60

The Acceleration of Gravity illustrated.

has fallen through a great height than if it has dropped merely a short distance.

A person may leap from a chair with impunity; if from a table, he receives a harder shock; if from a high window, a topmast of a ship, or the parapet of a high bridge, he most probably fractures bones; and if he fall from a balloon at a great height, his body will be literally dashed to pieces.

Meteoric stones, coming from great heights, bury themselves deep in the earth, by the force gradually acquired from gravitation. When the wood-cutters among the Alps launch an enormous tree from high up on the mountain side, along the smooth wooden trough prepared as a channel, it is seen plunging, in fewer minutes than it traverses miles, with terrific velocity and force into the lake below; this final effect has been produced by the continued action of gravity through the whole time of its descent.

The shock or blow of the ram of a pile-engine is not the effect of a momentary impulse given by the earth, but of an attraction continued through a space of perhaps twenty feet.

A common hammer in its instantaneous shock has the condensed effect of the arm and of gravity, as accumulated through its whole previous course.

There are some long-necked birds that fight and kill their prey by blows with their hard beaks. They draw back the head, bending the neck like a swan or serpent, and then dart it forward with a continued effort, till the strong, wedge-like beak reaches its destination almost with the velocity of a pistol bullet.

Water falling from a height acquires a power according to the extent of fall; its power to turn a mill depending on the "head," or height of the source of water pressure. A small stream falling a considerable height will, in course of time, hollow out a solid rock. We have a striking example of the acceleration of gravity in the great violence possessed by a mountain stream at the base of a mountain, compared with that high up on its side.

Soft snow in falling from the precipitous sides of the Alps gathers tremendous force as an avalanche, and with a sound like thunder rends and carries before it trees, rocks, and all other obstacles in its course.

But for the resistance of the air breaking the force of a waterfall, it would gradually penetrate deeply into the ground.

Any liquid falling from a reservoir forms a descending mass or stream, of which the bulk diminishes from above downwards in the

Acceleration of Gravity Uniform.

61

same proportion as the velocity increases, as is well exemplified in the pouring out of molasses or thick syrup. If the height of the fall be considerable, the bulky, sluggish mass which first escapes is gradually reduced, before it reaches the bottom, to a small thread, but the substance of that thread is moving proportionately faster, and fills the receiving vessel with surprising rapidity.

The same truth is exhibited on a vast scale in the Falls of Niagara ; when the broad river is seen first bending over the precipice a deep, gently moving mass, then becoming a thinner and a thinner sheet as it descends, until at last, surrounded by its foam or mist, it flashes like lightning into the deep below.

"Gravity is a uniformly accelerating force."

134. A thousand such instances might be given to show that gravity is an accelerating force, but they do not tell us what is the rate of acceleration or the rate of increase of the speed and momentum, or they do so only very roughly.

By the special contrivance known as Attwood's machine (see page 62), or more roughly by dropping a heavy body from successive heights and noting the times of fall, we find that the acceleration takes place at a uniform rate. That is to say, a falling body receives equal additions of velocity and of momentum during each successive unit of time.

If we let a stone drop vertically, as from the parapet of a bridge, we find it falls through about 16 feet in one second, through 64 feet in two seconds, through 144 feet in three seconds, and so on. But, after the body has fallen for a second, or through a space of 16 feet, it is found to have a velocity of 32 feet per second, and in two seconds it is found to have one of 64 feet, in three seconds of 96 feet, and so on. Thus the velocity at the end of two, three, four, &c., seconds is double, triple, quadruple, &c., what it is at the end of one second; that is, it increases by the same amount, viz. 32 feet per second, during each successive second of its fall.

Consequently the momentum increases in the same proportion, or the force of gravity is uniformly accelerating.

We measure the force of gravity, then, as we do any other force, by the quantity of motion it imparts to any mass (chosen as our unit or standard), such as a pound or an ounce, in a second (unit of time).

Usually it is expressed not exactly in this form, but in one easily convertible into it. The force of gravity is commonly quoted

62

Attwood's Machine.

as a velocity of 32 feet per second, for this reason, that any mass falling freely under the action of gravity acquires this velocity in one second from starting. Let us see why this is so.

"All unimpeded bodies fall equally fast."

135. It does not quite accord with popular notions that all bodies, light as well as heavy, should acquire the same velocity by the same duration of fall. For we see a feather, dropped along with a sixpence, lag behind in its descent, and we imagine its gravitating power less energetic. Weight for weight, however, this force is precisely the same for both; only the interference of the air has most effect on the feather, which, for the same velocity, has a much less quantity of motion than the sixpence. As we remove the obstruction of the air the discrepancy vanishes; thus, a piece of gold leaf falls more slowly to the ground than a half-sovereign, but, if the gold leaf be rolled into a small ball, it has now a smaller mass of air to oppose it, and it drops as quickly as the coin.

