24 PRINCIPLES OF ELEMENTARY DYNAMICS
nent, in the direction of the line AB, of the radial acceleration a' be called a. This a is the acceleration of the point P. Let d represent the displacement of P from the point C. We shall count d positive when above C and negative when below. We shall measure r outward from 0.
From the similar triangles P'QQ1 and P'001 we have
a d a' r
Representing the constant linear speed and angular speed of P' by v and w, respectively, we have
a'=--; and v = wr
Eliminating a' and v from these three equations, we get
a = -wed (31)
Since w is constant, it follows from this equation that the acceleration of P is proportional to its distance from the center of its path and is opposite in sign. That is, if a point moves with uniform speed in the circumference of a circle, the projection of the point on any straight line in the plane of the circle moves with simple harmonic motion of translation.
19. The Period of a Simple Harmonic Motion of Translation. - The theorem we have just proved will now be used for the determination of the value of the constant c in the defining equation of simple harmonic motion (30).
A comparison of (30) and (31) shows that c = w2. If the time of one revolution of P', that is, the time of one complete vibration of P, be denoted by T, we have
w = 2T radians per unit of time.
Hence, if a body of mass m is moving with simple harmonic motion of period T, then when the body is at a distance d from the middle of its path, the acceleration is directed toward the middle of the path and has the value
a[=-wed] _ - T ) d (32)