CENTRAL
, something relating to a centre. Thus we say central eclipse, central forces, central rule, &c.
Central Eclipse, is when the centres of the luminaries exactly coincide, and come in a line with the eye.
Central Forces, are forces having a tendency directly towards or from some point or centre; or forces which cause a moving body to tend towards, or recede from, the centre of motion. And accordingly they are divided into two kinds, in respect to their different relations to the centre, and hence are called centripetal, and centrifugal.
The doctrine of central forces makes a considerable branch of the Newtonian philosophy, and has been greatly cultivated by mathematicians, on account of its extensive use in the theory of gravity, and other physico-mathematical sciences.
In 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 external force or resistance should affect it, or act upon it. Hence, when a body at rest always tends to move, or when the velocity of any rectilinear motion is continnally accelerated or retarded, or when the direction of a motion is continually changed, and a curve line is thereby described, it is supposed that these circumstances proceed 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 moving; or by the ve- | locity gained or lost in the second case, or by the flexure of the curve described in the 3d ease: 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 of these kinds, which fall under our constant observation near the surface of the earth; 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 I. Newton has treated of central forces in lib. 1. sec. 2 of his Principia, and has demonstrated this fundamental theorem of central forces, viz, that the areas which revolving bodies describe by radii drawn to an immoveable centre, lie in the same immoveable planes, and are proportional to the times in which they are described. Prop. 1.
It is remarked by a late eminent mathematician, that this law, which was originally observed by Kepler, is the only general principle in the doctrine of centripetal forces; but since this law, as Newton himself has proved, cannot hold in cases where a body has a tendency to any other than one and the same point, there seems to be wanting some law that may serve to explain the motions of the moon and satellites which gravitate towards two different centres: the law he lays down for this purpose is, That when a body is urged by two forces tending constantly to two fixed points, it will describe, by lines drawn from the two fixed points, equal solids in equal times, about the line joining those fixed points, See Machin, on the Laws of the Moon's Motion, in the Postscript. See also a demonstration of this law by Mr. William Jones, in the Philos. Trans. vol. 59. Very learned tracts have also been since given, when the motion respects, not two only, but several centres, by many ingenious authors, and practical rules deduced from them for computing the places &c of planets and satellites; as by La Grange, De la Place, Waring, &c, &c. See Berlin Memoirs; those of the Academy of Sciences at Paris; and the Philof. Trans. of London.
M. De Moivre gave elegant general theorems relating to central forces, in the Philos. Trans. and in his Miscel. Analyt. pa. 231.—Let MPQ be any given curve, in which a body moves: let P be the place of the body at any time; S the centre of force, or the point to which the central force acting on the body is always directed; PG the radius of curvature at the point P; and ST perpendicular to the tangent PT; then will the centripetal force be everywhere proportional to the quantity SP/(GP X ST3). Vid. ut supra.
M. Varignon has also given two general theorems on this subject in the Memoirs of the Acad. an. 1700, 1701; and has shewn their application to the motions of the planets. See also the same Memoirs, an. 1706, 1710.
Mr. MacLaurin has also treated the subject of central forces very ably and fully, in his Treatise on Fluxions, art. 416 to 493; where he gives a great variety of expressions for these forces, and several elegant methods of investigating them.
Laws of Central Forces.
1. The following is a very clear and comprehensive rule, for which we are obliged to the marquis de l'Hôpital: Suppose a body of any determinate weight to revolve uniformly about a centre, with any given velocity; find from what height it must have fallen, by the force of gravity, to acquire that velocity; then, as the radius of the circle it describes is to double that height, so is its weight to its centrifugal force. So that, if b be the body, or its weight or quantity of matter, v its velocity, and r the radius of the circle described, also g = 16 <*>/12 feet; then, first 4g2 : v2 :: g : v2/4g the height due to the velocity v; and as the centrifugal force. And hence, if the centrifugal force be equal to the gravity, the velocity is equal to that acquired by falling through half the radius.
2. The central force of a body moving in the periphery of a circle, is as the versed sine AM of the indefinitely small arc AE; or it is as the square of that arc AE directly, and as the diameter AB inversely. For AM is the space through which the body is drawn from the tangent in the given time, and 2AM is the proper measure of the central force. But, AE being very small, and therefore nearly equal to its chord, by the nature of the circle
3. If two bodies revolve uniformly in different circles; their central forces are in the duplicate ratio of their velocities directly, and the diameters or radii of the circles inversely; that is F : f :: V2/D : v2/d :: V2/R : v2/r For the force, by the last article, is as | AE2/AB or AE2/D; and the velocity v is as the space AE uniformly described.
4. And hence, if the radii or diameters be reciprocally in the duplicate ratio of the velocities, the central forces will be reciprocally in the duplicate ratio of the radii, or directly as the 4th power of the velocities; that is, if V2 : v2 :: r : R, then F : f :: r2 : R2 :: V4 : v4.
5. The central forces are as the diameters of the circles directly, and squares of the periodic times inversely. For if c be the circumference described in the time t, with the velocity v; then the space c = tv, or v = c/t; hence, using this value of v in the 3d rule, it becomes F : f :: C2/DF2 : c2/dt2 :: D/T2 : d/t2 :: R/T2 : r/t2; since the diameter is as the circumference.
6. If two bodies, revolving in different circles, be acted on by the same central force; the periodic times are in the subduplicate ratio of the diameters or radii of the circles; for when F = f, then D/T2 = d/t2, and D : d :: T2 : t2, or T : t :: √D : √d :: √R : √r.
7. If the velocities be reciprocally as the distances from the centre, the central forces will be reciprocally as the cubes of the same distances, or directly as the cubes of the velocities. That is, if V : v :: r : R, then is F : f :: r3 : R3 :: V3 : v3.
