CENTRAL FORCES are those which cause a moving body to tend towards some point or center, called the center of force or motion. The doctrine of C. F. has for its starting-point the first law of motion—viz., that a body not acted on by any external force will remain at rest, or move uniformly in a straight line. It follows from this law that if a body in motion either changes its velocity or direction, some external force is acting upon it. The doctrine of C. F. considers the paths which bodies will describe round centers of force, and the varying velocity with which they will pass along in these paths. It investigates the law of the force round which a body describes a known curve, and solves the inverse problem, and many others, the general statement of which could convey no clear idea to the unmathematical reader. As gravity is a force which acts on all bodies from the earth's center, it affords the simplest general illustration of the action of a central force. If a stone be slung from a string, gravity deflects it from the linear path which it would otherwise pursue, and makes it describe a curved line which we know would, in vacuo, be a parabola. Again, the moon is held in her orbit round the earth by the action of gravity, which is constantly preventing her from going off in the line of the tangent to her path at any instant, which she would do, according to the first law of motion, if not deflected therefrom by any external force. To that property of matter by which it maintains its state of rest or motion, unless acted upon by other matter, has been given the name inertia.
We will now explain how, through the action of a central force, a body is made to describe a curved path. Suppose it to have moved for a finite time, and conceive the time divided into very small equal parts; and instead of the central force acting constantly, conceive a series of sudden impulses to be given to the body in the direction of the center, at the end of each of the equal intervals, and then observe what, on these suppositions, will happen. Let S (see Fig. 1) be the center, and let the original motion be from A, ou the line AB, which does not pass through S. In the first interval, the
body will move with a uniform velocity, say from A to B. In the second, if acted on by no force, it would move on in•AB pro duced to c, Bc being = AB. But when it arrives at B, it receives the first sudden im pulse towards S. By the composition ofvelocities (q.v.), it will move now with a new ...4 .
but still uniform velocity in BC instead of Be, BC being the diagonal of the parallelo gram of which the sides represent its im pressed and original velocity. Having reached C at the end of the second interval, it receives the second impulse towards S. A It will now move in CD instead of in BC produced. If, then, we suppose the periods Flg. 1.
of time to be indefinitely diminished in length, and increased in number, the broken line ABCD will become ultimately a continuous curve and the series of impulses a continuous force. This completes the explanation.
Going back, however, on our suppositions, we may here establish Newton's leading law of central forces. That the body must always move in the same plane, results from the absence of any force to remove it from the plane in which at any time it may be moving. The triangles ASB and BSC are clearly in the same plane, as the latter is on that in which lie the lines Bc and BS. Also, since the triangles ASB, BSc are equal, being on equal bases, AB, Be, and triangle BSC = triangle BSc, as they are between the same parallels, cC and BS, it follows (by Euclid I. 3') that ASB = BSC. So BSC = CSD; and so on. In other words, the areas, described in equal times by the line (called the radius vector) joining the center of force and the body, are equal. As this is true in the limit, we arrive, by the composition of the small equal areas, at the law: that the areas described by the lines drawn from the moving body to the fixed cen ter of force, are all in one plane, and proportional to the times of describing them. Very few of the laws of C. F. are capable of being proved like the preceding, without drawing largely on Newton's lemmas, with which we shall not suppose the reader to be acquainted.