A Treatise on Mechanics

(115.) Gravitation is the general name given to this attraction, by whatever masses of matter it may be manifested. As exhibited in the effects produced by the earth upon surrounding bodies, it is called “terrestrial gravity.”

As the attraction of the earth is directed towards its centre, it might be expected that two plumb-lines should appear not to be parallel, but so inclined to each other as to converge to a point under the surface of the earth. Thus, if A B and C D, fig. 23., be two plumb-lines, each will be directed to the centre O, where, if their directions were continued, they would meet. In like manner, if two bodies were allowed to fall from A and C, they would descend in the directions A B and C D, which converge to O. Observation, on the contrary, shows, that plumb-lines suspended in places not far distant from each other are truly parallel; and that bodies allowed to fall descend in parallel lines. This apparent parallelism of the direction of terrestrial gravity is accounted for by the enormous proportion which the magnitude of the earth bears to the distance between the two plumb-lines or the two falling bodies which are compared. If the distance between the places B, D, were 1200 feet, the inclination of the lines A B and C D would not amount to a quarter of a minute, or the 240th part of a degree. But the distance, in cases where the parallelism is assumed, is never greater than, and seldom so great as, a few yards; and hence the inclination of the directions A B and C D is too small to be appreciated by any practical measure. In the investigation of the phenomena of falling bodies, we shall, therefore, assume, that all the particles of the same body are attracted in parallel directions, perpendicular to an horizontal plane.

(116.) Since the intensity of terrestrial gravity increases as the square of the distance decreases, it might be expected that, as a falling body approaches the earth, the force which accelerates it should be continually increasing, and, strictly speaking, it is so. But any height through which we observe falling bodies to descend bears so very small a proportion to the whole distance from the centre, that the change of intensity of the force of gravity is quite beyond any practical means of estimating it. The radius, or the distance from the surface of the earth to its centre, is 4000 miles. Now, suppose a body descended through the height of half a mile, a distance very much beyond those used in experimental enquiries, the distances from the centre, at the beginning and end of the fall, are then in the proportion of 8000 to 8001, and therefore the proportion of the force of attraction at the commencement to the force at the end, being that of the squares of these numbers, is 64,000,000 to 64,016,001, which, in the whole descent, is an increase of about one part in 4000; a quantity practically insignificant. We shall, therefore, in explaining the laws of falling bodies, assume that, in the entire descent, the body is urged by a force of uniform intensity.

Although the force which attracts all parts of the same body during its descent in a given place is the same, yet the force of gravity, at different parts of the earth’s surface, has different intensities. The intensity diminishes with the latitude, so that it is greater towards the poles, and lesser towards the equator. The causes of this variation, its law, and the experimental proofs of it, will be explained, when we shall treat of centrifugal force, and the motion of pendulums. It is sufficient merely to advert to it in this place.

(117.) Since the earth’s attraction acts separately and equally on every particle of matter, without regard to the nature or species of the body, it follows that all bodies, of whatever kind, or whatever be their masses, must be moved with the same velocity. If two equal particles of matter be placed at a certain distance above the surface of the earth, they will fall in parallel lines, and with exactly the same speed, because the earth attracts them equally. In the same manner, a thousand particles would fall with equal velocities. Now, these circumstances will in no wise be changed if those 1000 particles, instead of existing separately, be aggregated into two solid masses, one consisting of 990 particles, and the other of 10. We shall thus have a heavy body and a light one, and, according to our reasoning, they must fall to the earth with the same speed.

