Relationship between SHM and MCU

Particle describing a uniform circular motion in the counterclockwise direction of a circle

In our Physics studies, we saw that Simple Harmonic Motion, also represented by the acronym (SHM), consists of a periodic and oscillatory motion. Therefore, we can say that simple harmonic motion has a direct relationship with uniform circular motion, represented by (MCU).

See the figure above. Suppose that a particle is making a uniform circular motion in the counterclockwise direction of a circle whose radius is R and that its projection thrown on the cosine axis will perform a simultaneous SHM.

In the illustration below, at a given time t, the projections of the vector R, the velocity vector and the centripetal acceleration vector of the MCU, for the particle located at a point P of the circular path, are exactly, at that same instant, the position, velocity and acceleration of the particle that performs a simple harmonic motion on the cosine axis.

Therefore, we can say that when the particle, in uniform circular motion, passes through points A and B, its vectors R and acceleration vector will be projected in their real sizes, and the velocity vector will be a point. Therefore, we can conclude that:

Extremes of SHM
R=x max =a
cp =a max
V=0

But when the particle, in MCU, is passing through points C and D, the vector v c will be projected in true magnitude, while the vectors R and a cp will have null projections. Therefore:

Equilibrium Position
c =V=V max
x=0
a=0

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