THE MOTION OF A FALLING BODY

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LECTURE XV. THE MOTION OF A FALLING BODY.

Introduction.—The First Law of Motion.—The Experiment of Galileo from the Tower of Pisa.—The Space is proportional to the Square of the Time.—A Body falls 16' in the First Second.—The Action of Gravity is independent of the Motion of the Body.—How the Force of Gravity is defined.—The Path of a Projectile is a Parabola.

INTRODUCTION.

483. Kinetics is that branch of mechanics which treats of the action of forces in the production of motion. We shall find it rather more difficult than the subjects with which we have been hitherto occupied; the difficulties in kinetics arise from the introduction of the element of time, into our calculations. The principles of kinetics were unknown to the ancients. Galileo discovered some of its truths in the seventeenth century; and, since his time, the science has grown rapidly. The motion of a falling body was first correctly apprehended by Galileo; and with this subject we can appropriately commence.

THE FIRST LAW OF MOTION.

484. Velocity, in ordinary language, is supposed to convey a notion of [Pg 231]rapid motion. Such is not precisely the meaning of the word in mechanics. By velocity is merely meant the rate at which a body moves, whether the rate be fast or be slow. This rate is most conveniently measured by the number of feet moved over in one second. Hence when it is said the velocity of a body is 25, it is meant that if the body continued to move for one second with its velocity unaltered, it would in that time have moved over 25 feet.

Fig. 66.

  485. The first law of motion may be stated thus. If no force act upon a body, it will, if at rest, remain for ever at rest; or if in motion, it will continue for ever to move with a uniform velocity. We know this law to be true, and yet no one has ever seen it to be true for the simple reason that we cannot realise the condition which it requires. We cannot place a body in the condition of being unacted upon by any forces. But we may convince ourselves of the truth of the law by some such reasoning as the following. If a stone be thrown along the road, it soon comes to rest. The body leaves the hand with a certain initial velocity and is not further acted upon by it. Hence, if no other force acted on the stone, we should expect, if the first law be true, that it would continue to run on for ever with the original velocity at the moment of leaving the hand. But other forces do act upon the stone; the attraction of the earth pulls it down; and, when it begins to bound and roll upon the ground, friction comes into operation, deprives the stone of its velocity, and brings it to rest. But let the stone be thrown upon a surface of smooth ice; when it begins to slide, the force of gravity is counteracted by the reaction of the ice: there is no other force acting upon the stone except friction, which is small. Hence we find that the stone will run on for a considerable distance. It requires but little effort of the imagination to suppose a lake whose [Pg 232]surface is an infinite plane, perfectly smooth, and that the stone is perfectly smooth also. In such a case as this the first law of motion amounts to the assertion that the stone would never stop.

486. We may, in the lecture room, see the truth of this law verified to a certain extent by Atwood's machine (Fig, 66). This machine has been devised for the purpose of investigating the laws of motion by actual experiment. It consists principally of a pulley c, mounted so that its axle rests upon two pairs of wheels, as shown in the figure; it being the object of this contrivance to enable the wheel to revolve with the utmost freedom. A pair of equal weights a, b, are attached by a silken thread, which passes over the pulley; each of the weights is counterbalanced by the other: so that [Pg 233]when the two are in motion, we may consider either as a body not acted upon by any forces, and it will be found that it moves uniformly, as far as the size of the apparatus will permit.

487. If we try to conceive a body free in space, and not acted upon by any force, it is more natural to suppose that such a body, when once started, should go on moving uniformly for ever, than that its velocity should be altered. The true proof of the first law of motion is, that all consequences properly deduced from it, in combination with other principles, are found to be verified. Astronomy presents us with the best examples. The calculation of the time of an eclipse is based upon laws which in themselves assume the first law of motion; hence, when we invariably find that an eclipse occurs precisely at the moment for which it has been predicted, we have a splendid proof of the sublime truth which the first law of motion expresses.

THE EXPERIMENT OF GALILEO FROM THE TOWER OF PISA.

488. The contrast between heavy bodies and light bodies is so marked that without trial we hardly believe that a heavy body and a light body will fall from the same height in the same time. That they do so Galileo proved by dropping a heavy ball and a light ball together from the top of the Leaning Tower at Pisa. They were found to reach the ground simultaneously. We shall repeat this experiment on a scale sufficiently reduced to correspond with the dimensions of the lecture room.

Fig. 67.

