(fig. 8) would serve to explain the motion of the earth, or any other planet, in a more or less circular path about the sun as a centre. The force acting in each case towards the sun, and changing the course of the planet from a straight line, is the force of gravity. EXERCISES. 1. Cohesion and gravity are sometimes regarded as the same force acting under somewhat different circumstances. Point out in what respects they resemble each other, and in what they differ. 2. A piece of wood is pushed down to some depth below the surface of a lake, and then released. Explain why the wood rises instead of falls. 3. Mention any facts you know to prove that gases are acted on by gravity. 4. Give some examples of gravity producing and destroying motion respectively. 5. Describe some simple experiments to prove that bodies supported so as to be unable to fall are still acted on by gravity. 6. How would you proceed to test a wall or other structure to ascertain if it were vertical? LESSON 12. THE LAW OF GRAVITATION. Although the force of gravity acts everywhere, and on all bodies, its magnitude or intensity is not the same in all cases. The law according to which this variation takes place was discovered and proved by Newton. He proved that the intensity of the mutual gravitation (or tendency to approach each other) of two bodies depends upon two things, viz.—the masses of the two bodies, and their distance from each other. The greater the masses of the bodies, the stronger is their mutual gravitation; the greater the distance by which they are separated, the weaker is their tendency to move towards each other. Gravitation is a property, not of heavy bodies merely, but of every body; and it is not a property of a body as a whole, but rather a property of each particle entering into the composition of the body. Every particle of matter in the universe attracts every other particle. Every particle of matter composing the earth attracts every particle of the moon or of a falling stone. What we usually regard as a single force acting on the moon or the stone, is the resultant of all these numberless small attractions, acting apparently on some one point, situated in a more or less central position in the body. Newton proved that for bodies spherical in shape, the resultant of these minute forces was a force equal to their sum, and acting from the centre of the sphere. The shape of the earth and of the moon, as well as of the sun and the planets generally, is so nearly that of a sphere, that this rule may be considered as practically true for all such bodies. The total action of all the particles composing the earth upon the moon, or a stone, is the same as if all these particles THE LAW OF GRAVITATION. 35 were collected together in one point at the centre of the earth. This will explain why bodies falling freely move in the direction of the centre of the earth, and why a string supporting a weight is pulled in such a direction. And in measuring the distances between two bodies, in order to ascertain over what distance gravity has to act, it is the distance between the centres of the bodies which must be measured. Thus the distance over which the resultant action of the earth is exerted on a stone near its surface, is about 4,000 miles: that is, equal to the radius of the earth. There is in every body a point at which the resultant of the attractive forces, acting on the particles of the body, may be considered as acting; and this point is called the centre of gravity of the body. This point has several important properties, to the consideration of some of which we shall devote the next lesson. It will readily be understood that as gravity acts upon the ultimate particles of which bodies are composed, the greater the quantity of matter in a body, the greater will be the resultant effect of gravity upon the body. This is proved by the simple fact that the weight of a body depends upon its mass; that if, for example, a body be weighed, and then a portion of it cut away, the weight of the remaining portion will be less in exactly the same proportion as the mass is less than before. The mutual gravitation of two bodies having masses of 1 and 2 respectively, would be 2 (that is, 1×2) as compared with the gravitation of two other bodies of masses 2 and 4, the gravitation of which would be represented by 8 (that is, 2× 4), or four times as great as in the former case, provided, of course, that the distance was the same in each case. In all cases of attraction-magnetic, electrical, &c.—the intensity decreases as the distance between the bodies is increased. And just in the same way the heat from a fire, or the light from a lamp, grows weaker as we recede from its source. The same is the case with gravity. If this be so, the weights of bodies, and the rates at which they fall, ought to be less as we go farther from the earth's surface; and there can be no doubt that such is the case. But the difference is so small at the different places to which we have access, that we cannot perceive it by ordinary means. We have explained that the force of gravity, acting between the earth and a stone on the earth's surface, must be considered as acting from the centre of the earth, a distance of about 4,000 miles. Compared with this distance the height of a room, of a house, or of a hill is so exceedingly small, tha' the weakening of the gravitation will be also exceedingly small. According to Newton's law of gravitation, if a body were removed to a distance of 4,000 miles above the earth's surface (that is, 8,000 miles from the centre of the earth) the force of gravity on it (and therefore its weight) would only be one quarter of what it was at the earth's surface. If the distance separating the body from the centre of the earth were increased to three times what it originally was, the effect of gravity on it would be only one-ninth part of its former amount. It will be observed that gravitation does not decrease in the same proportion as the distance is increased, but in the proportion of the square of the distance. At six times the distance, gravitation is reduced to th part of its former amount; at of the distance, to 16 times its former amount. The law of gravitation may now be stated generally as follows:-Every particle of matter in the universe attracts every other particle with a force proportional to the product of their masses, and varying inversely as the square of their distance from each other. Newton sought to prove the truth of this law by showing that, if gravitation varied as stated therein, the motion of the moon in its orbit could be explained. The moon is about 240,000 miles from the centre of the earth, that is to say, is 60 times as far away from the earth's centre as a body on the surface of the earth is. Gravitation at the distance of the moon should therefore be only the th part (that is, 60 x 60 times less) of its amount at the earth's surface. Newton showed that if gravity had this intensity at the distance of the moon, it would cause the latter to move out of a straight line, and towards the earth, about 15 feet in a minute, which is the amount required to account for the moon's motion round the earth. By means of this law of gravitation, it is possible for an astronomer to calculate by how much a comet will be deflected from its course by the attraction of a planet. Knowing the mass of the sun and the distance which a body on