certain amount of swing. Well, the pull between the earth and the stone soon sets the stone in quick motion; but the mass of the earth is comparatively so vast that the effect of the pull on it in producing motion is virtually nothing. 12. Gravitation affected by Distance.-If mass alone determined the attractive power of bodies, we might expect to see loose bodies near the earth fall towards the sun, whose mass is hundreds of thousands of times greater than that of the earth. But the earth prevails in virtue of its nearness. Our experience of nature generally would lead us to expect that the force of gravity would be lessened by distance, and we should be ready to conclude that the decrease would be in proportion to the distance that at twice the distance the force would be one half. It is found, however, that the rate is more rapid; that at twice the distance, the force is only th; at thrice the distance, it is only 4th, and so on; or generally, that the force of gravity diminishes as the square of the distance. It is necessary to observe here that the distance between bodies in the matter of gravity is counted, not from the nearest points, but from their centres or middle points. The distance of the globe A from B, is not ss but cc'. The reason is, that as the several particles of A are at different distances from any part of B, their united effect will depend on their mean distance. When the body is of regular shape, such as a globe, or a cube, and of uniform density, the mean distance will be that of the central point. 13. Bodies become lighter at a distance from the Earth.-According to the law stated in par. 12, bodies must become lighter as they are removed farther and farther from the earth. Let Fig. 1. B the circle (fig. 2) represent a section of the earth. Prolong the radius CA, making AB, BD, &c. each equal to CA (which in round numbers = 4000 miles). Then the weight of a body at B is to its B weight at A as 40002: 80002 11 16,000,000 64,000,000 1 : 4 at B; Hence 1 lb. at A becomes Fig. 2. or 40052 40002 16,040,025 : 16,000,000; that is, a little more than of its weight at the surface. In experiments of this kind the ordinary balance with scales and weights is of no avail, as the weights are equally affected with the thing to be weighed; a spring balance must be used. EXERCISE.-The polar diameter of the globe is 41,708,710 feet, and the equatorial diameter 41,848,380 feet; if a body weighing a ton at the equator were carried to the pole, what would it weigh? 14. The Law of Falling Bodies.-Gravity not only puts a body in motion, but continues to act on it with equal force after it is in motion, and thus is constantly adding to its speed. As the force of gravity continues unabated, it must give an equal addition to the speed on each successive second, and thus the motion is uniformly accelerated. The locomotive continues to urge the moving train; why is the speed not uniformly accelerated, as well as that of the falling stone? Let the student think out the answer for himself. All bodies, heavy and light, would fall equally fast, if the resistance of the air were removed. A guinea and a feather dropped in a vacuum fall to the bottom together. 15. Rate of Acceleration.-Observation tells us that when a body begins to fall from a state of rest, it descends 16 feet 1 inch in the first second of time. Suppose that no fall had ever been observed beyond that point, and that we had to find by reasoning what the body would do during the next second. The first consideration is, How far would the body fall, if gravity were to cease-in other words, What velocity has it acquired? It would not be unnatural to say 16 feet (we omit the 1 inch), since that is the space it fell in the second that is ended; but reflection shows that this cannot be the case. During the first half of the second, it made very little way, and the greater part of the distance was passed over in the last half; 16 feet expresses the mean velocity, or the velocity it had at the middle of the second, and at the end it must have been moving with a velocity just double the mean, or 32 feet. The velocity acquired, then, at the end of the first second is 32 feet; and if gravity were to cease, the body would, by the law of inertia, move over 32 feet in the next second. But gravity still acting, will make it fall another 16 feet in addition to the 32, or 48 feet in all; and will create as much velocity in addition as it created in the first second, so that if it were to cease at the end of the second second, the body would move 64 feet in the third second. Thus, during the second second, the body will fall through three spaces of 16 feet, and at the end of it will have its velocity double of what it was at the end of the first second. The whole space fallen during the two seconds will thus be four spaces of 16 feet. We arrive at this result by reasoning, and observation proves it correct. Without giving the steps of the investigation for the succeeding spaces of time, the results may be exhibited in the following table: The distances that bodies fall, then, do not increase simply as the times, but as the squares of the times. To find, therefore, how far a body will fall in any number of seconds, multiply the number of seconds by itself, and that product by 16. Thus in 7 seconds, a body will fall 7 x 7 x 16 784 feet. The height of a precipice might be roughly measured in this way, by observing how many seconds a stone takes to reach the bottom. = 16. Formulæ for Calculation.-The following formulæ are useful in many physical and mechanical problems : Putting h for the height or space fallen through in any number of seconds, v for the velocity at the end of that time, t the number of seconds, and g for 323, the velocity caused by gravity in one second, the relations of these quantities are expressed in the following formulæ, which enable us to calculate any one from the others: 16× t2, or (1) h = } g × t2 ; (2) v = = gxt and .. t = v If in formula (1) we substitute for t2 its value in (3)— 17. Gravitation extended from the Earth to the Heavenly Bodies.-When a body is made to move in a circle, it has a tendency at every point to proceed in a straight line, or fly off at a tangent, as it is called, and requires a constant force pulling it towards the centre, to keep it in its course. By observing the motions of the planets about the sun, astronomers, previously to the discoveries of Newton, had been led to believe that the planets are held in their paths by a central force proceeding from the sun, but how the force acted was not agreed upon. It was Newton who first showed that this force was identical with that which makes a stone fall to the ground. When the idea first occurred to him, it was only a conjecture, or hypothesis; but he proceeded to verify it, or put it to the test, in regard to the moon; and in this wise: mean 18. Hypothesis tested by the Moon.-The distance of the moon is sixty times the earth's radius. The force of the earth's attraction at that distance must therefore be 60 x 60 times less than at the surface of the earth, so that a body will fall in a second only the part of 16 feet, or about of an inch. If, then, it is the earth's attraction that keeps the moon in her orbit, her motion in one second must be drawn from the straight line by this amount, neither more nor less. Now we have the means of measuring whether this is the fact or not. Let the circle in fig. 3 represent the orbit moon, E being the earth and M the moon. of the The moon |