Show that the x component of the particle's position executes simple harmonic motion

Question

solving some excercise from Richard Haberman Mathematical Models, but I don't seems to know what to do to this problem.

Answer 1

First we observe that the angle covered, $\theta = \omega t$

So, we have this.

Intial position of the particle being '$A$' and the current position being '$B$'.

We see that the x-component of the position can be represented as $$\mathbf {r \cos \omega t}$$ where $r$ is the radius of the circle.

It is a $\cos$ function and is hence periodic. In fact this equation itself represents that the particle is executing Simple harmonic Motion in the x-component. But you could go ahead and double differentiate it with respect to time and notice that the acceleration is a function of position.

Now, armed with this understanding, I will go ahead and allow the OP to try out the y-component on their own.

Feel free to ask any clarifications!