Common experience, however, is not always consistent with this doctrine. What are called light substances, as feathers, gold-leaf, paper, &c., are observed to fall slowly and irregularly, while heavier masses, as solid pieces of metal, stones, &c., fall rapidly. Nay, there are not a few instances in which the earth, instead of attracting bodies, seems to repel them, as in the case of smoke, vapours, balloons, and other substances which actually ascend. We are to consider that the mass of the earth is not the only agent engaged in these phenomena. The earth is surrounded by an atmosphere composed of an elastic or aeriform fluid. This atmosphere has certain properties, which will be explained in our treatise on Pneumatics, and which are the causes of the anomalous circumstances alluded to. Light bodies rise in the atmosphere, for the same reason that a piece of cork rises from the bottom of a vessel of water; and other light bodies fall more slowly than heavy ones, for the same reason that an egg in water falls to the bottom more slowly than a leaden bullet. This treatise is not the place to give a direct explanation of these phenomena. It will be sufficient for our present purpose to show, that if there were no atmosphere, all bodies, heavy and light, would fall at the same rate. This may easily be accomplished by the aid of an air-pump. Having by that instrument abstracted the air from a tall glass vessel, we are enabled, by means of a wire passing air-tight through a hole in the top, to let fall several bodies from the top of the vessel to the bottom. These, whether they be feathers, paper, gold-leaf, pieces of money, &c. all descend with the same speed, and strike the bottom at the same moment.

(118.) Every one who has seen a heavy body fall from a height, has witnessed the fact, that its velocity increases as it approaches the ground. But if this were not observable by the eye, it would be betrayed by the effects. It is well known, that the force with which a body strikes the ground increases with the height from whence it has fallen. This force, however, is proportional to the velocity which it has at the moment it meets the ground, and therefore this velocity increases with the height.

When the observations on attraction in the last chapter are well understood, it will be evident that the velocity which a body has acquired in falling from any height, is the accumulated effects of the attraction of terrestrial gravity during the whole time of the fall. Each instant of the fall a new impulse is given to the body, from which it receives additional velocity; and its final velocity is composed of the aggregation of all the small increments of velocity which are thus communicated. As we are at present to suppose the intensity of the attraction invariable, it will follow that the velocity communicated to the body in each instant of time will be the same, and therefore that the whole quantity of velocity produced or accumulated at the end of any time is proportional to the length of that time. Thus, if a certain velocity be produced in a body having fallen for one second, twice that velocity will be produced when it has fallen for two seconds, thrice that velocity in three seconds, and so on. Such is the fundamental principle or characteristic of uniformly accelerated motion.

(119.) In examining the circumstances of the descent of a body, the time of the fall and the velocity at each instant of that time are not the only things to be attended to. The spaces through which it falls in given intervals of time, counted either from the commencement of its fall, or from any proposed epoch of the descent, are equally important objects of enquiry. To estimate the space in reference to the time and the final velocity, we must consider that this space has been moved through with varying speed. From a state of rest at the beginning of the fall, the speed gradually increases with the time, and the final velocity is greater still than that which the body had at any preceding instant during its descent. We cannot, therefore, directly appreciate the space moved through in this case by the time and final velocity. But as the velocity increases uniformly with the time, we shall obtain the average speed, by finding that which the body had in the middle of the interval which elapsed between the beginning and end of the fall, and thus the space through which the body has actually fallen is that through which it would move in the same time with this average velocity uniformly continued.

But since the velocity which the body receives in any time, counted from the beginning of its descent, is in the proportion of that time, it follows that the velocity of the body after half the whole time of descent is half the final velocity. From whence it appears, that the height from which a body falls in any proposed time is equal to the space through which a body would move in the same time with half the final velocity, and it is therefore equal to half the space which would be moved through in the same time with the final velocity.

(120.) It follows from this reasoning, that between the three quantities, the height, the time, and the final velocity, which enter into the investigation of the phenomena of falling bodies, there are two fixed relations: First, the time, counted from the beginning of the fall and the final velocity, are proportional the one to the other; so that as one increases, the other increases in the same proportion. Secondly, the height being equal to half the space which would be moved through in the time of the fall, with the final velocity, must have a fixed proportion to these two quantities, viz. the time and the final velocity, or must be proportional to the product of the two numbers which express them.

But since the time is always proportional to the final velocity, they may be expressed by equal numbers, and the product of equal numbers is the square of either of them. Hence, the product of the numbers expressing the time and final velocity is equivalent to the square of the number expressing the time, or to the square of the number expressing the final velocity. Hence we infer, that the height is always proportional to the square of the time of the fall, or to the square of the final velocity.

(121.) The use of a few mathematical characters will render these results more distinct, even to students not conversant with mathematical science.

