A pendulous body vibrates when it is suspended so that the centre of its mass is not placed directly under the point of suspension, because then the alternating influences of weight and velocity are constantly impressing it with motion. Weight carries it down as far as it can go towards the earth's attraction; acquired velocity then carries it onwards; but as the onward movement is constrained to be upward against the direction of the earth's attraction, that force antagonises, and at last arrests it, for velocity flags when it has to drag its load up-hill, and soon gives over the effort. The body swings down-hill with increasing rapidity, because weight and velocity are then both driving it; it swings up-hill with diminishing rapidity, because then weight is pulling it back in opposition to the force of velocity. Weight pulls first this way, then that way; velocity carries first this way, then that way: but the two powers do not act evenly and steadily together; they now combine with, and now oppose each other; now increase their influence together, and now augment and diminish it inversely and alternately; and so the suspended body is tossed backwards and forwards between them, and made to perform its endless dance.
It is related of Galileo, that he once stood watching a swinging lamp, hung from the roof of the cathedral at Pisa, until he convinced himself that it performed its vibratory movement in the same time, whether the vibration was one of wide or of narrow span. This traditionary tale is most probably correct in its main features, for the Newtons and Galileos of all ages do perceive great truths in occurrences that are as commonplace as the fall of an apple, or the disturbance of a hanging lamp. Trifles are full of meaning to them, because their minds are already prepared to arrive at certain conclusions by means of antecedent reflections. Simple and familiar incidents, thus accidentally associated with the history of grand discoveries, are the channels through which the accumulating waters at length descend, rather than the rills which feed the swelling of their floods. The orchard at Woolsthorpe, and the cathedral at Pisa, were outlets of this kind, through which the pent-up tide of gathering knowledge burst. If they had never offered themselves, the laws of universal gravitation and isochronous vibration would still have reached the world.
If the reader will hang up two equal weights upon nearly the same point of suspension, and by means of two strings of exactly the same length, he will have an apparatus at his command that will enable him to see, under even more favourable conditions, what Galileo saw in the cathedral at Pisa. Upon drawing one of them aside one foot from the position of rest, and the other one yard, and then starting them off both together to vibrate backwards and forwards, he will observe, that although the second has a journey of two yards to accomplish, while the first has but a journey of two feet, the two will, nevertheless, come to the end at precisely the same instant. As the weights swing from side to side in successive oscillations, they will always present themselves together at the point which is the middle of their respective arcs. This is what is called isochronous vibration—the passing through unequal arcs in equal periods of time.
At the first glance, this seems a very singular result. The careless observer naturally expects that a weight hung upon a string ought to take longer to move through a long arc than through a short one, if impelled by the same force; but the subject appears in a different light upon more mature reflection, for it is then seen, that the weight which performs the longer journey starts down the steeper declivity, and therefore acquires a greater velocity. A ball does not run down a steep hill and a more gently inclined one at the same pace; neither, therefore, will the suspended weight move down the steeper curve, and the less raised one, at equal rates. The weight which moves the fastest, of necessity gets through more space in a given period than its more leisurely companion does. The equality of the periods in which two weights vibrate, is perfect so long as both the unequal arcs of motion are short ones, when compared with the length of the suspending strings; but even when one of the arcs is five times longer than the other, ten thousand vibrations will be completed before one weight is an entire stride in advance of the other; and even this small amount of difference is destroyed when the arc in which the weights swing is a little flattened from the circular curve.
