Encyclopædia Britannica, Ninth Edition/Gravitation/I. Gravitation at the Surface of the Earth.
I. Gravitation at the Surface of the Earth.
IT is a matter of universal experience all over the earth that a heavy body tends to fall to the ground. Let us inquire into this in the first place by taking such a general view of the phenomenon as would be presented to an imaginary spectator who was sufficiently removed from the earth to be able to take a general view. Fig. 1 represents a section of the earth by a plane which is drawn through its centre 0. Then, the earth being sufficiently near a sphere for our purpose, we may regard the section PQRS as a circle, where the points P, Q, It, S are the intersections of the lines OA, OB, OC, OD with the surface of the earth. If a stone be dropped from a point A above the surface of the earth, it will fall to the ground at P. The spectator would also notice that, if a stone were dropped from B or C or D, it would fall upon the ground at the points Q, R, S re spectively. From A the stone would appear to fall downwards, from C it appears to move up wards, from B the spectator would see the stone moving to the left, while from D it appears
to move to the right. One feature of these motions could not fail to be noticed: they all tend to the centre of the earth. The spectator might therefore sum up his experi ence in the following statement : A body dropped from a point above the surface of the earth always falls in a straight line ivhich is directed to wards the centre of the earth.
§ 1. Attraction.—The familiar instance of the action of a magnet upon a piece of iron will suffice to illustrate what is meant by the word attraction. In virtue of certain properties possessed by the iron and the magnet, they are drawn together. The magnet draws the iron, arid the iron draws the magnet This particular kind of attraction is of a very special character, Thus, for example, the magnet appears to have no appreciable influence on a piece of wood or a sheet of paper, and has indeed no considerable influence on any known substance except iron. By the attraction of gravitation, every body attracts every other body, whatever be the materials of which each is composed. In this we see a wide difference between the attractions of gravitation and that form of attraction which is known as magnetic attraction. Nor is the contrast between the intensities of these two different attractions less striking. The keeper of a magnet is drawn to the magnet by two different forces of attraction. The first of these is the gravitative attraction, which, so far as we know at present, would be equally exerted, whether the magnetism were present or not. The second is the magnetic attraction. The latter is enormously greater than the former ; in fact, under ordinary circumstances as to intensity and dimensions, the intensity of the attraction of gravitation will not be nearly so much as a millionth part of the magnetic attrac tion. The intensity of the attraction of gravitation is indeed so small that, with one conspicuous exception, we can only become aware of its existence by refined and elaborate in quiries. That any two objects for example, two books lying on the table do actually attract each other, there can be no doubt whatever ; but the intensity of this is so small that the attractive force cannot overcome the friction of the table, and consequently we do not find that the books are drawn together. It has, however, been found that the in tensity of the attraction of gravitation between two masses is directly proportional to the product of those masses. Hence though the force is so small as to be almost inappreciable between two bodies of moderate dimensions, yet when the masses of the two bodies, or of even one of them, are enormously great, the intensity of the force will be sufficiently large to be readily discernible. In this way it is that the existence of the attraction of gravitation has been made known to us, and is, in fact, identified with our daily experience, indeed, with our actual existence. The mass of the earth is so enormous that the attraction of gravitation which, exists between it and an object near the surface is readily appreciable. It is this attraction of gravitation bstween the earth and any object which constitutes that force which is referred to when we speak of the weight of the body. It is the attraction of gravitation which causes bodies to fall to the surface of the earth ; and it is easy to show that the facts already presented with respect to the direction in which a falling body moves are readily explained by the supposition that the motions are due to an attractive influence exerted by the earth, or, to speak more correctly, to a mutual attraction subsisting between the earth and the body. Let be the centre of the earth, supposed to be a sphere, and let A be the position of a body above its surface. Then, when the body is released, the attraction must evi dently cause the body to move along the line OA. The line OA, in fact, is directed along a diameter of the sphere, and there is really no reason why the stone should move to one side of the line more than to another. We thus see that attraction would always tend to draw objects in a direction pointing towards the centre of the. earth. The observed facts are therefore explained by the supposition that the earth possesses a power of attraction.
