9, 16, the times of oscillations will be as 1, 2, 3, 4, these being the square roots of the former set of numbers.

Third law. If the length of the pendulum remains the same, but the substances are different, the duration of the oscillations is independent of the substance of which the pendulums are formed; that is, that whether of wood, or of ivory, or of metal, they all oscillate in the same length of time.

Fourth law. The duration of the oscillations of a given pendulum is inversely as the square root of the force of gravity in the place in which the observation is made.

58. Verification of the laws of the pendulum.-In order to verify the laws of the simple pendulum we are compelled to employ a compound one, the construction of which differs as little as possible from that of the former. For this purpose a small sphere of a very dense substance, such as lead or platinum, is suspended from a fixed point by means of a very fine thread. A pendulum thus formed oscillates almost like a simple pendulum, the length of which is equal to the distance of the centre of the sphere from the point of suspension.

In order to verify the isochronism of small oscillations, it is merely necessary to count the number of oscillations made in equal times, as the amplitudes of these oscillations diminish from pn to rq (fig. 50) say from three degrees to a fraction of a degree; this number is found to be constant.

That the time of vibration is proportional to the square root of