[Pg 235]  489. The apparatus used is shown in Fig. 67. It consists of a stout framework supporting a pulley h at a height of about 20 feet above the ground. This pulley carries a rope; one end of the rope is attached to a triangular piece of wood, to which two electro-magnets g are fastened. The electro-magnet is a piece of iron in the form of a horse-shoe, around which is coiled a long wire. The horse-shoe becomes a magnet immediately an electric current passes through the wire; it remains a magnet as long as the current passes, and returns to its original condition the moment the current ceases. Hence, if I have the means of controlling the current, I have complete control of the magnet; you see this ball of iron remains attached to the magnet as long as the current passes, but drops the instant I break the current. The same electric circuit includes both the magnets; each of them will hold up an iron ball f when the current passes, but the moment the current is broken both balls will be released. Electricity travels along a wire with prodigious velocity. It would pass over many thousands of miles in a second; hence the time that it takes to pass through the wires we are employing is quite inappreciable. A piece of thin paper interposed between the magnets and the balls will ensure that they are dropped simultaneously; when this precaution is not taken one or both balls may hesitate a little before commencing to descend. A long pair of wires e, b, must be attached to the magnets, the other ends of the wires communicating with the battery d; the triangle and its load is hoisted up by means of the rope and pulley and the magnets thus carry the balls to a height of 20 feet: the balls we are using weigh about 0·25 lb. and 1 lb. respectively.

490. We are now ready to perform the experiment. I break the circuit; the two balls are disengaged simultaneously; they fall side by side the whole way, and reach the ground together, where it is well to place a [Pg 236]cushion to receive them. Thus you see the heavy ball and the light one each require the same amount of time to fall from the same height.

  491. But these balls are both of iron; let us compare together balls made of different substances, iron and wood for example. A flat-headed nail is driven into a wooden ball of about 2"·5 in diameter, and by means of the iron in the nail I can suspend this ball from one of the magnets; while either of the iron balls we have already used hangs from the other. I repeat the experiment in the same manner, and you see they also fall together. Finally, when an iron ball and a cork ball are dropped, the latter is within two or three inches of its weighty companion when the cushion is reached: this small difference is due to the greater effect of the resistance of the air on the lighter of the two bodies. There can be no doubt that in a vacuum all bodies of whatever size or material would fall in precisely the same time.

492. How is the fact that all bodies fall in the same time to be explained? Let us first consider two iron balls. Take two equal particles of iron; it is evident that these fall in the same time; they would do so if they were very close together, even if they were touching, but then they might as well be in one piece: and thus we should find that a body consisting of two or more iron particles takes the same time to fall as one (omitting of course the resistance of the air). Thus it appears most reasonable that two balls of iron, even though unequal in size, should fall in the same time.

493. The case of the wooden ball and the iron ball will require more consideration before we realise thoroughly how much Galileo’s experiment proves. We must first explain the meaning of the word mass in mechanics.[Pg 237]

494. It is not correct to define mass by the introduction of the idea of weight, because the mass of a body is something independent of the existence of the earth, whereas weight is produced by the attraction of the earth. It is true that weight is a convenient means of measuring mass, but this is only a consequence of the property of gravity which the experiment proves, namely, that the attraction of gravity for a body is proportional to its mass.

495. Let us select as the unit of mass the mass of a piece of platinum which weighs 1 lb. at London; it is then evident that the mass of any other piece of platinum should be expressed by the number of pounds it contains: but how are we to determine the mass of some other substance, such as iron? A piece of iron is defined to have the same mass as a piece of platinum, if the same force acting on either of the bodies for the same time produces the same velocity. This is the proper test of the equality of masses. The mass of any other piece of iron will be represented by the number of times it contains a piece equal to that which we have just compared with the platinum; similarly of course for other substances.

496. The magnitude of a force acting for the time unit is measured by the product of the mass set in motion and the velocity which it has acquired. This is a truth established, like the first law of motion, by indirect evidence.

 497. Let us apply these principles to explain the experiment which demonstrated that a ball of wood and a ball of iron fall in the same time. Forces act upon the two bodies for the same time, but the magnitudes of the forces are proportional to the mass of each body multiplied into its velocity, and, since the bodies fall simultaneously, their velocities are equal. The forces acting upon the bodies are therefore proportional to their masses; but the force acting [Pg 238]on each body is the attraction of the earth, therefore, the gravitation to the earth of different bodies is proportional to their masses.

