This equation, which is the general mathematical expression for the law of gravitation, may be made to yield some curious results. Thus, if we select two bullets, each having a mass of 1 gram, and place them so that their centers are 1 centimeter apart, the above expression for the force exerted between them becomes
F = k {(1 × 1)/12} = k,
from which it appears that the coefficient k is the force exerted between these bodies. This is called the gravitation constant, and it evidently furnishes a measure of the specific intensity with which one particle of matter attracts another. Elaborate experiments which have been made to determine the amount of this force show that it is surprisingly small, for in the case of the two bullets whose mass of 1 gram each is supposed to be concentrated into an indefinitely small space, gravity would have to operate between them continuously for more than forty minutes in order to pull them together, although they were separated by only 1 centimeter to start with, and nothing save their own inertia opposed their movements. It is only when one or both of the masses m', m'' are very great that the force of gravity becomes large, and the weight of bodies at the surface of the earth is considerable because of the great quantity of matter which goes to make up the earth. Many of the heavenly bodies are much more massive than the earth, as the mathematical astronomers have found by applying the law of gravitation to determine numerically their masses, or, in more popular language, to "weigh" them.
The student should observe that the two terms mass and weight are not synonymous; mass is defined above as the quantity of matter contained in a body, while weight is the force with which the earth attracts that body, and in accordance with the law of gravitation its weight depends upon its distance from the center of the earth, while its mass is quite independent of its position with respect to the earth.
By the third law of motion the earth is pulled toward a falling body just as strongly as the body is pulled toward the earth—i. e., by a force equal to the weight of the body. How much does the earth rise toward the body?
38. The motion of a planet.—In [Fig. 20] S represents the sun and P a planet or other celestial body, which for the moment is moving along the straight line P 1. In accordance with the first law of motion it would continue to move along this line with uniform velocity if no external force acted upon it; but such a force, the sun's attraction, is acting, and by virtue of this attraction the body is pulled aside from the line P 1.
Knowing the velocity and direction of the body's motion and the force with which the sun attracts it, the mathematician is able to apply Newton's laws of motion so as to determine the path of the body, and a few of the possible orbits are shown in the figure where the short cross stroke marks the point of each orbit which is nearest to the sun. This point is called the perihelion.
Without any formal application of mathematics we may readily see that the swifter the motion of the body at P the shorter will be the time during which it is subjected to the sun's attraction at close range, and therefore the force exerted by the sun, and the resulting change of motion, will be small, as in the orbits P 1 and P 2.
On the other hand, P 5 and P 6 represent orbits in which the velocity at P was comparatively small, and the resulting change of motion greater than would be possible for a more swiftly moving body.
What would be the orbit if the velocity at P were reduced to nothing at all?