Periodicity of simple harmonic motion

The period \(T\) is defined as the time in which the oscillation resumes the same position:

\[x(t + T) = x(t).\]

Substituting the time dependence for the position, we get

\[A \cdot \sin\left(\omega \cdot [t + T] + \phi_0\right) = A \cdot \sin\left(\omega \cdot t + \phi_0\right),\]

\[\sin\left(\omega \cdot t + \omega \cdot T + \phi_0\right) = \sin\left(\omega \cdot t + \phi_0\right).\]

Since the sine function is periodic by \(2 \pi\), this is possible if the phases on the left and on the right differ in \(2 \pi\):

\[\omega \cdot t + \omega \cdot T + \phi_0 = \omega \cdot t + \phi_0 + 2 \pi,\]

\[\omega \cdot T = 2 \pi,\]

\[T = \frac{2 \pi}{\omega}.\]

The frequency of simple harmonic motion is thus

\[f = \frac{1}{T} = \frac{\omega}{2 \pi}.\]

That explains the choice of the constant of proportionality as \(\omega^2\) in the harmonic oscillator equation.