MECHANICS. 245 MECHANICS. pounded l^v geoiiietiioal addition or resolved into components. Since linear velocity is character- ized h a speed and by a direction, it can change in two independent ways: the speed can change, the direction remaining the same, e.g. a falling body: the direction can change, the speed remain- ing the same, e.g. a particle moving in a circle at a uniform rate. (In general, both speed and direction change, e.g. a vibrating sim|dt' pendu- lum.) There are therefore two independent types of linear acceleration. The three most interest- ing cases of linear acceleration are the following: ( 1 ) Motion in a straight line, constant ac- celeration. If the acceleration is positive, the speed increases ; if it is negative, the speed de- creases. Let the acceleration be called a, and the speed at any instant s„ ; then, t seconds later, the speed will be s = s„ -^- at, and the distance traversed in that time will be a? = s„f -H at". If t is eliminated from these equations it is seen that s- —So" = 2flj-. These formulte apply to a body falling freeh' toward the earth, in which case (/ = n80: to a body thrown vertically up- ward, in which case a =: — 980; and to many other illustrations. (2) Uniform motion in a circle. If the circle has a radius r, and if the constant speed is s, s^ the acceleration has for its numerical value — r and its direction at any instant is along the radius toward the centre from the point where at that moment the moving point is. This last fact is evident if the change in the velocity is considered. At any position in its path around the circle the moving point has a velociti/ along the tangent to the circle; the following instant this velocity is changed into the next tangent ; and to secure this change a small vector perpen- dicular to the first tangent must be added to the vector representing the first velocity. The proof that the numerical value of the acceleration is — will be found in all text-books on mechanics. r If the point makes N complete revolutions per second s=^2ir?"N; and the acceleration equals 4t'»'N". (.■?) Simple harmonic motion of translation. This is a vibratory motion, to and fro along a .straight line, such that, if distances from its middle point are called .r, the acceleration of the moving point when it is at a distance x from the centre has the numerical value irx, where n is a constant quantity, and its direction is toward the niiddh' ]Miint or centre. I To distances at one siilc of the centre are given positive values; at the other side negative.) This motion can be easily shown to be identical with that of the point which is the projection on a diameter of a point moving in a circle with uniform speed. It can be shown further that the period of this harmonic motion, that is, the time required for the point to go from one end of its path to the other and back again, is 2x/», where ir ^ 3.14Ifi. The length of the path is called the amplitude ; and the position of the vibrating point at any instant gives its pliasc. Thus there may be two vibrating points which have the same period and the same amplitude, but dilTer in phase — one lags apparently behinil the other. . pendvilum with a long supporting cord makes harmonic vibrations, if the amplitude is small ; so does any point of a violin string if the string is vibrating in its simplest mode; bo does a weight hanging from a rubber band or a spiral spring, if it is set vibrating in a vertical direction. R0T.TI0K. 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 fixed 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 a.xis. 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 Teigonometry) . This angular displacement can be completely pictured by a straight line in the proper direc- tion made to coincide with tlie axis of rotation and of a length proportional to the angle of rota- tion: such a line is called a rotm; or a localized vector, 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 fi.xed 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 difl'erent 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 case 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: (I) Position of axis fixed, constant angu- lar acceleration. If the constant acceleration is a, and if at any instant the angular speed is u„, the angular speed t seconds later will be M ^ W(| -|- a(. and the angle rotated through in that interval of time will be 8 = Wo' + , at". If t is eliminated from these two equations, it is seen that oi' — w „= = 2o9. This
