Spring-Mass SHM

A spring-mass oscillator consists of a block of mass m attached to an ideal spring of stiffness k. When displaced from equilibrium, the spring exerts a restoring force F = −kx that drives Simple Harmonic Motion (SHM).

The motion is described by the differential equation m(d²x/dt²) = −kx, whose solution is x(t) = A cos(ωt + φ), where A is amplitude and φ is the initial phase.

The angular frequency is ω = √(k/m). It depends ONLY on the spring constant and the mass — not on amplitude. Doubling the amplitude does NOT change the period.

The period T = 2π√(m/k) and frequency f = 1/T = (1/2π)√(k/m). Stiffer springs (larger k) oscillate faster; heavier blocks oscillate slower.

Velocity v(t) = −Aω sin(ωt + φ) is maximum at the equilibrium position (|v|_max = Aω) and zero at the extremes (turning points).

Acceleration a(t) = −Aω² cos(ωt + φ) is maximum at the extremes (|a|_max = Aω²) and zero at the equilibrium position. Acceleration always points toward equilibrium.

Energy oscillates between kinetic and potential, but the total mechanical energy E = ½kA² is constant (in the absence of friction).

On a smooth horizontal surface, gravity has no effect on the motion. On a vertical spring, gravity simply shifts the equilibrium position; the period remains T = 2π√(m/k).

Two springs in parallel give k_eq = k₁ + k₂ (stiffer combined system, higher ω). Two springs in series give 1/k_eq = 1/k₁ + 1/k₂ (softer combined system, lower ω).