498. We may here note the contrast between the attraction of gravitation and that of a magnet. A magnet attracts iron powerfully and wood not at all; but the earth draws all bodies with forces depending on their masses and their distances, and not on their chemical composition.

THE SPACE DESCRIBED BY A FALLING BODY IS
PROPORTIONAL TO THE SQUARE OF THE TIME.

499. We have next to discover the law by which we ascertain the distance a body falling from rest will move in a given time; it is not possible to experiment directly upon this subject, as in two seconds a body will drop 64 feet and acquire an inconveniently large velocity; we can, however, resort to Atwood’s machine (Fig. 66) as a means of diminishing the motion. For this purpose we require a clock with a seconds pendulum.

500. On one of the equal cylinders a I place a slight brass rod, whose weight gives a preponderance to a, which will consequently descend. I hold the loaded weight in my hand, and release it simultaneously with the tick of the pendulum. I observe that it descends 5" before the next tick. Returning the weight to the place from whence it started, I release it again, and I find that at the second tick of the pendulum it has travelled 20". Similarly we find that in three seconds it descends 45". It greatly facilitates these experiments to use a little stage which is capable of being slipped up and down the scale, and which can be clamped to the scale in any position. By actually placing the stage at the distance of 5", 20", or 45" below the point from which the weight starts, the coincidence of [Pg 239]the tick of the pendulum with the tap of the weight on its arrival at the stage is very marked.

501. These three distances are in the proportion of 1, 4, 9; that is, as the squares of the numbers of seconds 1, 2, 3. Hence we may infer that the distance traversed by a body falling from rest is proportional to the square of the time.

502. The motion of the bodies in Atwood’s machine is much slower than the motion of a body falling freely, but the law just stated is equally true in both cases so that in a free fall the distance traversed is proportional to the square of the time. Atwood’s machine cannot directly tell us the distance through which a body falls in one second. If we can find this by other means, we shall easily be able to calculate the distance through which a body will fall in any number of seconds.

A BODY FALLS 16' IN THE FIRST SECOND.

503. The apparatus by which this important truth maybe demonstrated is shown in Fig. 67. A part of it has been already employed in performing the experiment of Galileo, but two other parts must now be used which will be briefly explained.

504. At a a pendulum is shown which vibrates once every second; it need not be connected with any clockwork to sustain the motion, for when once set vibrating it will continue to swing some hundreds of times. When this pendulum is at the middle of its swing, the bob just touches a slender spring, and presses it slightly downwards. The electric current which circulates about the magnets g (Art. 489) passes through this spring when in its natural position; but when the spring is pressed down by the pendulum, [Pg 240]the current is interrupted. The consequence is that, as the pendulum swings backwards and forwards, the current is broken once every second. There is also in the circuit an electric alarm bell c, which is so arranged that, when the current passes, the hammer is drawn from the bell; but, when the current ceases, a spring forces the hammer to strike the bell. When the circuit is closed, the hammer is again drawn back. The pendulum and the bell are in the same circuit, and thus every vibration of the pendulum produces a stroke of the bell. We may regard the strokes from the bell as the ticks of the pendulum rendered audible to the whole room.

505. You will now understand the mode of experimenting. I draw the pendulum aside so that the current passes uninterruptedly. An iron ball is attached to one of the electro-magnets, and it is then gently hoisted up until the height of the ball from the ground is about 16'. A cushion is placed on the floor in order to receive the falling body. You are to look steadily at the cushion while you listen for the bell. All being ready, the pendulum, which has been held at a slight inclination, is released. The moment the pendulum reaches the middle of its swing it touches the spring, rings the bell, breaks the current which circulated around the magnet, and as there is now nothing to sustain the ball, it drops down to the cushion; but just as it arrives there, the pendulum has a second time broken the electric circuit, and you observe the falling of the ball upon the cushion to be identical with the second stroke of the bell. As these strokes are repeated at intervals of a second, it follows that the ball has fallen 16' in one second. If the magnet be raised a few feet higher, the ball may be seen to reach the cushion after the bell is heard. If the magnet be lowered a few feet, the ball reaches the cushion before the bell is heard.[Pg 241]

506. We have previously shown that the space is proportional to the square of the time. We now see that when the time is one second, the space is 16 feet. Hence if the time were two seconds, the space would be 4 × 16 = 64 feet; and in general the space in feet is equal to 16 multiplied by the square of the time in seconds.

