(2) Uniform motion in a circle. If the circle has a radius r, and if the constant speed is s, the acceleration has for its numerical value .92 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 velocity 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 'mist 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.
If the point makes N complete revolutions per second s — 2 and the acceleration equals (3) 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 x, the acceleration of the moving point when it is at a distance a. from the centre has the numerical value nix, where n is a constant quantity, and its direction is toward the middle point or centre. (To distances at one
side 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 he shown further that the period of this harmonie motion, that is, the time required for the point to go from one end of its path to the other and hack again, is where 3.1416. The length of the path is called the amplitude; and the position of the vibrating point at any instant gives its phase. Thus there may be two vibrating points which have the same period and the same amplitude, but differ in phase—one lags apparently behind the other.
A pendulum 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; so does a weight hanging from a rubber band or a spiral spring, if it is set vibrating in a vertical direction.