ROTATION. It can be shown by geometry that if a figure of any shape with one point fixed is displaced in any way by any series of rota tions, the final position may be reached from the initial one by a single rotation around an axis passing through the fixed point. The simplest mode of describing such a displacement is to imagine a plane section through the figure per pendicular to this axis, to take in this plane a line fixed in space and one fired in the figure, and then to measure the rotation by the change in the angle made with the former line by the latter as the figure turns around the axis. Three things are then necessary for the representation of the angular displacement : (1) The position of the axis; (2) its direction—a line in one direc tion will represent rotation in the direction of the hands of a watch, while one in the opposite direction will represent opposite rotation ; (3) the numerical value of the angle of displacement, measured as just described.
(The numerical value of the angle between two lines is obtained by describing a circle of any radius R with the point of intersection of the lines as the centre, measuring, the length of the arc. A, intercepted between the two lines, and dividing A by R. See TRIGONOMETRY ) This angular displacement can he completely pictured by a straight line in the proper direc tion made to coincide with the axis of rotation and of a length proportional to the angle of rota tion: such a line is called a rotor, or a localized rector. because it is a vector placed in a definite position.
If a rotation around a fixed axis is considered, the angular speed is the rate of change of the angle formed by the line fixed in space and that fixed in the figure, as described above. The angu lar velocity in this case is the angular speed around the given axis in a definite sense of ro tation; it is therefore a rotor. If a figure with One point fixed is given simultaneously two angu lar velocities around two different axes, the re sultant angular velocity will be a rotor which is the geometrical sum of the two component rotors. Angular acceleration. is the rate of change of angular velocity; and there are two, independent types: (1) the position of the axis fixed, but the angular speed changing; (2) the angular speed constant, but the position of the axis changing. A door or gate when opening or clos ing is an illustration of the first type; while a spinning top generally furnishes an illustration of the second, because, when the axis of the top is not vertical, it is moving so as to describe a cone in space. Actually in the ease of a spin ning top the angular speed is decreasing owing to friction, so it is an illustration of the combina tion of the two types.
The three most interesting cases of rotation are the following: (1) Position of axis fixed, constant angu lar acceleration. If the constant acceleration is a, and if at any instant the angular speed is the angular speed t seconds later will be w = + at, and the angle rotated through in that interval of time will be 0 = +A at'. If t is eliminated from these two equations, it is seen that co:— w 2 a0. This motion is illustrated by a dy-wbeel or grind stone coming to rest under a constant friction or being set in motion at a uniform rate.
It is evident from the above definition of the numerical value of an angle that if the linear speed and acceleration of any point at a distance H from the axis are s and a, they are connected with the angular speed and acceleration of the whole figure by the relations s = llw, a = (2) Angular speed constant, but the position of the axis describing a cone at a uniform rate.
This motion is illustrated, as explained above, by a spinning top. A piece of apparatus which furnishes a more accurate illustration consists es sentially of a heavy wheel whose axle is so sup ported that it can turn freely within a circular ring which is fastened rigidly to a metal rod carrying sliding weights at its further end; this rod is pivoted at its middle point so as to be free to turn in any direction; and the axle of the wheel is set in the same line as this rod. This instrument is called a 'gyroscopic pendulum.' ( For a description of one made out of a bicycle wheel, see Physical &Tic w, vol. x. p. 43, IDOL) To produce the desired motion, balance the wheel and its ring by means of the sliding weights until the rod is horizontal, set the wheel in rapid ro tation, and disturb the balance slightly by adding a small weight to either portion of the rod. The rod will immediately begin to move around in a horizontal plane; and thus the position of the axis of rotation of the wheel will change, and will describe a plane—the limiting form of a cone. The reason for this change is that there is compounded with the angular velocity of the wheel around its own axis another one due to the disturbed balance of the rod which would of itself make the whole apparatus rotate around a horizontal axis, i.e. turn over as the extra weight pulls its side down. This added angular velocity is about an axis at right angles to that of the wheel, and both lie in a horizontal plane: the two angular velocities will compound therefore to form an angular velocity about an axis in the same horizontal plane. but in a position different from that of the axis of the wheel before it was dhbturbed. As fast as this axis takes up its new position, it is again disturbed ; and so the mothm is a continuous change of position of the axis of the wheel in a horizontal plane. (This ease in rotation corresponds, therefore, perfectly to the one i t translation of motion of a point in a circle at a uniform speed.) In the actual use of the gyroscopic pendulum there are other phe nomena depending upon the properties of matter in motion; the above description is designed to be a purely kinematic one.
(3) Simple harmonie motion of rotation. This motion is illustrated by the to and fro rotation of an ordinary clock pendulum or by the vibrations of any body set swinging through small arcs when suspended on a horizontal axis, also by the bal nnee•wheel of a watch. Let, as before, two lines he taken in a plane at right angles to the axis, one fixed in the figure. the other in space, but so chosen that they eoineide when the vibrating figure is in its central position. Then, if 0 is the angular displaevinent at any insrtant of the line fixed in the figure from the one fixed in space, the angular acceleration equals ni'0, where m is a constant quantity. and the direction of the axis of the neceleration is such as always to produce an angular velocity toward the posi tion of equilibrium. The period of a complete vibration may be shown to be The ampli tude is the extreme angle turned through by the line fixed in the figure; the phase at any given instant depends upon the position of this line at that instant.
ix GENERAL. Translation and rotation are particular types of motion, and in general the motion of a figure includes both. It may be proved, however, by geometry that the most gen eral displacement of a figure, produced by any number of motions, may be reduced to a com bination of a translation along a certain line and a rotation around it as an axis; such a combina tion is called 'screw-motion.'