MECHANICS. 246 MECHANICS. motion is illustrated by n lly-wheel 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 K from the axis are * and a, they are connected with the angular speed and acceleration of the whole figure by the relations s = Rw, a = Ra. (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 lino as this rod. This instrument is called a -gyroscopic jjcndulum.' ( For a description of one ma<le out of a bicycle wheel, see Physical Review, vol. x. p. 43, 1901.) To produce the desired motion, balance the wheel and its ring by means of the sliding weights until the rod is liorizontal, 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 inuiiediately 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 angidar 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 al)Out 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 i)Iai!e. but in a position dilTerent from that of the axis of the wheel before it was disturbed. As fast as this axis takes up its new position, it is again disturbed; and so the motion is a continuous change of jiosition of the axis of the wheel in a horizcmtal jilane. (This case in rotation corresponds, therefore, i)erfectly to the one ih translation of motion of a point in a circle at a uniform speed.) In the actual use of the gyroscopic pendtilum 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 harmonic motion of rotation. This motion is illustrated by the to and fro rotation of an ordinary clock pendulum or l>y the vilirations of any body set swinging through small arcs when suspended on a liorizontal axis, also by the bal- nnei> whiKd of a watch. T.et. as before, two lines be taken in a plane at right angles to the axis, one fixed in the figure, the otiier in space, but so chosen that they coincide when the vilirating figure is in its central position. Then, if d is the angular dlsplacenieiit at any instant of the line fixed in the tiguro from the one fixed in spnco, the angular acceleration equals m" 8, ■where m is a constant quantity, and the direction of the axis of the accelerntion is such ns always to produce an angular velocity toward the posi- tion of equilibrium. The period of a complete vibration may be shown to be lir /m. 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. JIoTioN IN 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.' DYNAMICS. Kinematics is a science which is concerned with geometrical ideas alone; it is the application of logical principles to certain definitions and axioms; it is not concerned with any appeal to experience. On the other hand, dynamics is fundamentally a science based on our experience of certain sensations associated with the idea of matter ; and the object of the science is to make ; such an analysis of the facts of observation and , experience as will lead to the statement of a few principles from which all observed phenomena may be predicted. It is possible to have a science based entirely on definitions — which are sug- gested by observations, however — and to show that all observed ])henomena can be regarded as consequences of these definitions, if they are iden- tified with actual physical quantities which ap- peal to our senses. Such a science is called 'theoretical dynamics.' In the following treat- ment statics is considered as a special case of kinetics. TuANSLATioN. The simplest properly of mat- ter (q.v.) is illustrated by an experiment due to Galileo. If a ball rolls down an inclined jilane and then meets another plane inclined in the op- posite direction, the ball will roll up it with a constantly decreasing velocity; the less inclined this second plane is, the less is the rate of change of the velocity of the ball as it rolls up; therefore, if the plane is perfectly horizontal, there is every reason for believing that the cause of the observed decreasing velocity of the ball is friction, and that if there were no friction the velocity of (lie liall would not change. In other words, it is tliought to be a general law of nature that a portion of matter free from all external actions will maintain its state of motion un- altered. If, however, the motion of one portion <if mat- ter is inlluenced by the presence of another piece of matter, it is oliserved tli.it the effect is mutual. The simplest case of two bodies influencing each other's motion is illustrated by two billiard balls striking when rolling on a smooth table, i.e. a surface free from friction; by a man standing on a board which rests on siiinoth ice, and then jumping off; by a bullet fired from a gun; etc. One law applies to all such cases: if m, and m, are the m'assesof the twopiecesof matter which are supposed to be so small as to be called 'particles,' r, and r, their linear velocities at any instant, V, and V. their linear velocities at any later time, then m,v, + m,Vt = m,V, -f OTjV,, provided there are no external actions, that is, provided that the only cause of the change in the
