### Connection Between SHM and Uniform Circular Motion

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)