507. By the help of this rule we are sometimes enabled to ascertain the height of a perpendicular cliff, or the depth of a well. For this purpose it is convenient to use a stop-watch, which will enable us to measure a short interval of time accurately. But an ordinary watch will do nearly as well, for with a little practice it is easy to count the beats, which are usually at the rate of five a second. By observing the number of beats from the moment the stone is released till we see or hear its arrival at the bottom, we determine the time occupied in the act of falling. The square of the number of seconds (taking account of fractional parts) multiplied by 16 gives the depth of the well or the height of the cliff in feet, provided it be not high.

THE ACTION OF GRAVITY IS INDEPENDENT
OF THE MOTION OF THE BODY.

508. We have already learned that the effect of gravity does not depend upon the actual chemical composition of the body. We have now to learn that its effect is uninfluenced by any motion which the body may possess. Gravity pulls a body down 16' per second, if the body starts from rest. But suppose a stone be thrown upwards with a velocity of 20 feet, where will it be at the end of a second? Did gravity not act upon the stone, it would be at a height of 20 feet. The principle we have stated tells us that gravity will draw this stone towards the earth [Pg 242]through a distance of 16', just as it would have done if the stone had started from rest. Since the stone ascends 20' in consequence of its own velocity, and is pulled back 16' by gravity, it will, at the end of a second, be found at the height of 4'. If, instead of being shot up vertically, the body had been projected in any other direction, the result would have been the same; gravity would have brought the body at the end of one second 16' nearer the earth than it would have been had gravity not acted. For example, if a body had been shot vertically downwards with a velocity of 20', it would in one second have moved through a space of 36'.

509. We shall illustrate this remarkable property by an experiment. The principle of doing so is as follows:—Suppose we take two bodies, a and b. If these be held at the same height, and released together, of course they reach the ground at the same instant; but if a, instead of being merely dropped, be projected with a horizontal velocity at the same moment that b is released, it is still found that a and b strike the floor simultaneously.

510. You may very simply try this without special apparatus. In your left hand hold a marble, and drop it at the same instant that your right hand throws another marble horizontally. It will be seen that the two marbles reach the ground together.

511. A more accurate mode of making the experiment is shown by the contrivance of Fig. 68.

Fig. 68.

In this we have an arrangement by which we ensure that one ball shall be released just as the other is projected. At a b is shown a piece of wood about 2" thick; the circular portion (2' radius) on which the ball rests is grooved, so that the ball only touches the two edges and not the bottom of the groove. Each edge of the groove is covered with tinfoil c, but the pieces of tinfoil on the two sides must not communicate. One edge is connected with one pole of the battery k, and the other edge with the other pole, but the current is unable to pass until a communication by a conductor is opened between the two edges. The ball g supplies the bridge; it is covered with tinfoil, and therefore, as long as it rests upon the edges, the circuit is complete; the groove is so placed that the tangent to it at the lowest point b is horizontal, and therefore, when the ball rolls down the curve, it is projected from the bottom in a horizontal direction. An india-rubber spring is used to propel the ball; and by drawing it back when embraced by the spring, I [Pg 244]can communicate to the missile a velocity which can be varied at pleasure. At h we have an electro-magnet, the wire around which forms part of the circuit we have been considering. This magnet is so placed that a ball suspended from it is precisely at the same height above the floor as the tinned ball is at the moment when it leaves the groove.

512. We now understand the mode of experimenting. So long as the tinned ball g remains on the curve the bridge is complete, the current passes, and the electro-magnet will sustain h, but the moment g leaves the curve, h is allowed to fall. We invariably find that whatever be the velocity with which g is projected, it reaches the ground at the same instant as h arrives there. Various dotted lines in the figure show the different paths which g may traverse; but whether it fall at d, at e, or at f, the time of descent is the same as that taken by h. Of course, if g were not projected horizontally, we should not have arrived at this result; all we assert is that whatever be the motion of a body, it will (when possible) be at the end of a second, sixteen feet nearer the earth than it would have been if gravity had not acted. If the body be projected horizontally, its descent is due to gravity alone, and is neither accelerated nor retarded by the horizontal velocity. What this experiment proves is, that the mere fact of a body having velocity does not affect the action of gravity thereon.

513. Though we have only shown that a horizontal velocity does not affect the action of gravity, yet neither does a velocity in any direction. This is verified, like the first law of motion, by the accordance between the consequences deduced from it and the facts of observation.

