Simple harmonic motion and circular motion

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Simple harmonic motion (SHM) is closely related to uniform circular motion, as the latter can be viewed as two-dimensional SHM. When an object moves in a circle, its projection onto a Cartesian coordinate system reveals that the x and y coordinates follow SHM equations, specifically x=R*cos(wt) and y=R*sin(wt). This demonstrates that uniform circular motion consists of two SHMs with the same frequency but a phase difference of π/2, and the components are perpendicular to each other. The typical representation of SHM can be interpreted as the x component of circular motion. Understanding this relationship enhances the comprehension of both motion types.
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why simple harmonic motion is projected as or compared with uniform circular motion ?
 
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Because uniform circular motion is essentially "two dimensional simple harmonic motion."

Consider the shadow of an object moving in a circle when the light is shined "edge on". (This essentially "collapses" or "ignores" one of the dimensions of the motion)
 
Well if one considers a cartesian coordinate system with origin at the center of the circle of the uniform circular motion, then can prove fairly easily that the x and y coordinates (of the particle that does circular motion), are doing simple harmonic motion. That is it will be x=R*cos(wt) and y=R*sin(wt) where w the angular velocity, R the radius of the circle.

So you can view uniform circular motion as composition of two simple harmonic motions of the same frequency, with phase differnce pi/2, and (thats important) of direction perpendicular to each other.
 
The usual form the solution is given in, that is in terms of a shifted cosine, can be interpreted as the x component of a body in uniform circular motion.
 

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