What is Phase?

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.

Understanding Phase Difference

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: $y = A\sin \omega t$

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: $y = A\cos \omega t$ for top position or $y =  - A\cos \omega t$ 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 $x = A\cos \omega t$ and $y = a\sin \omega t$ each follow the defining relationship for SHM as ordinary differential equations of $\frac{{{d^2}x}}{{d{t^2}}} =  - {\omega ^2}x$ and $\frac{{{d^2}y}}{{d{t^2}}} =  - {\omega ^2}y$ respectively.

Q4: In, the diagram above, what is the phase difference between blue and magenta color y direction motion?

A4: the phase difference is 90 degrees, or more precisely blue lead red by 90 degrees angle.