Mathematical Representation of Simple Harmonic Motion:

At any time t it takes the position B and OB makes an angle θ with x-axis as illustrated. Let OC be the projection of OB along x-axis.

After that we have            OC = x = OB cos θ = r cos ω t

The motion of C along x-axis may be termed as Simple Harmonic Motion.

We have,             x = r cos ω t

 By differentiating w. r. t. to time,

dx/ dt  =- r ω sin ω t  . . . velocity of C

By differentiating again w. r. t. to time,

d ²x/ dt ²  = a_x  =- r ω² cos ω t

a_x  = - ω²  x (. . . acceleration of C)

or          

Negative sign mention the acceleration is oriented towards origin.

From the figure it is apparent that the diametrical projection of such a point oscillates between AOD and back to A as the particle rotates w.r.t. the circle. ω is called as the natural circular frequency. The circle is called auxiliary circle.

The radius is called as amplitude of the motion. The time needed to complete one oscillation is called as the period of the motion.

Periodic time, T = 2 π /ω

Frequency is described as the number of oscillations per second.