Consider an object experiencing uniform circular motion. What are the equations giving x and y as a function of time for such an object?

If the object starts at x = A and y = 0, and has an initial velocity in the +y direction, then the equations giving the x and y coordinates of its position as a function of time are:

x(t) = R cos(θ)

y(t) = R sin(θ)

In uniform circular motion the angular velocity is constant, so:

θ = ωt

The equations for x and y become:

x(t) = R cos(ωt)

y(t) = R sin(ωt)

The equation for x(t) for our object experiencing uniform circular motion looks the same as the one we derived for the mass on the spring experiencing simple harmonic motion.

Clearly there's a parallel here - when A = R and the two ω's are equal, the two motions in the x-direction are identical (given appropriate initial conditions).

A light shining in the +y direction would cast a shadow of the circular motion object - that shadow would exactly match the motion of the object undergoing simple harmonic motion.

What happens if, at t = 0, the objects are not at their extreme +x position? All we need to do is modify our equation, shifting it by the appropriate phase angle, δ. The equation for simple harmonic motion becomes:

x(t) = A cos(ωt + δ)

Instead of plugging in θ = ωt we're using θ = ωt + θ_o, and we're calling θ_o δ instead.