LINEAR SIMPLE HARMONIC OSCILLATOR (LHO)

A Linear Harmonic Oscillator is any system obeying Hooke's law, $F\propto x$, written $F=-kx$, where the negative sign shows the restoring force always opposes the displacement. This linear relationship holds only for small displacements; for very large stretching forces the amplitude grows large enough that the restoring force picks up higher powers of $x$, and the oscillation becomes non-linear -- this unit restricts itself entirely to the linear (Hooke's-law-obeying) regime. Applying Newton's second law to a mass $m$ under this force gives $m\dfrac{d^2x}{dt^2}=-kx$, i.e. $\dfrac{d^2x}{dt^2}=-\dfrac{k}{m}x$. Comparing with the general SHM equation identifies the natural (angular) frequency of the oscillator as $\omega=\sqrt{k/m}\ \text{rad s}^{-1}$. The general solution of this differential equation is $x(t)=A\sin(\omega t+\varphi)$ or $x(t)=A\cos(\omega t+\varphi)$ (or, most generally, $A\sin(\omega t+\varphi)+B\cos(\omega t+\varphi)$), with the constants fixed by the initial conditions. The following sub-sections apply this s …