Rotation. When all points of a body are moving with the same Velocity (q.v.) the motion is one of pure translation, and is easy to comprehend. When, however, this condition is not fulfilled there must exist the kind of motion known as rotation. As simple examples, take the whirling of a flywheel or the motion of the hands of a watch. In such cases we readily see that there is, in the rotating body, a row of points which does not itself move. This row of points is called the axis of rotation, and every other point in the body describes a circle about it. To specify the motion completely we must know not only the position of this axis, but also the rate of rotation and the sense, clockwise or counterclockwise, with which the body is rotating about the axis. The rate of rotation may be measured by the number of revolutions made in a chosen time. It is more scientific, however, to measure it in terms of the angular speed. If the body is rotating uniformly the angular speed is the angle described in unit time by any plane drawn in the body parallel to or containing the axis of rotation: e.g. with the unit of time one sidereal day, the earth's angular speed about its axis is or ; but with the second as the unit of time the angular speed is a quarter of a minute of arc, or .000073 in radians.
In a simple geometric way a given rotation may be represented by a directed line taken of length numerically equal to the angular speed, and drawn along the axis of rotation in that direction which is related to the sense of rotation exactly as the to-and-fro motion of a right-handed screw is to the rotational motion of the screw. Such a directed quantity of definite length and of definite line position is called by Clifford a rotor. It is a Vector (q.v.) under the restriction that its lie in space is limited to a particular straight line.
So long as the axis of rotation is fixed with reference to lines which appear steady to us, there is no difficulty in apprehending the character of the motion. Take, however, the case of a carriage wheel or boy's hoop rolling along the road. Here we may regard the wheel as rotating about an axis drawn through the centre, while the axis is at the same time travelling forward with a definite linear speed—i.e. we may regard the motion as a combination of translation and rotation. In this particular case we may, however, represent the motion at each instant as one of pure rotation about an axis coinciding with the instantaneous line of contact of the wheel with the road. For, with rolling and no slipping, this line of contact with the road is for the moment at rest. And it is almost self-evident that, if at any instant there exists in rigid connection with a moving body an axis momentarily at rest, the instantaneous motion must be of the character of a rotation about this axis. The above is a simple example of what holds generally in uniplanar motion—i.e. motion in which every point of the body moves in a plane perpendicular to a fixed direction. The general theorem is that any uniplanar displacement whatever (which is not a pure translation) can be effected by a pure rotation about a determinate axis. Since any given motion may be regarded as consisting of a succession of displacements, it follows that any such uniplanar motion can be effected by a succession of rotations about instantaneous axes whose successive positions in space and in the body are determinate.
In uniplanar motions generally it is clear that the instantaneous axis of rotation, however much it may move both in space and in the body, must always remain parallel to the same direction. If discontinuous motion be excluded—and all natural motions are continuous—this instantaneous axis will pass continuously from position to position. It will trace out cylindrical surfaces, one in space and one in the body; and at any given instant these surfaces will touch along the line which is for the moment the instantaneous axis. It is not difficult to show that the complete motion of the body may be represented by the rolling of one of these surfaces upon the other. In the simple case of the carriage wheel the rolling surfaces are evidently the circumference of the wheel and the plane of the road. These theorems in uniplanar motion have many interesting applications in the kinematics of machinery (see Minchin's Uniplanar Kinematics, Clarendon Press, 1882).
If the motion is not uniplanar it is no longer possible in general to represent it by a succession of pure rotations. There is, however, a very remarkable theorem, which can be proved without difficulty, but which is hard of apprehension and even of acceptance. It is that after any displacement whatever of a body in space there is, in the body or rigidly connected with it, a line of points which is simply shifted along its own line in space. The whole displacement may then be effected by means of a sliding along this line together with a pure rotation about it—in other words, by a definite screw motion with reference to this line as axis (see SCREW). Even in the simpler case, when by fixing one point of a body we quite exclude translation, it is not easy to grasp the significance of the fact that after any displacement there is always one row of points which occupy exactly the same positions as before the displacement. From this theorem it follows that, however such a body may be moving, there is momentarily a line which is at rest. This line is the instantaneous axis of rotation. It always passes through the fixed point, and will as it shifts in time describe two conical surfaces, one in space and the other in the body. Any given continuous motion can then be effected by the rolling of one determinate conical surface fixed in the body upon another fixed in space. As a familiar example take an ordinary spinning-top. Here to the eye there is in general a rotation of the top about its axis of figure, while at the same time the top executes a conical motion about a vertical line through the point of support. In reality, however, at any instant of time the top is subject to one rotation about an instantaneous axis, which coincides neither with the axis of figure nor with the vertical line. This instantaneous axis executes a definite conical motion, both in the body and in space. Clerk-Maxwell (see his collected papers) devised a very ingenious and simple optical method for observing the position of the instantaneous axis, and so studying experimentally its motions with reference to the top. It should be mentioned in conclusion that infinitely small rotations are resolved and compounded according to the same laws as velocities and forces, so that we may regard the instantaneous angular velocity of a rotating body as made up of component angular velocities about any three chosen axes. It is thus that the subject is usually treated analytically. Such a treatment, however, is essentially artificial; and for a natural treatment we must go to geometry or to the Calculus of Quaternions (q.v.).