SHM as a projection of circular motion :
Let a particle be moving uniformly on a circle of radius A with angular speed ω.
If at t = 0, particle starts its motion from x-axis and in time t , it goes to p , then we have
x = OQ = A Cosθ = A Cosωt
y = OR = A Sinθ = A Sinωt
where Q and R feet of the perpendiculars drawn from P on diameter along the X-axis and the Y-axis respectively
From above equation it can be said that Q and R performing SHM about O along the x-axis and the Y-axis respectively with the same angular speed ω.
From figure (b) centripetal acceleration of the particle at P is Aω2.
Resolving this acceleration along PR and PQ
aR = ω2Asinωt = ω2x
aQ = ω2Acosωt = ω2y
The direction of aR is opposite to X and the direction aQ is opposite to Y.
Therefore, also from acceleration point of view, it can be said that Q and R are performing SHM.