The Connection between Circular Motion and Simple Harmonic Motion

In this simulation, you can examine the connection between uniform circular motion and simple harmonic motion.
This connection is one way to justify the basic equation of motion for an object experiencing simple harmonic motion.
x = A cos (ωt)
Applied to mass 1,
x₁ = A₁ cos (ω₁t)
This equation also corresponds to one dimension of uniform circular motion.
In this simulation, we even go beyond the equation above a little, and explore the phase angle, φ.
In that case, the equation of motion becomes x₂ = A₂ cos (ω₂t + φ).
There are analogous equations for the velocity and the acceleration as a function of time.
In Simple harmonic motion, the restoring spring force is
F = -kx = -kAcos (ωt)
Using Newtons Second Law, F = ma
-kAcos (ωt) = m(-Aω² cos (ωt))
Therefore, k/m=ω²
This means the spring constant, k divided by the mass, m of the spring mass system is equal to the angular frequency squared, ω² of the simple harmonic motion.