Centrifugal and Centripetal Force.—We have shown that a body continually drawn t6 a center, if it has an original motion in a line that does not pass through the center, will describe a curve. At each point in the curve, it tends, through its inertia,' to recede from the curve, and proceed in the tangent to it at that point. It always tends to move in a straight line in the direction in which it may at any time be moving, and that line, by the definitions of a tangent and of curvature, is A. D the tangent to the curve'at the point. At the point A (see fig. 2), Art it will endeavor to proceed in AD: if hindered it, it would actually proceed in that line, so as, in the time in which it describes the arc of the curve AE, to reach the point D, and thus recede the length DE from the curve; but being continually drawn out of its direction into a curve by a force to a center, it falls below the point D by the distance DE. The force which Fig. 2. draws it through this distance is called the centripetal force, and that which would make it recede in the same time through the distance DE from the curve is called the centrifugal force. It may be remarked that the centrifugal force is not, like the centripetal, an impressed or external force act ing on the body. It is simply the assertion of the body's inertia under the circumstances produced by the centripetal force.
Many familiar illustrations will occur to the reader of the action of what is called the centrifugal force. A ball fastened to the end of a string, and whirled round, will, if the motion is made sufficiently rapid, at last break the string, and fiv off. A glass of water may be whirled so rapidly that, even when the mouth is pressed downwards, the water will still be retained in it, by the centrifugal force pressing it up against the bot tom of the glass. The centrifugal action will be found to increase with the velocity.
In all cases of a body moving in a circle, the force, it can be proved, varies as the square of the velocity of the body at the moment, and in the inverse ratio of the radius. As in this case the velocity varies as the radius inversely, it follows that the force is as the inverse cube of the radius. As in the case of circular motion the body always is at the same distance from the center, it follows that the centrifugal and centripetal forces are equal at all points of a circular orbit. The general law for all orbits is, that the centrifugal force varies as the inverse cube of the distance from the center. As the attractive force of gravitation varies as the inverse square of the distance, it may hence be shown that the centrifugal force gives perfect security, notwithstanding the con stant attraction of the sun, that the planets, so far as that attraction is concerned, will never fall into the sun.
The doctrine of C. F. owes more to Kepler and sir Isaac Newton, of whose philos ophy it makes a considerable branch, than to all the rest of the philosophers, though almost all the leading mathematicians have contributed to it. The doctrine of centrif ugal forces was first mentioned by Huygens, at the end of his Horologium Oscillatorium, published in 1673; but Newton was the first who fully handled the doctrine, at least so far as regards the conic sections.