So, too, the feather and sixpence, when dropped within a long tube or other glass vessel from which the air has been removed, are seen to fall side by side.

The reason of this is obvious. A regiment of soldiers marches no more quickly along the road than a single man, where each merely carries his own burden and has his own power within him. But in the push against difficulties the weak goes to the wall. So a million of particles, bound side by side in one common mass, gravitate to the earth by virtue of the power innate in each individual particle, and it is the interference of the air which bears harder on a few particles than a large company. Could we live on a globe devoid of atmosphere, then the lightest dust would fall at once to the ground as quickly as we see a stone fall.

"Attwood's Machine."

136. We should infer, then, that, if it were possible to increase the mass of a body without at the same time, and in the same degree, increasing the attracting force of the earth for it, there would no longer be the same rate of fall. It is so.

This is shown by a beautiful, though simple, contrivance, invented in the end of last century, by Mr. Attwood, of Cambridge. Essentially it consists of a pulley, P, or wheel with a grooved edge, of about six inches diameter, over which are balanced two equal weights, a, b, attached to the ends of a long silk string. The

Attwood's Machine.

63

pulley has its axle, d, laid on what are called friction wheels, e, f, to

give great delicacy of motion. We may thus look on a, b, as two masses freed from the action of the earth's gravity, since each just counteracts the other's weight.

Suppose that a and b are masses of one pound each; then, if we hook a mass of an ounce, c to a, the action of the earth will cause

C.

P

&

to move down; but c has to move not only its own mass, but also the masses a and b, which gives altogether thirty-three times the mass of Hence the motion produced will be correspondingly slow, and the velocity attained at the end of a second will be only rd part of the usual free velocity (32 feet), that is, only about a foot per second. In this way the rate at which a mass falls to the earth can be made as slow as we please, so that we can study most accurately the above laws as to the quantity of motion. This gives to Attwood's machine a great importance, as it enables us to verify experimentally several of the abstract laws of motion.

Fig. 9.

137. Attwood's machine is in principle just a very delicate kind of weighing-beam. If we have in the two scale-pans of a nice balance two weights exactly equal—say, a metal pound and a pound of sugar-the addition of a little sugar to the one side will make that scale move slowly down, and all the more slowly according as we increase the counterpoising weights in the scales; the same addition will have less effect with two pounds weight in each scale, still less if we have five pounds.

138. By means of this contrivance, then, as well as by others of more recent invention, we can calculate to any degree of nicety the heights fallen through in one, two, three, &c., seconds by a heavy body, and so we can trace the changes upon its velocity. It is deduced from the average of a great number of trials, that a body drops through 16'1 feet in the first second of its fall, and has acquired in this time a velocity just double, i.e., of 32'2 feet in a second; that is to say, it would continue its course at the rate of 32*2 feet per second afterwards, if the earth's attraction were instantly to cease at the end of one second. The reason of this is manifest; for at the end of the first half-second, the velocity would only be 161 feet, and at any instant before the middle of the second, its velocity would be just as much less than 16'1 feet as it is greater at

64

The Laws of Falling Bodies.

the corresponding instant after; so that the velocity on the average is 161 feet per second. Or it may appear more plain in this form :— Suppose we take ten seconds in place of one; the velocity at the end of ten seconds is 322 feet per second, and the space fallen through is known (by a simple calculation to be explained presently) to be just 1610 feet, that is a space which it would have passed through in the same time with a uniform velocity of 161 feet, or just half of its final velocity. Now, at the end of five seconds its velocity would be just 161 feet per second; and, a second before that, it would be as much less than 161 feet as it is greater than that a second after. Thus the velocity acquired, after the middle of the time, above the average, exactly compensates for the previous lack of velocity below the average. It is therefore a necessary consequence, from the uniformity of the acceleration, that the final velocity be double of the average one for any time.

139. In the next second the body falls through 32'2 feet in virtue of the velocity already acquired, and also through an additional 161 by the continued action of gravity; or, in all, three times as far as in the first second. So that, in two seconds, it falls altogether four times as far as in one second.

At the end of two seconds the velocity acquired is twice as great as at the end of one, or is at the rate of 64'4 feet per second.

Thus, during the third second the body falls through 64°4 feet and other 161, in all 80'5 feet, or five times as far as in the first second. In three seconds, therefore, it has descended nine times as far as in one second. And so on.

Thus the spaces fallen through in successive intervals of one second each, are in the proportion of the odd integers-1, 3, 5, 7, 9, &c.

And the whole spaces passed through at the end of 1, 2, 3, &c., seconds, counting from the commencement of the fall, are proportional to the squares of these numbers—that is, are as 1, 4, 9, 16, 25, &c.

These facts may be presented in the following tabular form :—

Whole space fallen through to end of 16 feet 64 feet 144 feet 256 feet.