Simple Harmonic Oscillator

2 min read#oscillations

A mass pulled back by a force proportional to its displacement — the linear ideal whose sinusoidal motion underlies every other oscillator.

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Simple Harmonic Oscillator

The simple harmonic oscillator (SHO) is the purest oscillation there is: a system whose restoring force grows in exact proportion to how far it has been displaced. For a mass m on a spring of stiffness k, Newton's second law reads

m\ddot x = -kx,

the minus sign saying the force always points back toward equilibrium. Divide by m and define the natural angular frequency \omega_0 = \sqrt{k/m} to get the equation in its canonical form, \ddot x = -\omega_0^2 x.

The sinusoidal solution

Any function whose second derivative is its own negative (up to a constant) is a sine or cosine, so the general motion is

x(t) = A\cos(\omega_0 t + \varphi),

with amplitude A and phase \varphi fixed by the initial position and velocity. The motion is isochronous: the period T = 2\pi/\omega_0 = 2\pi\sqrt{m/k} depends on the mass and stiffness but not on the amplitude. A gentle wobble and a violent swing take exactly the same time — a fact that made the harmonic oscillator the heart of mechanical clocks.

x(t) = cos(t): the canonical harmonic motion
One full cycle every 2π. Amplitude sets the height; it never changes the period.

Energy flows back and forth

The total energy is constant, but it sloshes between two forms — kinetic and potential:

E = \underbrace{\tfrac12 m\dot x^2}_{\text{kinetic}} + \underbrace{\tfrac12 k x^2}_{\text{potential}} = \tfrac12 k A^2.

At the turning points all the energy is potential; flying through the center it is all kinetic. Because energy never leaves, the trajectory in State Space is a closed ellipse traced forever — the Phase Portrait of a center.

See also