CENTRAL forcer, the powers which cause a moving body to tend towards, or recede from, the center of motion.
If a body A (plate III. Miscel. fig. 10.) be suspended at the end of a string A C, moveable about a point C, as a centre, and in that position it receive an impulse in an horizontal direction, it will be there by compelled to describe a circle about the central point. While the circular mo tion continues, the body will certainly en deavour to recede from the center, which is called its centrifugal force, and arises from the horizontal impetus. With this force it acts upon the fixed center-pin, and that, by its immobility, re-acts with an equal force on the body, by means of the string, and solicits it towards the center of motion ; whence it is called the centripetal force ; and when we speak of either or both indefinitely, they are called the central ffirces of the revolving body.
The doctrine of central forces makes a considerable branch of the Newtonian philosophy, and has been greatly cultiva ted by mathematicians, on account of its extensive use in the theory of gravity, and other physical and mathematical sciences.
Tn this doctrine it is supposed, that matter is equally indifferent to motion or rest ; or that a body at rest never moves itself; and that a body in motion never of itself changes either the velocity or the direction of its motion ; but that every motion would continue uniformly; and its direction rectilinear, unless some exter nal force or resistance should affect it, or act upon it. Hence, when a boclrat rest always tends to move, or when the veloci ty. of any rectilinear motion is continually accelerated or retarded, or when the di rection of a motion is continually changed, and a curve line is thereby described, it is supposed that these circumstances pro ceed from the influence of some power that acts incessantly ; which power may be measured, in the first case, by the pressure of the quiescent body against the obstacle which prevents it from mov ing, or by the velocity gained or lost in the second case, or by the flexure of the curve described in the third case ; due regard being had to the time in which these effects are produced, and other circumstances, according to the principles of mechanics. Now the power
or force of gravity produces effects of each these kinds, which fall under our constant observation near the surface of the earths for the same power which renders bodies heavy, while they are at rest, accelerates their motion when they descend perpendicularly ; and bends the track of the motion into a curve line, when they are projected in a direction oblique to that of their gravity. But we can judge of the forces or powers that act on the celestial bodies by effects of the last kind only. And hence it is, that the doctrine of central forces is of so much use in the theory of the planetary motions.
Sir Isaac Newton has tr ited of central forces in his Principia, and has demon strated this fundamental theorem, viz. that the areas which evolving bodies describe, by radii drawn to an immoveable centre, he in the same immoveable planes, and are proportional to the times in which they are described.
The theory of this species of motion is comprised in the following propositions. 1 When two or more bodies revolve at equal distances from the center of the circle they describe, but with unequal velocities, the central forces necessary to retain them will be to each other as the squares of their velocities. That is, if one revolves twice as fast as the other, it will require four times the retaining force the other does ; if with three times the ve locity, it will require nine times the force to retain it in its orb, &c.
2. When two or more bodies move with equal velocities, but at unequal distances from the center they revolve about, their central forces must be inversely as their distances. That is, by how many times greater the distance a body revolves at is from the center, so many times less force will retain it.