§ 2. Movement of a Falling Body.—Our knowledge of the force of gravitation being ultimately founded on observation and experiment, it will be convenient at this point to de scribe the experiments by which a knowledge of the laws of motion of a falling body may be ascertained. We shall first describe these experiments, and then we shall discuss the laws to which we are conducted by their aid. A beginner is apt to be surprised when he is told that a heavy body and a light body will fall to the ground in the same time if let drop from the same height. Yet nothing can be easier than to prove this important fact experimentally. Take a piece of cork in one hand and a bullet in the other, and drop these two objects at the same moment from the same height. They will reach the ground together. Nor will the results be different if we try a stone and a piece of wood. If, however, one of the objects were a feather and the other were a stone, then no doubt the latter would reach the ground long before the former. But this arises from a cause quite different from gravity. It is the resistance of the air which retards the motion of the feather. Even the stone is retarded to a certain extent by the resistance of the air; but the feather, on account of its greater surface in proportion to its mass, is much more retarded. If we could get rid of the influence of the air, the stone and the feather would be found to fall to the ground in the same time. This can actually be verified by performing the experiment (or a similar one) in a space from which the greater portion of the air has been withdrawn by the aid of an air-pump. But the same thing can also be shown in a much more simple manner. Lay a small flat feather upon the top of a penny piece held horizontally. Then let the penny fall ; it will be followed with equal rapidity by the feather, which will be found to remain in contact with the penny throughout the entire descent. In this case the penny piece displaces the air, and thus to a great extent shields the feather from the resistance to which it would be exposed without such pro tection ; it is thus found that the two objects fall to the ground from the same height at the same time. The various experiments to which we have referred suffice to establish the very important result that the time occupied by a body in falling to the surface of the earth, if dropped from a point above it, is independent of the mass of the body as well as of the materials of which the body is composed. There are, no doubt, certain apparent exceptions to the generality of this statement. The law, as we have stated it, does surely not apply to the case of a balloon or a live bird. In each of these cases the air is made, directly or indirectly, to supply a force which overcomes the force of gravity and neutralizes its effects; but if there were no air, then the balloon and the bird would fall to the ground in precisely the same time as a 56 B) weight would do when dropped from the same height. It will not be necessary for us to introduce any further reference to the resistance of the air, and we shall discuss the phenomena presented by falling bodies as they would occur in a space from which the air has been removed. We have by these considerations cleared the way for a very important quantitative determination. Taking a given interval of time, for example, one second, we see that the height through which a heavy body will fall in one second depends neither upon the mass of the body nor on the materials of which it is composed. This is therefore a constant at any given place on the earth s surface for every description of body, and it is of fundamental importance to determine that quantity accurately. By an indirect method, founded on pendulum observations, it is possible to deter mine this quantity with far greater accuracy than would be attainable by actually making the experiment. The value as thus found is slightly different at different parts of the earth though constant at each one. At any part of the United Kingdom it may be taken as 16 - 1 feet. When the distance which the falling body moves over in the first second has been ascertained, it is possible to find the distance which will be accomplished in two seconds, or indeed in any number. The difficulty of the question arises from the circumstance that, as the velocity of the falling body is gradually increasing, the distance moved over in the second second is greater than it was in the first, and generally that the distance in any second is greater than the distance accomplished in any previous second. Imagine the "lift" in a hotel to be a room 16 1 feet high ; then when the lift is at rest, a stone will take one second to fall from the top of the room to the floor. But now suppose the experiment to be repeated, when the lift is either ascending or descending. It will be found that no matter what be the velocity of the lift, provided it remains uniform for a second, and no matter whether the lift be ascending or descending, the stone will still take exactly one second to fall from the ceiling to the floor. To illustrate the important conclusions which can be drawn from this experiment, let us make some suppositions with reference to the velocity of the lift. Suppose that the lift is descending with a velocity of 5 feet per second. Then since it is found that the stone will reach the floor in one second, it is manifest that during that second the stone must actually have fallen through a distance equal to the height of the room augmented by the 5 feet through which the floor of the room has descended. The total dis tance traversed by the stone is therefore 16-1 + 5 = 21-1 feet. It is, however, to be observed that at starting the stone must necessarily have had the same velocity as the lift, i.e., 5 feet per second. The observed facts can therefore be explained by supposing that the stone retained its initial velocity