Position, velocity and acceleration

In looking for a solution for the harmonic oscillator equation, we have already obtained the most important kinematical quantities describing the motion.

The position at time \(t\):

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

The velocity at time \(t\):

\[v(t) = A \cdot \omega \cdot \cos\left(\omega \cdot t + \phi_0\right).\]

The acceleration at time \(t\):

\[a(t) = -A \cdot \omega^2 \cdot \sin\left(\omega \cdot t + \phi_0\right) = -\omega^2 \cdot x(t).\]

The quantity \(\omega \cdot t + \phi_0\) is called the phase of the oscillation:

\[\phi(t) = \phi_0 + \omega t.\]

The phase describes the current state of harmonic motion; eg, \(\phi = 0\) represents the beginning of a cycle (equilibrium position), \(\phi = \pi / 2\) is the extreme position and \(\phi = \pi\) is the middle of a cycle (returning to the equilibrium position).