Let S = the height from which the body falls, expressed in feet.

V = the velocity at the end of the fall in feet per second.

T = the number of seconds in the time of the fall.

g = the number of feet through which a body would fall in one second.

It will therefore follow that the velocity acquired in one second will be 2g, and the velocity acquired in T seconds will therefore be 2g × T; so that

V = 2g × T  [1]

Since the space which a body falls through in T seconds is found by multiplying the space it falls through in one second by T², we shall have

S = g × T²  [2]

from which, combined with [1] we deduce

S = V²/4g     [3]

S = 1/2V × T  [4]

By these formularies, if the height through which a body falls freely in one second be known, the height through which it will fall in any proposed time may be computed. For since the height is proportional to the square of the time, the height through which it will fall in two seconds will be four times that which it falls through in one second. In three seconds it will fall through nine times that space; in four seconds, sixteen times; in five seconds, twenty-five times, and so on. The following, therefore, is a general rule to find the height through which a body will fall in any given time: “Reduce the given time to seconds, take the square of the number of seconds in it, and multiply the height through which a body falls in one second by that number; the result will be the height sought.”

The following table exhibits the heights and corresponding times as far as 10 seconds:

Each unit in the numbers of the first row expresses a second of time, and each unit in those of the second row expresses the height through which a body falls freely in a second.

(122.) If a body fall continually for several successive seconds, the spaces which it falls through in each succeeding second have a remarkable relation among each other, which may be easily deduced from the preceding table. Taking the space moved through in the first second still as our unit, four times that space will be moved through in the first two seconds. Subtract from this 1, the space moved through in the first second, and the remainder 3 is the space through which the body falls in the second second. In like manner if 4, the height fallen through in the first two seconds, be subtracted from 9, the height fallen through in the first three seconds, the remainder 5 will be the space fallen through in the third second. To find the space fallen through in the fourth second, subtract 9, the space fallen through in the first three seconds, from 16, the space fallen through in the first four seconds, and the result is 7, and so on. It thus appears that if the space fallen through in the first second be called 1, the spaces described in the second, third, fourth, fifth, &c. seconds, will be expressed by the odd numbers respectively, 3, 5, 7, 9, &c. This places in a striking point of view the accelerated motion of a falling body, the spaces moved through in each succeeding second being continually increased.

(123.) If velocity be estimated by the space through which the body would move uniformly in one second, then the final velocity of a body falling for one second will be 2; for with that final velocity the body would in one second move through twice the height through which it has fallen.

(124.) Since the final velocity increases in the same proportion as the time, it follows that after two seconds it is twice its amount after one, and after three seconds thrice that, and so on. Thus, the following table exhibits the final velocities corresponding to the times of descent:

The numbers in the second row express the spaces through which a body with the final velocity would move in one second, the unit being, as usual, the space through which a body falls freely in one second.

(125.) Having thus developed theoretically the laws which characterise the descent of bodies, falling freely by the force of gravity, or by any other uniform force of the same kind, it is necessary that we should show how these laws can be exhibited by actual experiment. There are some circumstances attending the fall of heavy bodies which would render it difficult, if not impossible, to illustrate, by the direct observation of this phenomenon, the properties which have been explained in this chapter. A body falling freely by the force of gravity, as we shall hereafter prove, descends in one second of time through a height of about 16 feet1; in two seconds, it would, therefore, fall through four times that space, or 64 feet; in three seconds, through 9 times the height, or 144 feet; and in four seconds, through 256 feet. In order, therefore, to be enabled to observe the phenomena for only four seconds, we should command an height of at least 256 feet. But further; the velocity at the end of the first second would be at the rate of 32 feet per second; at the end of the second second, it would be 64 feet per second; and towards the end of the fall it would be about 120 feet per second. It is evident that this great degree of rapidity would be a serious impediment to accurate observation, even though we should be able to command the requisite height. It appears therefore that the number expressed by g in the preceding formulæ is 16·083.

