and B are two equal cylindrical weights suspended from the ends of the thread, 13 which rests in a groove on the edge of the wheel bed. S is a small stage which can be screwed upon the graduated shaft, at any particular division at which it is designed to stop the descent of the weight. G is a clock, attached to the principal pillar, which beats seconds, in order to mark the rate of descent. The weights A B are, commonly, so adjusted, that, by placing on the top of the cylindrical weight A a weight Ô of a quarter of an ounce, the weight A will descend through three inches in one second. Thus we have obtained an accelerating force, which is sixty-four times less than that of gravity, and yet which retains all the characteristic peculiarities of that force. In fact it is the force of gravity correctly represented in miniature. (38.) We shall now show how this machine is applied to establish by experiment the laws which regulate the descent of heavy bodies and which have been already explained. Ex. 1. To establish these laws by experiment, a ring R is provided, at tached to a block E, which can be fixed by a screw to any division of the graduated shaft. A bar of metal fis also provided, weighing a quarter of an ounce, and longer than the diameter of the ring R. Let the ring R be fixed by the screw to any division of the scale, and let the stage S be so fixed, that when the weight A rests upon it, the top of the weight will be six inches exactly below the ring R. This done, let the weight A be elevated by drawing down the weight B until the top of the weight A is exactly three inches above the ring R. Holding the weight A in this position, let the bar F be placed upon it, and observing the beats of the clock, let the weight A commence its descent with any beat. It will be found that the stroke of the bar F on the ring R will exactly coincide with the next beat, and that the stroke of the weight A on the stage S will coincide precisely with the succeeding beat. It will be observed that the accelerated motion of the weight. A for the first second, and before the bar strikes the ring, is entirely owing to the action of the force of gravity on the bar (36). When the bar is taken off the weight A by the ring at the end of the first second, this cause of acceleration ceases, the action of gravity is suspended, and the weight A moves on to the stage S with the velocity which it had acquired at R. Now we have seen that this velocity was such that it moved through six inches in one second. Ex. 2. Again, let the stage be placed so that when the weight A rests upon it, the top of the weight will be twelve inches from the ring R, and let the weight B be depressed until the top of the weight A is twelve inches above the ring R. This done, let the bar F be placed on the weight A, and let that weight be disengaged at the moment of any beat of the clock; it will be observed that the stroke of the bar Fupon the ring R will coincide exactly with the third beat, the descent through twelve inches being made in two seconds, and that the stroke of the weight A upon the stage S will coincide precisely with the fourth beat, the weight moving through the twelve inches below the ring with the velocity it has acquired in two seconds. Ex. 3. Now let the stage S be once more removed, and placed so that, when the weight A stands upon it, the top of the weight will be eighteen inches below the ring R. Let the weight B be depressed until the top of the weight A is twenty-seven inches above the stage S. Let the bar F be then placed upon the weight A as before, and permitting the weight to commence its descent with the first beat of the pendulum, the bar will strike the ring R with the fourth beat, and the weight A will strike the stage S with the fifth beat. The weight, therefore, descends through twenty-seven inches with an accelerated motion in three seconds, and at the end of that time has acquired such a velocity, as to move through eighteen inches in a second. (39.) Now let us review the results of these three experiments. By the first it appears, that the velocity acquired in one second is such as to make the weight A move at the rate of six inches per second. By the second experiment it appears, that the velocity acquired in two seconds is twelve inches per second; and by the third experiment it appears, that the velocity acquired in three seconds is eighteen inches per second. Thus the velocities acquired in one, two, and three seconds, are as six, twelve, and eighteen, which numbers are as one, two, and three. Hence the law before explained, that" the velocities acquired are as the time of acquiring them," is verified. In the first experiment the weight fell through three inches in one second; in the second experiment it fell through twelve inches in two seconds, and in the third it fell through twenty-seven inches in three seconds. Now the numbers three, twelve, and twenty-seven are as one, four and nine, which are the squares of one, two, and three. Hence the law already explained, that "the spaces fallen through are proportional to the squares of the times," is verified. In the first experiment it was shown that the velocity acquired in falling through three inches, was such as would carry the weight in the same time through six inches when continued uniform and without further increase. In the second experiment it was shown that with the velocity acquired in falling through twelve inches in two seconds, the weight A would move through twelve inches in one second, and it would, therefore, move through twentyfour inches in two seconds. In like manner, in the third experiment, it appeared that with the velocity acquired in falling through twenty-seven inches in three seconds, the weight A moved through eighteen inches in one second; and, therefore, would move through fifty-four inches in three seconds. Each of these experiments, therefore, verifies the law, that, "with the velocity which a body acquires in any time, it would, if that velocity were continued uniform, move through twice that space in the same time." Also by the first experiment it appeared that the space fallen through in the first second of the descent was three inches. By the second experiment it appeared, that the space fallen through in the first two seconds was twelve inches. It consequently follows, that the space fallen through in the second second must have been nine inches. By the third experiment the space fallen through in three seconds was twentyseven inches. Taking