SHM Chapter 03 circle to explain phase

Example 3: Uniform Circular motion’s one dimensional projection

Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular speed ω around a circle of radius A centered at the origin of the x−y plane, then its motion along each coordinate is simple harmonic motion with amplitude A and angular frequency ω.

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Q1: given that, a circular motion can be described by x = A cos(ω t)  and y A sin(ω t) what is the y-component model-equation that can describe the motion of a uniform circular motion?

A1: y = Asin (ωt)

Q2: When the x-component of the circular motion is modelled by x = A cos(ω t)  and y A sin(ω t) suggest an model-equation for y.

A2: y = Acos (ωt) for top position or y = - Acos (ωt) for bottom position

Q3: explain why are the models for both x and y projection of a uniform circular motion, a simple harmonic motion?

A3: both x = A cos(ω t)  and y A sin(ω t) each follow the defining relationship for SHM as ordinary differential equations of   ${\displaystyle \frac{d^{2}x}{{dt}^2}=-{\omega }^{2}x}$ and  ${\displaystyle \frac{d^{2}y}{{dt}^2}=-{\omega }^{2}y}$ respectively.

Run Model:

https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMcircle/SHMcircle_Simulation.html