But there is yet another surprise to be encountered. Hang a weight of a pound upon one of the strings, and a weight of two pounds upon the other, and set them vibrating in arcs of unequal length as before, and still their motions will be found to be isochronous. Unequal weights, as well as equal ones, when hung on equal strings, will swing through arcs of unequal length in equal periods of time. This seeming inconsistency also admits of a satisfactory explanation. It has been stated, that the motion of swinging bodies is caused by the earth's attraction. But what are the facts that are more particularly implied in this statement? What discoveries does the philosophic inquirer make when he looks more narrowly into it? For the sake of familiar illustration, let it be imagined that a man stands at the top of the Monument of London, with two leaden bullets in his hand, each weighing an ounce, and that he drops these together. They go to the earth, because the earth's mass draws them thither; and since the two bodies exactly resemble each other, and start at the same instant upon their descent, they must of course both strike the pavement beneath simultaneously. There can be no reason why one should get down before the other, for the same influence causes the fall of each. The entire mass of the huge earth attracts each bullet alike, and the bullets, therefore, yield like obedience to the influence, and fall together to the ground.
But now, suppose that the two bullets were to be all at once fused into one, and that this combined mass were then dropped from the top of the Monument as a single bullet, would there then be any reason why the two ounces of lead should make a more rapid descent than they would have made while in separate halves? Clearly not. There is but the same earth to attract, and the same number of particles to be drawn in each case, and therefore the same result must ensue. Each particle still renders its own individual obedience, and makes its own independent fall, although joined cohesively to its neighbours. It is the mass of the attracting body, and not the mass of the attracted body, that determines the velocity with which the latter moves. The greater mass of an attracted body expends its superior power, not in increasing its own rate of motion, but in pulling more energetically against the attracting mass. Every particle of matter when at rest resists any attempt to impress it with motion. The amount of this resistance is called its inertia. When many particles are united together into one body, they not only, therefore, take to that body many points upon which the earth's attraction can tell, but they also carry to it a like quantity of resistance or inertia, which must be overcome before any given extent of motion can be produced. If the earth's force be but just able to make particle 1 of any body go through 200 inches in a second, it will also be but just able to make particles 2, 3, and 4 do the same; consequently, whether those particles be separate or combined together, their rate of travelling will be the same. Hence all bodies descend to the earth with exactly the same velocities, however different their natures may be in the matter of weight, always provided there be no retarding influence to act unequally upon their different bulks and surfaces. It is well known that even a guinea and feather will fall together when the atmospheric resistance is removed from their path.
The reader will now, of course, see that what is true of the motion of free bodies, must also be true of the motion of suspended ones, since the same terrestrial attraction causes both. There is no reason why the two-pound weight in the experiment should vibrate quicker than the one-pound weight, just as there is no reason why a two-ounce bullet should fall quicker than a one-ounce bullet. Here, also, there are only the same number of terrestrial particles to act upon each separate particle of the two unequal weights. Hence it is that the vibrations of unequal weights are isochronous when hung on strings of equal lengths.
Thus far our dealings have been with what has seemed to be a very single-purposed and determined agent. We have hung a weight upon a piece of string and set it swinging, and have then seen it persisting in making the same number of beats in the same period of time, whether we have given it a long journey or a short one to perform; and also whether we have added to or taken from its mass. But now we enter upon altogether new relations with our little neophyte, and find that we have reached the limits of its patience.
Take three pieces of string of unequal lengths—one being one foot long; the second, four feet; and the third, nine feet. Hang them up by one extremity, and attach to each of the other ends a weight. Then start the three weights all off together vibrating, and observe what happens. The several bodies do not now all vibrate in the same times as in the previous experiments. By making the lengths of the strings unequal, we have introduced elements of discord into the company. The weight on the shortest string makes three journeys, and the weight on the next longest string makes two journeys, while the other is loitering through one.
This discrepancy, again, is only what the behaviour of the vibrating masses in the previous experiments should have taught the observer to anticipate. Each of the weights in this new arrangement of the strings, has to swing in the portion of a circle, which, if completed, would have a different dimension from the circles in which the other weights swing. The one on the shortest string swings in the segment of a circle that would be two feet across; the one on the longest string swings in the segment of a circle that would be eighteen feet across. Now, if these two weights be made to vibrate in arcs that shall measure exactly the twelfth part of the entire circumference of their respective circles, then one will go backwards and forwards in a curved line only half a foot long, while the other will move in a line four feet and a half long.