514. We may summarize these results by saying that no matter what be the material of which a particle is composed, whether it be heavy or [Pg 245]light, moving or at rest, if no force but gravity act upon the particle for t seconds, it will then be 16t² feet nearer the earth than it would have been had gravity not acted.

Fig. 69.

  515. A proposition which is of some importance may be introduced here. Let us suppose a certain velocity and a certain force. Let the velocity be such that a point starting from a, Fig. 69, would in one second move uniformly to b. Let the force be such that if it acted on a particle originally at rest at a, it would in one second draw the particle to d; if then the force act on a particle having this velocity where will it be at the end of the second? Complete the parallelogram a b c d, and the particle will be found at c. By what we have stated the force will equally discharge its duty whatever be the initial velocity. The force will therefore make the particle move to a distance equal and parallel to a d from whatever position the particle would have assumed, had the force not acted; but had the force not acted, the particle would have been found at b: hence, when the force does act, the particle must be found at c, since b c is equal and parallel to a d.[Pg 246]

HOW THE FORCE OF GRAVITY IS DEFINED.

516. From the formula

we learn that a body falls through 64' in 2 seconds; and as we know that it falls 16' in the first second, it must fall 48' in the next second. Let us examine this. After falling for one second, the body acquires a certain velocity, and with that velocity it commences the next second. Now, according to what we have just seen, gravity will act during the next second quite independently of whatever velocity the body may have previously had. Hence in the second second gravity pulls the body down 16', but the body moves altogether through 48'; therefore it must move through 32' in consequence of the velocity which has been impressed upon it by gravity during the first second. We learn by this that when gravity acts for a second, it produces a velocity such that, if the body be conceived to move uniformly with the velocity acquired, the body would in one second move over 32'.

517. In three seconds the body falls 144', therefore in the third second it must have fallen

144' - 64' = 80';

but of this 80' only 16' could be due to the action of gravity impressed during that second; the rest,

80' - 16' = 64',

is due to the velocity with which the body commenced the third second.

518. We see therefore that after the lapse of two seconds gravity has communicated to the body a velocity of 64' per second; we should similarly find, that at the end of the third second, the body has a [Pg 247]velocity of 96', and in general at the end of t seconds a velocity of 32t. Thus we illustrate the remarkable law that the velocity developed by gravity is proportional to the time.

519. This law points out that the most suitable way of measuring gravity is by the velocity acquired by a falling body at the end of one second. Hence we are accustomed to say that g (as gravity is generally designated) is 32. We shall afterwards show in the lecture on the pendulum (XVIII.) how the value of g can be obtained accurately. From the two equations, v = 32t and s = 16t² it is easy to infer another very well known formula, namely, v² = 64s.

THE PATH OF A PROJECTILE IS A PARABOLA.

520. We have already seen, in the experiments of Fig. 68, that a body projected horizontally describes a curved path on its way to the ground, and we have to determine the geometrical nature of the curve. As the movement is rapid, it is impossible to follow the projectile with the eye so as to ascertain the shape of its path with accuracy; we must therefore adopt a special contrivance, such as that represented in Fig. 70.

b c is a quadrant of wood 2" thick; it contains a groove, along which the ball b will run when released. A series of cardboard hoops are properly placed on a black board, and the ball, when it leaves the quadrant, will pass through all these hoops without touching any, and finally fall into a basket placed to receive it. The quadrant must be secured firmly, and the ball must always start from precisely the same place. The hoops are easily adjusted by trial. Letting the ball run down the quadrant two or three times, we can see how to place the first hoop in its right position, and secure it by [Pg 248]drawing pins; then by a few more trials the next hoop is to be adjusted, and so on for the whole eight.

521. The curved line from the bottom of the quadrant, which passes through the centres of the hoops, is the path in which the ball moves; this curve is a parabola, of which f is the focus and the line a a the directrix.

Fig. 70.

It is a property of the parabola that the distance of any point on the curve from the focus is equal to its perpendicular distance from the directrix. This is shown in the figure. For example, the dotted line [Pg 249]f d, drawn from f to the centre of the lowest hoop d, is equal in length to the perpendicular d p let fall from d on the directrix a a.

522. The direction in which the ball is projected is in this case horizontal, but, whatever be the direction of projection, the path is a parabola. This can be proved mathematically as a deduction from the theorem of Art. 515.

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