# Circle of reference in Simple Harmonic Motion

A circle of reference is a graphical representation that facilitates the understanding of various SHM relationships. Consider the figure given below in which a point Q moves anti-clockwise around a circle of radius A with a constant angular velocity ω (in rad.s-1).

Here in the figure
OQ→position of Q relative to O
θ→angle OQ vector makes with the positive x-axis
This vector whose horizontal component represents the actual motion is called a
phasor

Let P be the point representing the projection of point Q (known as the reference point) on the horizontal diameter of the circle (known as the reference circle). As the reference point revolves, the point P moves back and forth along the horizontal diameter always keeping below or above the point Q. The motion of this point P is comparable to that of a body moving under the influence of elastic restoring force and executing SHM in the absence of frictional forces.
From the figure given above, displacement of point P at any given time t is the distance OP or x, and from the figure
x=Acosθ
The angular velocity of circular motion is
ω = angle swept time-taken = θ t
so,
θ=ωt
and
x=Acosωt                                               (1)
The figure given below shows the velocity in SHM using the circle of reference.

The velocity of the reference point Q would be the component of Q’s velocity parallel to the x-axis since P is always directed below or above the reference point. So,
vP=-vsinθ = -ωAsinωt                            (2)
since v=ωA is the tangential velocity of motion of reference point Q. The negative sign is introduced because the direction of velocity is towards the left. When Q is below horizontal diameter, the velocity of P is towards the right, but since sinθ is negative at such points, the minus sign is still needed.
The figure given below shows the acceleration in SHM using the circle of reference.