What are the equations for x(t) and y(t) for a mass in uniform circular motion?
If the mass starts at x = A and y = 0, and has an initial velocity in the +y direction, then the x and y coordinates of its position as a function of time are:
x(t) = A cos θ y(t) = A sin θ
In uniform circular motion the angular velocity is constant, so: θ = ωt
The equations for x and y become: x(t) = A cos ωt y(t) = A sin ωt
The equation for x(t) for uniform circular motion is the same as that for a mass on the spring!
A light shining in the +y direction would cast a shadow of the circular motion that would exactly match the motion of the object undergoing SHM.
What happens if, at t = 0, the objects are not at their extreme +x position? All we need to do is to shift the equation by the appropriate phase angle, φ. Then equation for SHM becomes:
x(t) = A cos(ωt + φ)