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

v2

a'=--; and v = wr

r

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.

C[= W2]=\2712

T/I

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

2
a[=-wed] _ - T ) d   (32)

SIMPLE HARMONIC MOTION

This body is urged toward the middle of the path by a fore

(2 r) 2

F[=ma]=-m T d   (8

It follows that the period of a simple harmonic motion of trans lation is

T = 27r V- d   (34)

a

the minus sign indicating that the displacement and the acceleration are in opposite directions.

20. Simple Harmonic Motion of Rotation. - When a body rotates back and forth with a motion such that the angular acceleration is always directed toward a position of equilibrium and is always proportional to the angular displacement of the body from that position, the body is said to have a simple harmonic motion of rotation. The defining equation of simple harmonic motion of

rotation is   a = -b¢   (35)

where b is a positive constant.

When a body is vibrating about any axis with simple harmonic motion of rotation, any point on a line fixed in the body, and not on the axis of vibration, moves back and forth in the arc of a circle with a linear acceleration of which the component tangent to the arc has a magnitude which is proportional to the linear displacement of the point, measured along the arc, from the position of equilibrium. Hence the value of the period of a simple harmonic motion of rotation as well as the value of the angular speed, acceleration, and displacement at any time can be obtained from the values of the period, linear speed, acceleration and displacement of a point on a line fixed in the body.

Angular Acceleration and Period. - Substituting in (32) values of linear acceleration and linear displacement in terms of the corresponding angular quantities, (17) and (7), we find that

a = - ()2   (36)

Solving for T and substituting for a its value from (19)

T = 2 rr ~- -a0- = 2 rV - L~ = 21r~- S   (37)

where L is the torque acting upon a body of moment of inertia K with respect to the axis of vibration, at the instant when the angu-