EJSS Circular Motion to Simple Harmonic Motion Model
Example 3: Uniform Circular motionís 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 ? 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 ?.
Q1: given that, a circular motion can be described by x = A cos(? t) and y A sin(? t) what is the y-component model-equation that can describe the motion of a uniform circular motion?
A1: y = Asin (?t)
Q2: When the x-component of the circular motion is modelled by x = A cos(? t) and y A sin(? t) suggest an model-equation for y.
A2: y = Acos (?t) for top position or y = - Acos (?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(? t) and y A sin(? t) each follow the defining relationship for SHM as ordinary differential equations of d 2 x d t 2 = - ? 2 x and d 2 y d t 2 = - ? 2 y respectively.