of 5 feet per second, and that gravity acted upon the stone so as to draw it 16-1 feet nearer the earth than it would have been had gravity not acted. On the other hand, suppose that at the time when the experiment was made the lift was ascending with an uniform velocity of 5 feet per second. Then the actual distance travelled by the stone in falling will be less than the height of the ceiling by the distance through which the floor has been raised, i.e., 16 1 -5 = ll g l feet. Observation nevertheless shows that the time occupied in falling from the ceiling to the floor is still one second. The observed facts can be explained by remembering that at the moment of starting, the stone must actually have had the same velocity as the lift, i.e., an upward velocity of 5 feet per second. If there fore gravity had not acted, the stone would in one second have ascended through a vertical distance of 5 feet. The observations are therefore explained by supposing that gravity in this case also draws the body 16 1 feet nearer the earth in one second than the body would have been had gravity not acted. By suitable contrivances it is possible to ascertain that a body dropped from rest will in a time of two seconds move over a space of 6 4 4 feet. We have already seen that during the first second the body will fall 16 I feet. It follows that in the second second the space described by a body falling freely from rest is 64 4 - 16 1 = 48 3. It is thus obvious that the space described in the second second is three times as great as the space described in the first second. To what is this difference to be ascribed 1 At the commencement of the first second the body was at rest; at the conclusion of the first second the body had attained a certain velocity, and with this velocity the body com menced its motion during the second second. The total distance of 48 3 feet accomplished during the second second is partly due to the velocity possessed by the body at the commencement, and partly to the action of gravity during that second. By the principle just ex plained, we are able to discriminate the amounts due to each cause. It appears, from the experiments already re ferred to, that during the second second as during the first the effect of gravity is simply to make the body 16 1 feet nearer the earth than it could otherwise have been. But the body moves altogether 48-3 feet in the second second, and as the action of gravity during that second will only account for 16 - l feet, it follows that the residue, amounting to 48 3 - 16 1 == 32 - 2 feet, must be attributed to the velocity accumulated during the first second. We are therefore led to the very important result that a body falling freely from rest in the United Kingdom will have acquired a velocity of 32 "2 feet per second when one second has elapsed. It need not be a matter for surprise that, though at the close of the first second the velocity acquired is 32 2, the distance moved over during that second is only 16 1. It will be remembered that the body starts from rest, and that while in the act of falling its velocity is gradually increasing. The body, therefore, moves much further in the last half of the second than it did in the first half, and consequently the total distance travelled must be less than the distance which would have been accomplished had the body been moving during the whole second with the velocity acquired at its termination. It might not be easy to arrange a direct experiment to show how far the body will fall during the third second ; we can, however, deduce the result by reasoning from what we have already learned, Let us suppose that the lift already referred to is descending with an uniform velocity of 32 2 feet per second. A body let drop from the ceiling during the motion, will, as before, reach the floor in one second. The body will therefore have acquired, relatively to the moving lift, a velocity of 32 "2 feet per second. But the lift is itself in motion with a velocity of 32 2 feet per second. The actual velocity of the body must be measured by its velocity relatively to the lift, added to the velocity of the lift itself. It therefore appears that the body which, when it commenced to fall, had a velocity of 32 2 feet per second, acquires an equal amount during its fall, so that at its close the body actually had a velocity of 32 2 + 32 2 = 64 4 feet per second. A body falling freely from rest acquires a velocity of 32 2 feet in the first second; it follows that at the close of the first second the body is in the same condition as if it were let fall from the ceiling of the lift, under the circumstances just described. The motion during the third second is therefore commenced with the velocity of 64 4 feet, and in consequence of this initial velocity alone a distance of 64 - 4 feet will be accomplished in the third second. To this must be added IG l feet, being the additional distance due to the action of gravity, and there fore we have for the distance through which a body falling freely from rest will move in the third second, 6 4 4 + IG l = 80 - 5. Similar reasoning will show that the velocity ac quired at the close of the third second is G4 4 + 32 2 = 9G G. With this velocity the fourth second is commenced, and therefore the distance accomplished during the fourth second is 96-6 + 16-1 = 1127. The results at which we have arrived may be summarily stated in the following propositions : A body falling freely from rest acquires a velocity which is equal to the product of 32 2 and the number of seconds during which the motion has lasted. A body falling freely from rest moves over spaces propor tional to the consecutive odd mimbers (1, 3, 5, 7, &c.) in each of the consecutive seconds during which the motion lasts. A body falling freely from rest will, in a given number of seconds, move over a distance ivhich is found by multiplying the square of the number of seconds by IG l.
