Popular Science Monthly/Volume 11/May 1877/Gravitation, and How it Works
By GRANVILLE F. FOSTER.
"The force of gravity acts on bodies directly in proportion to the quantity of matter in each." "The force of gravity decreases in the reciprocal proportion of the square of the distance."—(Orr's "Circle of the Sciences," vol. vi., p. 1.)
AMONG students of natural philosophy no facts are more frequently misunderstood than those pertaining to the laws of gravitation. It is readily admitted that if a body A exerts on B a certain force of attraction, if A's mass be doubled, then will A's attractive influence on B be doubled also, but the fact is not so apparent that any two bodies, whatever their disparity of mass, or however great their distance apart, will attract each other with precisely equal forces; and that if, for instance, the mass of A be doubled, not only will A's attraction for B be doubled, but at the same time B's attraction for A will be doubled also. The pen I hold in my hand attracts the sun with precisely the same amount of force that the sun attracts the pen, and, if either the mass of the pen or sun be doubled, the mutual attraction will be doubled also. The first law of gravitation most certainly teaches that the earth, so insignificantly small as compared with the sun, both in volume and mass, attracts the sun with a force exactly equal to that which, being by the sun exerted on itself, reduces it to obedience, and compels it to make its annual revolution. So, too, the moon and the earth mutually and equally attract each other.
The fact that the forces of attraction between two bodies are equal may be easily explained as follows: Let there be five bodies. A, B, C, D, E, and let A be so situated as to be at equal distances from the other four: then it is evident that the forces which measure the mutual attractions of (A and B), (A and C), (A and D), and (A and E), are equal. Calling the force which A exerts on B, or B exerts on A, one, then will the sum of the forces which B, C, D, and E exert on A be equal to four, but the sum of A's attractions for B, C, D, and E, will also be equal to four, since A's attraction for B is in no way either increased or diminished by the fact that at the same time it also exerts an attraction on C, D, or E. Now, let B, C, D and E, be united into one mass, F, and it will be readily perceived that the truth of the foregoing statements cannot thereby be affected.
As a general formula the law of gravitation may be enunciated as follows: "If one of the masses contain m units of mass, and the other one unit, the force will be m times as great as though they were both units of mass; but if the second body contain n units of mass, the attraction will be n times as great as before; that is, m n divided by the square of the distance between the bodies."
Now, suppose A and F free to move, then on meeting A will have moved over four-fifths of the distance between A and F, and F during the same time will have moved over one-fifth of the same distance; that is to say, the velocity of A has just been equal to four times that of F, and this is just what might have been expected from what is known of the laws of force. Suppose A and F to be placed where friction and other obstacles to motion do not exist, the velocities of the bodies will be indirectly as their masses, if the respective forces exerted on the bodies be equal; that is, a force which would propel a body with a certain velocity would propel another body one-quarter of the mass of the former with four times the velocity. In the case supposed, since A is one-quarter of the mass of F, a given force must necessarily move A over four times the space and with four times the velocity that it is able to move F, and when A and F meet the momenta of A and F will be respectively equal.
The truth that two bodies mutually and equally attract each other is also abundantly proved in astronomy. Take the case of the earth and moon. The earth by its attraction compels the moon to make around it as a centre her monthly revolution; but it is equally true that the moon compels the earth to move around the centre of gravity of the earth and moon, which centre, on account of the earth's mass being over eighty times that of the moon, is distant from the earth's centre a little over 2,000 miles, and this motion of the earth is performed in precisely the time of the lunar revolution, namely 27 3 days. Now, it will require but little reflection to perceive that to move the earth in a circle with a radius of a little more than 2,000 miles, and the moon in a circle with a radius of nearly 240,000 miles, would require equal forces. The same thing is true of the sun, which is obliged by the combined forces of the planets to revolve around the centre of gravity of the solar system, and on making the necessary calculations we find that the force exerted on the planets by the sun just equals the force exerted by the planets on the sun.
Weight has been defined as the measure of the earth's attraction. A body weighing one pound attracts the earth and is attracted by it with a force of one pound, but the same body at the sun's surface would attract the mass of the sun with a force of twenty-seven pounds, since its weight has been increased twenty-seven times by the sun's attraction.
We have hitherto considered the mutual attraction of two bodies, but now let a third be introduced, as, for instance, in the case of A and F, let G be placed at equal distances from A and F, and let the relative masses of A and F be as stated before in this paper: then will the force which measures the mutual attraction of F and G be equal to four times the force which measures the mutual attraction of G and A, or, in other words, F will attract G with four times the force that A will attract G. Lastly, let G's mass equal A's mass, and let G be placed at double the distance from F that A has been placed: then, according to the second law of gravitation, the units of force which measure the mutual attraction of A and F will be four times the force which measures the mutual attraction of G and F.