shall serve to compare other bodies with; for solids and liquids, water is taken as the unit or standard of comparison. And we say that iron is about seven, lead eleven, quicksilver thirteen times as heavy as it; or that they have specific gravities expressed by those numbers. The unit of comparison for gaseous and vaporous bodies is atmospheric air. When an unsupported body is allowed to fall, its path is in a vertical line. If a body be suspended by a thread, the thread represents the path in which that body would have moved. It occupies a vertical direction, or is perpendicular to the position which would be occupied by a surface of stagnant water. Such a thread is termed a plumb-line. It is of constant use in the arts to determine horizontal and vertical lines. AQ OB If in two positions, A B, Fig. 92, on the earth's surface, plumb-lines were suspended, it would be found that, though they are perpendicular as respects that surface, they are not parallel to one another, but incline, at a small angle, A C B, to each other. If their distance be one mile, this convergence would amount to one minute; and if it be sixty miles, the convergence will be one degree. Now, as the plumb-line indicates the path of a falling body, it is easily understood that on different parts of the earth's surface the paths of falling bodies have the inclinations just described. A little consideration shows that the descent of such bodies is in a line directed to the centre, C, of the earth. Fig. 92. That centre we may therefore regard as the active point, or seat of the whole earth's attractive influence. When examinations with plumb-lines are made in the neighbourhood of mountain masses, those masses exert a disturbing agency on the plummet, drawing the line from its true vertical position. But this is nothing more than what ought to place on the theory of universal gravitation; for that theory asserting that all masses exert an attractive influence, the results here pointed out must necessarily ensue, and the lateral action of the mountains correspondingly draw the plummet aside. CHAPTER XIX. THE DESCENT OF FALLING BODIES. Accelerated Motion-Different bodies fall with equal Velocities-Laws of Descent as respects Velocities, Spaces, Times-Principle of Attwood's Machine-It verifies the laws of Descent-Resistance of the Atmosphere. OBSERVATION proves that the force with which a falling body descends depends upon the distance through which it has passed. A given weight falling through a space of an inch or two may give rise to insignificant results: but if it has passed through many yards those results become correspondingly greater. Gravity being a force continually in operation, a falling body must be under its influence during the whole period of its descent. The soliciting action does not take effect at the first moment of motion and then cease, but it continues all the time, exerting, as it were, a cumulative effect. The falling body may be regarded as incessantly receiving a rapidly-recurring series of impulses, all tending to drive it in the same direction. The effect of each one is therefore added to those of all its predecessors, and a uniformly accelerated motion is the result. Falling bodies are, therefore, said to descend with a uniformly accelerated motion. As the attraction of the earth operates with equal intensity on all bodies, all bodies must fall with equal velocities. A superficial consideration might lead to the erroneous conclusion that a heavy body ought to descend more quickly than a lighter. But if we have two equal masses, apart from each other, falling freely to the ground, they will evidently make their descent in equal times or with the same velocity. Nor will it alter the case at all if we imagine them to be connected with each other by an inflexible line. That line can in no manner increase or diminish their time of descent. The space through which a body falls in one second of time varies to a small extent in different latitudes. It is, however, usually estimated at sixteen feet and one-tenth. As the effect of gravity is to produce a uniformly accelerated motion, the final velocities of a descending body will increase as the times increase;. thus, at the end of two seconds, that velocity is twice as great as at one; at the end of three seconds three times as great; at the end of four, four times, and so on. Therefore the final velocity at the end The spaces through which the body descends in equal successive portions of time increase as the numbers 1.3.5.7, &c.! that is to say, as the body descends through sixteen feet and one-tenth in the first second, the subsequent spaces will be and these numbers are evidently as 1.3.5, &c. The entire space through which a body falls increases as the squares of the times. Thus, the entire space is, and these numbers are evidently as 1.4.9, &c., which are themselves the squares of the numbers 1.2.3, &c. If a body continued falling with the final velocity it had acquired after falling a given time, and the operation of gravity were then suspended, it would descend.in the same length of time through twice the space it fell through before relieved from the action of gravity. The following Table embodies the results of the three first laws : Times Final velocities. Space for each time Whole spaces 1.2.3.4.5.6.7, &c. 1.3.5.7.9.11.13. &c. It would not be easy to confirm these results by experiments directly made on falling bodies, the space described in the first second being so great (more than sixteen feet), and the spaces increasing as the squares of the times. There is an instrument, however, known as Attwood's machine, in which the force of gravity being moderated without any change in its essential characters, we are enabled to verify the foregoing laws. 2 A The principle of Attwood's machine is this. Over a pulley, A, Fig. 93, let there pass a fine silk line which suspends at its extremities equal weights, b c. These weights, being equally acted upon by gravity, will of course have no disposition to move; but now to one of the weights, c, let there be added another much smaller weight; these conjointly, preponderating over b, will descend, b at the same time rising. It might be supposed that the small additional weight, under these circumstances, would fall as fast as if it were unsupported in the air; but we must not forget that it has simul Fig. 93. taneously to bring down with it the weight to which it is attached, and