§ 3. Values of g.—The velocity acquired by a body in one second is usually denoted by the symbol g. The following are values of g at different parts of the earth (adapted from Everett On C.G.S. Units, p. 12): Latitude. Value of g in Feet per Second. Length in Feet of Pendulum beating Seconds. Equator 32-091 S 2514 Latitude 45 45 32-173 3-2597 Munich 48 9 32-181 3-2607 Paris 48 50 32-183 3-2609 Greenwich 51 29 32-191 3-2616 Gottingen 51 32 32-191 3 2616 Berlin 52 30 32-194 3-2619 Dublin 53 21 32-196 3-2621 Manchester 53 29 32-196 3-2622 Belfast 54 36 32-199 3 2625 Edinburgh 55 57 32-203 3-2629 Aberdeen 57 9 32-206 3-2632 Pole . 90 32 255 3-2682 The value of g in feet at a station of which the latitude is A, and winch is h feet above the level of the sea, is in feet g = 32 -173 -0-082 cos 2A-0-000003&. The length of the pendulum in feet which vibrates in one second is 1 = 3 -2597 -0-0083 cos 2A-0-OOOOOOA. g is really the excess of gravitation over the centrifugal force arising from the earth s rotation. The value of gravitation alone is given by the following expression : 32-225-0-026 cos 2A.
§ 4. Algebraical Formulæ.—The employment of the symbols and operations of algebra will enable us to express very concisely the results at which we have arrived. Let v denote the velocity acquired in t seconds by a body -which has been dropped from a state of rest. Let s denote the number of feet over which the body has moved. The laws we have arrived at may be thus expressed : v = gt ; s = 5gP. From these equations we can eliminate t and obtain v- = 2gs. This expresses the velocity acquired in terms of the distance through which the force has acted. We have hitherto considered the movement of a falling body which was simply dropped. It remains to determine the elfect on the movement of the body which would be produced by a certain initial velocity. Let us for simplicity take the case of a body thrown vertically downwards, and calculate the distance through which the body will move in a certain time, as well as the velocity which it will acquire. In the act of throwing the hand moves with a certain velocity, and the body when released starts off with that velocity. It will thus be observed that the act of throwing is merely to impart initial velocity to the body. Let v 1 be the initial velocity with which the body leaves the hand. Then the velocity of the body at the moment of starting is precisely the same as it would have been had it been dropped from rest v -r-g seconds previ ously. The velocity acquired at the end of t seconds is therefore the same as would have been acquired by a body which fell from rest for a period of (v -^-g + t) seconds ; whence we have = v + gt. The distance must obviously be equal to the difference between the distance through which the body would drop from rest in (v -^-g + t} seconds and the distance through which a body would drop from rest in v -i-g seconds ; whence The case of a body projected vertically iipwards seems at first to present somewhat greater difficulties, but this is not really the case. Such problems can always be readily solved by the help of the following general principle : A body moving vertically for t seconds will, at the end of that time, be 5yt 2 feet nearer the earth tkanit would have been had gravity not acted. If the body be projected vertically upwards with an initial velocity v , then, if the influence of gravity were suspended, the body would in t seconds ascend to a height v t in accordance with the first law of motion. The effect of gravity will be to reduce the height actually obtained by the amount -5yt 2 . Whence wo have s v t- 5gt 2 This expression may be written in the form s=v *+2g- -5g(t -v +g) It is therefore obvious that the greatest altitude h is attained when t = v +g ; in which case h = v ^2g, or v * = 2gh. As an illustration we may take the case of a body thrown verti cally upwards with an initial velocity of 40 feet per second, and in quire where that body will be at the end of two seconds. Had gravity not acted, the body would, in two seconds, have ascended to a height 2 x 40 = 80 feet. The action of gravity will reduce this by "5</2 2 = 2<7 feet, and hence the actual height of the body will be 80 - 2g feet, = 1 5 6 if g be taken at 32 2.
