The Mass-Spring Oscillator

Newton's second law for a mass on a spring is m x'' = -kx: the restoring force is always proportional to displacement and points back toward equilibrium. Dividing by m gives x'' = -(k/m) x, the defining equation of simple harmonic motion. Its solution is a sine or cosine whose angular frequency is omega = √(k/m) — you can check this by plugging x = A cos(omega t) back into the equation and seeing that it only works when omega² = k/m. From omega you get the period directly: T = 2 π / omega = 2 π √(m/k).

This tells you exactly why a heavier mass swings more slowly: more mass means more inertia resisting the spring's pull, so it takes longer to complete each swing even though the force law is identical. A stiffer spring (larger k) does the opposite — it yanks the mass back harder, shortening the period. Neither the amplitude nor the initial speed shows up anywhere in omega or T, which is the single most surprising fact in this topic.

This result sets up everything else in the unit. The same omega = √(k/m) reappears when you compute energy, since the oscillator trades kinetic and potential energy back and forth at a rate fixed by this same frequency. It also reappears, in disguise, in the pendulum equation and in driven oscillations, where matching an external driving frequency to this natural omega produces resonance.