Harmonic oscillator

By Hooke's law a spring exerts the force \(-ky\) on the mass, and by Newton's second law \(my''=-ky\). Dividing by \(m\) gives \(y''+\omega^2y=0\) with \(\omega^2=k/m\). The characteristic equation \(r^2+\omega^2=0\) has roots \(\pm i\omega\), so \(y=c_1\cos\omega t+c_2\sin\omega t\), which can also be written \(y=A\cos(\omega t-\varphi)\) with amplitude \(A=\sqrt{c_1^2+c_2^2}\) and phase \(\varphi\).

The period is \(T=2\pi/\omega=2\pi\sqrt{m/k}\) and the frequency is \(f=1/T\) cycles per unit of time. A larger mass oscillates more slowly; a stiffer spring, faster. Remarkably, \(T\) does not depend on the amplitude: the initial conditions \(y(0)\) and \(y'(0)\) only fix \(c_1=y(0)\) and \(c_2=y'(0)/\omega\), that is, how far it swings and with what phase, but not how fast.

When the spring hangs vertically, equilibrium is where \(mg=kL\), with \(L\) the stretch; measuring \(y\) from there removes the weight from the equation and leaves \(\omega^2=g/L\). The model is an idealization: without friction the energy \(\tfrac12my'^2+\tfrac12ky^2\) is conserved and the motion lasts forever. Adding damping and an external force leads to \(my''+cy'+ky=F(t)\), where the characteristic equation returns with other discriminants and the particular-solution methods come into play.