ROTATION (Rota, a wheel). The popular conception of a body in rotation is vague, except only in the case in which the rotation is made about an immoveable axis. This subject has accordingly been usually treated by mathematical methods; and mathematicians content with their results, and with their power of interpreting them, did nothing towards the improvement of the manner of presenting the elementary view of rotation. M. Poinsot first divested the subject of its previous complexity, in a 3lemoir read to the Academy of Sciences, May 19, 1834.
There is this parallel between the conception we form of the simple motion of a point and that of a solid body, namely, that each has a case of peculiar simplicity, by which others are rendered more easy to describe. A point may move in a straight line, or may preserve its direction unaltered; a body may revolve round a fixed axis, or each point may preserve its circle of revolution unaltered. But owing to the comparative simplicity of the motion of a point, it is easy [Thnreetosi] to carry with us, when it moves in a curve, the idea of its still having a different direction at every point of the motion, namely, that of the TANGENT of the curve. It is not so easy to see that whenever a body moves about a fixed point, no matter how irregularly, there is always, at every instant of the motion, some one axis which is, for that instant, at rest. This notion of an instantaneous axis of repose, not continuing to be such for any finite time—answering to that of an instantaneous direction in curvilinear motion, which does not continue for any finite time to represent thu direction—must be first distinctly formed, before any satisfactory account of the rotation of a body can be given.
Let us suppose a uniform sphere, with a fixed centre, but otherwise free to move in any way. Let a succession of forces act upon it, gradual or not, in such a manner that it will never move round one axis for any finite time during the contiuuance of their action. At a certain moment, let all the forces cease entirely, leaving the sphere to itself. It is easy enough to see that from and after the moment of discontinuance, the sphere will move round an axis which remains unaltered. Them must then. at the very moment of discontinuance
of the forces, have been an axis which was for that moment at rest, namely, the axis on which the motion is to continue after the forces Cease. In this way, knowing that curvilinear motion would become rectilinear the moment that the deflecting forces are removed, we may form an idea of the tangent of a curve, the lino of direction for the time being.
One of M. I'oiusot'e remarkable propoeitious is the following :—Any motion of a system round a fixed point may be attained by cutting a cone (in the most general sense of the word) out of the body, with the fixed point for a vertex, and fixing in space another cone for it to roll upon, also with the fixed point for a vertex.
Thus if, In the adjoining diagram, the cone A be made to roll upon the cone n, both being supposed destitute of impenetrability, so that the contact of the curves of a and B can always be made, and if the syetern out of which A II cut be then restored (also without im penetrability), there will be a complete geometrical representation of one possible motion of the system about c. Moreover, there is no possible motion which might not be represented in the same manner by properly choosing the cones A and n, and the axis of repose for the instant is the line in which the two cones touch.
If we suppose no fixed point in the system, so that motion of trans lation, as well as of rotation, Is possible, 31. Poinsot hakgiven another j Lined translation and rotation is the screw-like motion, in which a uniform motion of translation is accompanied by a uniform motion of rotation round a line parallel to the motion of translation. N. Poinsot has shown that every motion of a system must be, at any one instant, either a simple motion of translation, or one of rotation, or the screw like motion above described. That is to say, at every point of time in the motion of a system there exists a line (whether internal or external to the material system matters not, so long as they are immoveably connected) along which the system is at that instant sliding, while all the rest of the motion at that instant is simple rotation about that Clipping axis.