8. If the velocities be reciprocally in the subduplicate ratio of the central distances, the squares of the times will be as the cubes of the distances: for if V2 : v2 :: r : R, then is T2 : t2 :: R3 : r3.
9. Wherefore, if the forces be reciprocally as the squares of the central distances, the squares of the periodic times will be as the cubes of the distances; or when F : f :: r2 : R2, then is T2 : t2 :: R3 : r3.
Exam. From this, and some of the foregoing theorems, may be deduced the velocity and periodic time of a body revolving in a circle, at any given distance from the earth's centre, by means of its own gravity. Put g = 16 1/12 feet, the space described by gravity, at the surface, in the first second of time, viz = AM in the foregoing fig. and by rule 2; then, putting r = the radius AC; it is the velocity in a circle at its surface, in one second of time; and hence, putting c = 3.14159 &c, the circumference of the earth being 2cr = 25,000 miles, or 132,000,000. feet, it will be √(2gr) : 2cr :: 1″ : c√ 2r/g = 5078 seconds nearly, or 1h 24m 38, the periodic time at the circumference: Also the velocity there, or √(2gr) is = 26000 feet per second nearly. Then, since the force of gravity varies in the inverse duplicate ratio of the distance, by rules 8 and 9, it is √R : √r :: v or the velocity of a body revolving about the earth at the distance R; and √r3 : √R3 :: t or 5078″ : 5078 √(R3/r3) = T the time of revolution in the same. So if, for instance, it be the moon revolving about the earth at the distance of 60 semidiameters; then R = 60r, and the above expressions become V = 26000√(1/60) = 3357 feet per second, or 38 1/7 miles per minute, for the velocity of the moon in her orbit; and T = 5078√(R3/r3) = 2360051 seconds or 27 3/10 days nearly, for the periodic time of the moon in her orbit at that distance.
Thus also the ratio of the forces of gravitation of the moon towards the sun and earth may be estimated. For, 1 year or 365 1/4 days being the periodic time of the earth and moon about the sun, and 27 3/10 days the periodic time of the moon about the earth, also 60 being the distance of the moon from the earth, and 23920 the distance from the sun, in semidiameters of the earth, by art. 5 it is ; that is, the proportion of the moon's gravitation towards the sun, is to that towards the earth, as 2 2/9 to 1 nearly.
Again, we may hence compute the centrifugal force of a body at the equator, arising from the earth's rotation. For, the periodic time when the centrifugal force is equal to the force of gravity, it has been shewn above, is 5078 seconds, and 23 hours, 56 minutes, or 86160 seconds, is the period of the earth's rotation on its axis; therefore, by art. 5, as 861602 : 50782 :: 1 : 1/289, the centrifugal force required, which therefore is the 289th part of gravity at the earth's surface. Simpson's Flux. pa. 240, &c.
Also for another example, suppose A to be a ball of 1 ounce, which is whirled about the centre C, so as to describe the circle ABE, each revolution being made in half a second; and the length of the cord AC equal to 2 feet. Here then t = 1/2, r = 2, and it having been sound above that c√(2R/g) = T is the periodic time at the circumference of the earth when the centrifugal force is equal to gravity; hence then, by art. 5, as R/T2 : r/t2 :: F or 1 : f, which proportion becomes = the centrifugal force, or that by which the string is stretched, viz, nearly to ounces, or 10 times the weight of the ball.
Lastly, suppose the string and ball be suspended from a point D, and describes in its motion a conical surface ADB; then putting DC = a, AC = r, and AD = h; and putting F = 1 the force of gravity as before; then will the body A be affected by three forces, viz, gravity acting parallel to DC, a centrifugal force in the direction CA, and the tension of the string, or force by which it is stretched, in the direction DA; hence these three powers will be as the three sides of the triangle ADC respectively, and therefo re as CD or a : AD or h :: 1 : h/a the tension of the string as compared with the weight of the body. Also AC or a : AC or r :: 1 : (2c2r)/(gt2) the general expression for the centrifugal force above-found; | hence, gt2 = 2ac2 and so t = c√(2a)/g = 1.108√a = the periodic time. And
10. When the force by which a body is urged towards a point is not always the same, but is either increased or decreased as some power of the distance; several curves will thence arise according to that power. If the force decrease as the squares of the distances increase, the body will describe an ellipsis, and the force is directed towards one of its foci; so that in every revolution the body once approaches towards it, and once recedes from it; also the eccentricity of the ellipse is greater or less, according to the projectile force; and the curve may sometimes become a circle, when the eccentricity is nothing; the body may also describe the other two conic sections, the parabola and hyperbola, which do not return into themselves, by supposing the velocity greater in certain proportions. Also if the force increase in the simple ratio as the distance increases, the body will still describe an ellipse; but the force will in this case be directed to the centre of the ellipse; and the body, in each revolution, will twice approach towards it, and again twice recede from that point.
Central Rule, is a rule or method discovered by Mr. Thomas Baker, rector of Nympton in Devonshire, which he published in his Geometrical Key, in the year 1684, for determining the centre of a circle which shall cut a given parabola in as many points as a given equation, to be constructed, has real roots; which he has applied with good success in the construction of all equations as far as the 4th power inclusive.
The Central Rule is chiefly founded on this property of the parabola; that if a line be inscribed in the curve perpendicular to any diameter, the rectangle of the segments of this line, is equal to the rectangle of the intercepted part of the diameter and the parameter of the axis.
The Central Rule has the advantage over the methods of constructing equations by Des Cartes and De Latteres, which are liable to the trouble of preparing the equations by taking away the second term; whereas Baker's method effects the same thing without any previous preparation whatever. See also Philos. Trans. N° 157.