its surface will be from the centre, it is possible to calculate the weight which a body would have on the sun, compared with what it has on the earth. The weight of a body on the sun would, we are told, be 28 times as great as on the earth, while on the moon it would only be one-sixth as great as on the earth. This law also explains why the tide raised by the moon is more than twice as great as that raised by the sun, the smaller distance between the earth and moon more than making up for the enormously greater mass of the sun. Since the time of Newton it has been several times proved by direct experiment that all bodies have a tendency to gravitate towards each other. And it is only because the mass of the earth is so enormously great compared with the masses of bodies on its surface, and therefore its attraction so much greater than theirs, which prevents us from more readily observing this general property of bodies to gravitate towards one another. It has been recently shown that when a large mass of lead is brought close under a small body hung from a very delicate balance, the body weighs slightly more than when the lead is removed; thus proving an attraction between the body and the lead. It has also been proved that a plumb-line does not hang in an exactly vertical line near a mountain, but that the weight at the end of the string is drawn slightly towards the mountain, by the mutual attraction between them. EXERCISES. 1. Find the weight of a mass of iron at a distance of 4,000 miles above the surface of the earth, which on the surface weighs one hundred-weight CENTRE OF GRAVITY. 37 2. Why should a body on the surface of the moon weigh less than the same body on the surface of the earth? What would be the weight on the moon's surface of a body which on the surface of the earth weighed 3 cwt. ? 3. If the tide raised by the moon's action alone is 2 times as great as that raised by the sun alone, compare the weight of the tide raised by the two bodies acting in conjunction with each other (spring tides) with that of the tide raised by them when acting against each other (neap tides). 4. A pendulum was allowed to swing for a certain time at the top of a mine, and then for the same length of time at the bottom, and the number of swings in each case were counted. How could it be ascertained from these experiments whether the intensity of gravity was different in the two places? 5. Why does a piece of lead weigh more than a piece of wood the same size? If both bodies were taken to a place where the intensity of gravity was less than in England, which would lose the more in weight? LESSON 13. CENTRE OF GRAVITY. We have already explained the centre of gravity of a body to be that point at which the resultant of the several forces, due to the action of gravity on the particles of the body, may be supposed to act. In the case of a thin rod of uniform thickness and density, the middle point of the rod will obviously be the centre of gravity. In experimenting with such a rod-a wooden ruler or lead pencil, for example-it will be found that the rod will balance on the finger if the latter be placed under the middle point of the rod, but that it will not balance in any other position. If a string is tied round the middle part of the rod and held up, the rod will hang balanced horizontally. But if the string is attached to any other part of the rod, the latter will not rest horizontally but will incline more or less until the middle point or centre of gravity is vertically below the string. And if any body whatever be suspended by a string and allowed to hang freely from the point of suspension, it will be found that there is one position, and one only, in which the body will hang steadily, and that if it be displaced from this position it will return to it again. And if the position of the centre of gravity of the body is known, it will be seen when the body is at rest to be vertically under the string; or, in other words, a straight line drawn in continuation of the direction of the string would pass through the centre of gravity of the body. It appears, therefore, that in order to prevent a body from falling, its centre of gravity must be supported either from above or below. A simple practical method of determining the position of the centre of gravity of a body is founded on the principle just stated. For example, suppose it is required to find the centre of gravity of an irregularly shaped piece of cardboard (fig. 9). Suspend it from 4 с any point a by means of a piece of string, or by a pin through the card, so as to let it freely take its natural position. Then from the point of suspension a draw a vertical line acb on the card; the direction of which can be most readily ascertained by means of a small plumb-line, e.g., a small weight hung to the end of a piece of string. This line will pass through the centre of gravity of the cardboard. Then suspend it from any other point, such as d, and again draw a vertical line dce from the point of suspension, which line will also pass through the centre of gravity. The centre of gravity being in both these lines, can only be at the point c, where they cross. And it will be found that from whatever point the body is suspended, a vertical line drawn from the point of suspension will always pass through the point c. It need scarcely be pointed out that the centre of gravity will not be on the surface of the body, but mid-way between the two faces of the cardboard. Fig. 9. The centre of gravity of regularly shaped bodies, of uniform thickness and material, can frequently be most readily found by geometrical construction. Thus the centre of gravity of a circular disc, or a sphere, is at the centre of each; of a square or oblong, at the intersection of the diagonals. If a hole were made through the cardboard, represented in fig. 9, at the centre of gravity c, and a smooth knitting needle inserted and held horizontally, it would be found that the card would remain in equilibrium on the needle in any position. But if the needle were inserted through a hole made in any other part of the card, such as a or d, the card would only remain in equilibrium in two positions, viz., when the centre of gravity was directly above, or directly below, the point of support. When the card was balanced with its centre of gravity vertically above the point of support, its equilibrium would be very unstable or unsteady; the least displacement to the one side or the other of its position of equilibrium would cause it to fall down into the more steady or stable position of equilibrium, where its centre of gravity would be vertically below the point of support. After any displacement from this latter position the body returns to the same position again when released. A body is said to be in stable equilibrium when, after displacement, it returns to its original position, with its centre of gravity as low as possible. A body is said to be in unstable equilibrium when, after a slight displacement, it does not recover its position, but moves into some more stable position, with its centre of gravity lower than before. A body is said to be in neutral equilibrium when, like the cardboard with the needle through its centre of gravity, it remains in equilibrium in any position. The centre of gravity of a body always tends to get to its lowest position; and a body will be in stable or unstable equilibrium ac |