It occurred to Mr. George Attwood, a mathematician and natural philosopher of the last century, that all the phenomena of falling bodies might be experimentally exhibited and accurately observed, if a force of the same kind as gravity, viz. an uniformly accelerating force, be used, but of a much less intensity; so that while the motion continues to be governed by the same laws, its quantity may be so much diminished, that the final velocity, even after a descent of many seconds, shall be so moderated as to admit of most deliberate and exact observation. This being once accomplished, nothing more would remain but to find the height through which a body would fall in one second, or, what is the same, the proportion of the force of gravity to the mitigated but uniform accelerating force thus substituted for it.

(126.) To realise this notion, Attwood constructed a wheel turning on its axle with very little friction, and having a groove on its edge to receive a string. Over this wheel, and in the groove, he placed a fine silken cord, to the ends of which were attached equal cylindrical weights. Thus placed, the weights perfectly balance each other, and no motion ensues. To one of the weights he then added a small quantity, so as to give it a slight preponderance. The loaded weight now began to descend, drawing up on the other side the unloaded weight. The descent of the loaded weight, under these circumstances, is a motion exactly of the same kind as the descent of a heavy body falling freely by the force of gravity; that is, it increases according to the same laws, though at a very diminished rate. To explain this, suppose that the loaded weight descends from a state of rest through one inch in a second, it will descend through 4 inches in two seconds, through 9 in three, through 16 in four, and so on. Thus in 20 seconds, it would descend through 400 inches, or 33 feet 4 inches, a height which, if it were necessary, could easily be commanded.

It might, perhaps, be thought, that since the weights suspended at the ends of the thread are in equilibrium, and therefore have no tendency either to move or to resist motion, the additional weight placed upon one of them ought to descend as rapidly as it would if it were allowed to fall freely and unconnected with them. It is very true that this weight will receive from the attraction of the earth the same force when placed upon one of the suspended weights, as it would if it were disengaged from them; but in the consequences which ensue, there is this difference. If it were unconnected with the suspended weights, the whole force impressed upon it would be expended in accelerating its descent; but being connected with the equal weights which sustain each other in equilibrium, by the silken cord passing over the wheel, the force which is impressed upon the added weight is expended, not as before, in giving velocity to the added weight alone, but to it together with the two equal weights appended to the string, one of which descends with the added weight, and the other rises on the opposite side of the wheel. Hence, setting aside any effect which the wheel itself produces, the velocity of the descent must be lessened just in proportion as the mass among which the impressed force is to be distributed is increased; and therefore the rate of the fall bears to that of a body falling freely the same proportion as the added weight bears to the sum of the masses of the equal suspended weights and the added weight. Thus the smaller the added weight is, and the greater the equal suspended weights are, the slower will the rate of descent be.

To render the circumstances of the fall conveniently observable, a vertical shaft (see fig. 24.) is usually provided, which is placed behind the descending weight. This pillar is divided to inches and halves, and of course may be still more minutely graduated, if necessary. A stage to receive the falling weight is moveable on this pillar, and capable of being fixed in any proposed position by an adjusting screw. A pendulum vibrating seconds, the beat of which ought to be very audible, is placed near the observer. The loaded weight being thus allowed to descend for any proposed time, or from any required height, all the circumstances of the descent may be accurately observed, and the several laws already explained in this chapter may be experimentally verified.

(127.) The laws which govern the descent of bodies by gravity, being reversed, will be applicable to the ascent of bodies projected upwards. If a body be projected directly upwards with any given velocity, it will rise to the height from which it should have fallen to acquire that velocity. The earth’s attraction will, in this case, gradually deprive the body of the velocity which is communicated to it at the moment at which it is projected. Consequently, the phenomenon will be that of retarded motion. At each part of its ascent it will have the same velocity which it would have if it descended to the same place from the highest point to which it rises. Hence it is clear, that all the particulars relative to the ascent of bodies may be immediately inferred from those of their descent, and therefore this subject demands no further notice.

To complete the investigation of the phenomena of falling bodies, it would now only remain to explain the method of ascertaining the exact height through which a body would descend in one second, if unresisted by the atmosphere, or any other disturbing cause. As the solution of this problem, however, requires the aid of principles not yet explained, it must for the present be postponed.