We call a harmonic motion when it can be described by harmonic time functions (sine or cosine), which are so named because of their graphical representation:
When this happens, the movement is called Simple Harmonic Movement (MHS).
To simplify the study of this movement, it can be analyzed as a projection of a uniform circular motion about an axis. Like this:
Elongation time function
Imagine a particle moving over a circle of radius A that we will call oscillation amplitude.
Placing the x-axis in the center of the circle describing the Uniform Curvilinear Motion and comparing the displacement in the Simple Harmonic Motion:
Using what we already know about MCU and projecting the angular displacement on the x axis we can deduce the time function of the shift in the Simple Harmonic Motion:
Using the cosine trigonometric ratio of the angle to obtain the value of x:
This is the exact position the particle is in the figure shown, if we consider that in the MCU this angle varies with time we can write φ as a function of time using the angular displacement time function:
Then we can substitute this function in the x-axis projected MCU equation and have the elongation time function, which calculates the position of the particle that describes an MHS at a given time. t.