from this the space fallen through in the first two seconds, which is twelve inches, the remainder, fifteen inches, is the space fallen through in the third second. Thus the spaces described in the first, second, and third seconds of the fall are three, nine, and fifteen inches respectively, which are as the numbers one, three, and five. This verifies the law before explained, that "the spaces described by a falling body in the successive equal intervals are as the odd integers." Since the heights from which bodies (40.) WE have stated that, at a given place upon the surface of the earth, the force of gravity acts on all bodies in lines which are parallel to each other, and perpendicular to an horizontal, or level plane. When it acts upon a single body, it does not act, as it were, by a single effort, but impresses a separate force upon each particle of the body; and its total effect is composed of the sum of all its effects thus produced upon the particles. Now there is in the body a certain point, at which, if the attraction of gravity impressed a single force, equal in intensity to the sum of all its separate actions on the component parts of the body, the ultimate effect would be the same as it is under the system of separate action which really obtains. This point, the existence of which we shall prove experimentally, is called the centre of gravity. (41.) If the attraction of gravity were confined in its action to one particular point, there are certain effects which would very evidently ensue. First, if that point were supported or fixed, the body would rest in any position whatever in which it should be placed. For the only cause which we suppose to affect it so as to produce motion, acts upon a point which we suppose fixed. Secondly. If the body be perfectly free to move, the point on which the attraction acts will commence to move in the direction of that attraction, and in this case will, therefore, commence to move in a line perpendicular to an horizontal plane. Thirdly. If the body be suspended by any point different from that at which alone the attraction of gravity is supposed to act, it will only remain at rest in two positions, viz. when the attracted point is immediately under or immediately over the point of suspension. If the attracted point be in any the point of suspension, all its parts P. The attraction Next, let the point sue. Lastly, let C be in a position neither directly above nor below the fixed point P. Draw B will only proP, which will fig. 11. B C D' perpendicular to an horizontal plane, and parallel to C D, and taking fig. 12. D other position, the body will move round any portion Co from C, draw the to two separate attractions represented in intensity and direction by the lines Cn and Cm, and its effect is the same as the united effects of these two would be. Now it is obvious, that a force acting from C, in the direction Cn, would have no effect in producing motion, but would be resisted by the fixed point P, against which it would press, while the other force C m, perpendicular to C P, would tend to turn the body round C P, so as to bring the point C to the line PD, directly below the point of suspension P, in which position, after some oscillations, it would rest. (42.) From this investigation, it follows, that if the parallel actions of the force of gravity on the particles of a body be capable of being represented by an equivalent force, acting at a single point, that point may be determined by the properties which we have just explained. Let a body which is bounded by two parallel planes, be suspended from any point taken at pleasure in it. It will be found that there is but one position in which it will hang steadily at rest, and without swinging. To the point of suspension let a plumb line be attached, and let the line in which it touches the plane surface of the suspended body be marked. Let the body be now suspended from some other point in its plane surface, and let another fine be drawn upon it in the direction of the plumb line. This process being applied to any number of different points in the surface of the body, and a number of such lines being drawn upon it in the direction of the plumb line, it will be found that all these lines will intersect each other in the same point. It follows, therefore, that this point has the property mentioned in (41.), of settling itself vertically under the point of suspension when the body is in equilibrium. Next let the point thus determined be made the point of suspension, and it will be found that the body will rest in any position in which it may be placed, and that it will not, under any circumstances, vibrate or swing. Again, let the body be suspended by any point different from that which we have here determined, and let it be so placed that this point shall be placed vertically over the point of suspension. It will be found that the body will remain in equilibrium so long as its position is not changed; but upon the least impulse which moves the point in question from its position, it will turn round the point of suspension, and settle, after some vibrations, into the position directly under the point of suspension. The point, the existence and properties of which are thus established, is then the centre of gravity. In the preceding experiment, we have selected a body bounded by parallel planes, for the purpose of simplifying the experimental process. Strictly speaking, the centre of gravity is not at the intersection of the lines determined by the plumb line on the plane surface, but, if a line be drawn perpendicular to the plane surface through the body, it will be at the middle point of this line. If we could conveniently pierce the dimensions of a body by straight lines, the centre of gravity of any body, whatever be its figure, could be found experimentally by the same process. If it be successively suspended by several points, and pierced by straight lines, in each case passing in a vertical direction through the point of suspension, it would be found that, however numerous these lines might be, they would all intersect in one point, which would be the centre of gravity of the body. (43.) By these properties of the centre of gravity, mechanical problems respecting the effects of the weights of bodies are susceptible of considerable simplifica tion; for, instead of taking into consideration the separate effects of the attraction of gravitation on the several particles of which a body is composed, it will be |