§ 5. Motion of a Projectile.—We have hitherto referred only to the motion of a falling body in a vertical line ; it will now be neces sary to examine some cases in which the motion of the body is not so restricted. From a point on the mast of a steamer g-^-2 feet above the deck a ball is dropped, which falls upon the deck at a certain point. When the steamer is at rest the time taken by the ball to fall will of course be one second, and its path will be vertical ; but when the steamer is moving with uniform velocity it is found that the ball still falls precisely on the same spot of the deck as when the steamer was at rest, and that the time occupied in the descent is still one second. It is obvious that in this case the ball does not move in a straight line at all, but in a curved path due to the motion of the vessel compounded with the actual falling motion. We there fore see that the effect of the motion of the vessel on the ball was to project that ball with a certain initinl horizontal velocity, but that notwithstanding that initial velocity the ball still reaches the deck in one second. We are therefore led to the general conclusion that A body projected horizontally will, at the end of t seconds, hare fallen through a space of 16 1 t 2 feet. An experiment illustrating this result maybe made in an exceedingly simple manner. Take a marble in each hand, and throw one of the marbles horizontally at the same time as you drop the other from the same height ; you will find that the two marbles reach the ground together. Suppose for simplicity that the height at the moment when the marbles are released is 4 feet, then the time taken by one of the marbles in falling is half a second. But as both marbles reach the ground together, the experiment has really proved to us that a body at the height of 4 feet from the ground will if pro jected horizontally reach the ground in half a second. This is equally true whatever be the magnitude of the velocity, i.e., whether it be 5, 10, or any other number of feet per second. We have now studied the effect of gravity upon a body which has been projected either in a vertical line or in a horizontal line. We have found that in each case the effect of gravity is to bring the body ^gt 1 feet nearer the surface of the earth in t seconds than it would have been had gravity not acted. We are therefore tempted to inquire whether the same statement would not be true for a body projected in any direction. In every way in which this suggestion can be tested it has been found to be verified, and there cannot there fore be the slightest doubt that it is true. To illustrate this principle we may apply it to the case of a body projected in any direction, and deduce the form of the path in which the body moves. Let (fig. 2) be the point from which the body is projected, and let P 5 OP be the direction in which the body would move after projection if it were not for gravi tation. In consequence of the first law of mo tion, we should find that if it had not been for the action of gravity the ball would reach P : in one second, P 2 in two seconds, &c. , where m> _ P P P p 1 M r 2 1 2 1 3 Gravitation will, how ever, make the body swerve from the direc tion OP^, &c., so that at the end of one
second the body is really found at the point Aj, at the end of two seconds at A 2 , at the end of three seconds at A 3 , &c. The curve drawn through the points A 1( A 2 , A 3 , &c. , which is actually de scribed by the body, can be readily constructed. Take, for example, t 3. If gravity had not been acting, the body would in three seconds have reached the point P 3 . We can find where the body actually is by taking a point A 3 , which is vertically beneath P 3 at the distance 16 1 x 3" feet. Similarly we can find where the body is after any other specified number of seconds, and thus we obtain the points Aj, A. 2 , &c. The equation of the curve is thus found. Take the line OP as the axis of x, and let x denote the number of seconds during which the motion has lasted ; then, if y denote the vertical distance through which the body has been deflected by gravity, we must have This curve, being of the second degree, represents a conic section ; and as the highest terms form a perfect square, the conic section must be a parabola.