also to lift the opposite one. By its gravity, therefore, it does descend, but with a velocity which is less in proportion as the difference between the two weights to which it is affixed is less than their sum. It gives us a force precisely like gravity-indeed it is gravity itself-operating under such conditions as to allow a moderate velocity. To avoid friction of the axle of the pulley, each of its ends rests upon two friction-wheels, as is shown at Q, Fig. 94; P is the pillar which supports the pulley. One of the weights is seen at b, the other moves in front of the divided scale, c d. This last weight is made to preponderate by means of a rod. There is a shelf which can be screwed opposite any of the divisions of the scale, and the arrival of the descending weight at that point is indicated by the sound arising from its striking. A clock, R, indicates the time which has elapsed. To enable us to fulfil the condition of suspending the action of gravity at any moment, a shelf, in the form of a ring, is screwed upon the scale at the point required. Through this the descending weight can freely pass, but the rod which caused the preponderance is intercepted. The equality of the two weights, is, therefore, reassumed, and the action of gravity virtually suspended. By this machine it may be shown that, in order that the descending weight shall strike the ring at intervals of 1, 2, 3, 4, &c., seconds, counting from the time at which its fall commences, the ring must be placed at distances from the zero of the scale, which are as the numbers 1, 4, 9, 16, &c.; and these are the squares of the times. And in the same manner may the other laws of the falling of bodies be proved. [Galileo himself made experiments regarding the free descent of bodies, which were subsequently repeated by Piccioli and Grimaldi from the tower of Degli Assinelli in Bologna. Dechalles has, however, made the most accurate observations on the subject. The spaces through which bodies fall are always smaller than we might be led to expect from theory. This difference depends, however, solely upon the resistance of the air, which increases in accordance with the square of the velocity. In the fallingmachine of Attwood, and the falling-channel, the resistance of the air does not influence the results.-Professor Müller on Physics and Meteorology, Lecture 10.] When a body is thrown vertically upward, it rises with an equably retarded motion, losing 323 feet of its original velocity every second. If in vacuo, it would occupy as much time in rising as in falling to acquire its original velocity, and the time expended in the ascent and descent would be the same. M. Forces which, like gravity, in this instance produce a retardation of motion, are nevertheless designated as accelerating forces. Their action is such that, if it were brought to bear on a body at rest, it would give rise to an accelerated motion. In rapid movements taking place in the astmosphere, a disturbing agency arises in the resistance of the air-a disturbance which becomes the more striking as the descending body is lighter, or exposes more surface. If a piece of gold and a feather are suffered to drop from a certain height, the gold reaches the ground much sooner than the feather. Thus, if in a tall air-pump receiver we allow, by turning the button, a, Fig. 95, a gold coin and a feather to drop, the feather occupies much longer than the coin in effecting its descent; and that this is due to the resistance of the air is proved by withdrawing the air from the receiver, and, when a good vacuum is obtained, making the coin and the feather fall again. It will now be found that they descend in the same time precisely, as shown in the above diagram. Fig. 95. Nor is it alone light bodies which are subject to this disturbance it is common to all. Thus, it was found that a ball of lead, dropped from the dome of St. Paul's Cathedral, in London, occupied 43 seconds in reaching the pavement, the distance being 272 feet. But in that time it should have fallen 324 feet, the retardation being due to the resistance of the air. It has been observed that the force of gravity is not the same on al parts of the earth. The distance fallen through in one second at the pole is 16.12 feet; but at the equator it is 16.01 feet. This arises from the circumstance that the earth is not a perfect sphere, its polar diameter being shorter than its equatorial, and, therefore, bodies at the poles are nearer to its centre than at the equator. Thus, in Fig. 96, let NS represent the globe of the earth, N and S being the north and south poles, respectively. Owing to its polar being shorter than its equatorial diameter, bodies situated at different points on the surface may be at very different distances from the centre, and the force of gravity exerted upon them may be correspondingly very different. Fig. 96. CHAPTER XX. MOTION OF INCLINED PLANES. Case of a Horizontal, a Vertical, and an Inclined Plane-Weight expended partly in producing pressure and partly motion-Law of Descent down Inclined Planes-Systems of Planes- Ascent up Planes Parabolic theory of Projectiles-Disturbing agency of the Atmosphere-Resistance to Cannon Shot-Ricochet-Ballistic Pendulum. WHEN a spherical body is placed on a plane set horizontally, its whole gravitation is expended in producing a pressure on that plane. If the plane is set in a vertical position the body no longer presses upon it, but descends vertically and unresisted. At all intermediate positions which may be given to the plane, the absolute attraction will be partly expended in producing a pressure upon that plane, and partly in producing an accelerated descent. The quantities of force thus relatively expended in producing the pressure and the motion will vary with the inclination of the plane: that portion producing pressure increasing as the plane becomes more horizontal, and that producing motion increasing as the plane becomes more vertical. Let there be a ball descending on the surface of an inclined plane, A B, Fig. 97, and let the line, de, represent its weight or absolute gravity. By the parallelogram of forces we may decompose this into two other forces, d f, and d g, one of which is perpendicular to the plane, and the other parallel to it. The first, therefore, is expended in producing pressure upon the plane, and the second in producing motion down it. The following are the laws of the descent of bodies down inclined planes: |