Simple Harmonic Motion

The setup

An object of mass m is connected to a spring with spring constant k. The spring exerts a restoring force F = -k·x (Hooke's law), where x is displacement from equilibrium.

From Newton's 2nd law: m·ẍ = -k·x → ẍ = -(k/m)·x = -ω²·x

where ω = √(k/m). This is the differential equation of simple harmonic motion.

The solution

The general solution is:

x(t) = A·cos(ω·t + φ)

where A is the amplitude (max displacement from equilibrium) and φ is the phase (depends on initial conditions). Equivalently: x(t) = a·cos(ωt) + b·sin(ωt) for arbitrary constants a, b.

Velocity and acceleration:

x(t) = A·cos(ωt + φ)
v(t) = -A·ω·sin(ωt + φ)
a(t) = -A·ω²·cos(ωt + φ) = -ω²·x(t)  ✓

Period and frequency

Quantity	Symbol	Formula	Unit
Angular frequency	ω	√(k/m)	rad/s
Period	T	2π/ω = 2π·√(m/k)	s
Frequency	f	1/T = ω/(2π)	Hz
Amplitude	A	(initial conditions)	m

Energy in SHM

Total mechanical energy:

E = ½k·x² + ½m·v² = ½k·A² (constant)

Distribution oscillates between KE and PE:

Position	x	v	KE	PE
At equilibrium	0	±A·ω (max)	½kA² (max)	0
At ±A (turning points)	±A	0	0	½kA² (max)
At ±A/2	±A/2	±A·ω·√3/2	(3/4)·½kA²	(1/4)·½kA²

Common SHM systems

System	ω formula	What's restoring it?
Mass on spring	√(k/m)	Hooke's law spring force
Simple pendulum (small angle)	√(g/L)	Gravity component
Physical pendulum	√(mgd/I)	Gravity torque about pivot
Torsional pendulum	√(κ/I)	Spring constant of twisted wire
LC circuit	1/√(LC)	Inductor + capacitor
Molecule (diatomic)	√(k_eff/μ)	Bond elasticity (μ = reduced mass)
Particle in parabolic potential	√(k/m) where U = ½kx²	Quantum harmonic oscillator

Damping — real-world oscillation

With friction or air resistance proportional to velocity (F_drag = -b·v):

m·ẍ + b·ẋ + k·x = 0

Three regimes based on damping ratio ζ = b/(2√(mk)):

Regime	Condition	Behavior
Underdamped	ζ < 1	Oscillates with exponentially decaying amplitude
Critically damped	ζ = 1	Returns to equilibrium in shortest time, no oscillation
Overdamped	ζ > 1	Slow exponential decay, no oscillation

Engineers tune damping (shock absorbers, door dampers) to be critically damped for fastest settling without overshoot.

JavaScript — SHM simulation

// Analytical solution: x(t) = A·cos(ωt + φ)
function shm(amplitude, omega, phase, t) {
  return {
    x: amplitude * Math.cos(omega * t + phase),
    v: -amplitude * omega * Math.sin(omega * t + phase),
    a: -amplitude * omega * omega * Math.cos(omega * t + phase)
  };
}

// Mass-spring period
function springPeriod(mass, k) {
  return 2 * Math.PI * Math.sqrt(mass / k);
}

console.log(`1 kg, k = 100 N/m: T = ${springPeriod(1, 100).toFixed(2)} s`); // 0.63
console.log(`1 kg, k = 25 N/m: T = ${springPeriod(1, 25).toFixed(2)} s`);   // 1.26 (4× softer = 2× longer)

// Energy distribution
function shmEnergy(mass, k, amplitude, x) {
  const PE = 0.5 * k * x * x;
  const v = Math.sqrt(k / mass) * Math.sqrt(amplitude * amplitude - x * x);
  const KE = 0.5 * mass * v * v;
  return { PE, KE, total: PE + KE };
}

console.log(shmEnergy(1, 100, 0.1, 0));    // x=0: all KE (~0.5 J)
console.log(shmEnergy(1, 100, 0.1, 0.1));  // x=A: all PE (~0.5 J)

// Damped oscillator simulation
function dampedShm(m, k, b, x0, v0, dt = 0.001, t_max = 10) {
  let x = x0, v = v0;
  const result = [];
  for (let t = 0; t < t_max; t += dt) {
    const F = -k * x - b * v;  // spring + damping
    const a = F / m;
    v += a * dt;
    x += v * dt;
    if (Math.abs(t * 100 - Math.round(t * 100)) < dt) {
      result.push({ t, x, v });
    }
  }
  return result;
}

// Underdamped: oscillates while decaying
const underdamped = dampedShm(1, 100, 1, 0.1, 0);
// Critically damped: b = 2√(mk) = 20
const critical = dampedShm(1, 100, 20, 0.1, 0);
// Overdamped: b = 100
const overdamped = dampedShm(1, 100, 100, 0.1, 0);

// Inspect amplitude decay
console.log('Underdamped at t=1:', underdamped.find(p => p.t > 1)?.x);

Where SHM shows up

• Mechanical systems. Springs, pendulums, swinging signs, vibration in machinery and bridges.
• Acoustics and music. Air molecules in sound waves, instrument strings (with boundary modes), drum heads.
• Electromagnetic. LC circuits oscillate at 1/√(LC); used in radio tuning, oscillators in circuits.
• Optics. Light is electromagnetic oscillation; Maxwell's equations give wave equation, solutions are SHM-like.
• Quantum mechanics. Quantum harmonic oscillator — exact eigenstates of parabolic potential. Foundation of phonon, photon, and field theory.
• Atomic and molecular vibrations. IR spectroscopy probes vibrational modes (~10¹³ Hz). Each bond is approximately a spring with characteristic ω.
• Engineering — vibration analysis. Buildings, vehicles, aircraft analyzed for natural frequencies; resonance avoidance is critical.

Common mistakes

• Confusing angular frequency ω with frequency f. ω = 2πf. ω in rad/s, f in Hz. Don't mix them in formulas.
• Forgetting it's only valid for SMALL displacements. Real springs deviate from F = -kx for large x. Pendulums deviate from SHM for large angles (>~15°).
• Using SHM for non-linear systems. Many "oscillating" systems aren't SHM — non-linear pendulum, double pendulums, anharmonic potentials. SHM is small-amplitude approximation.
• Wrong period formula. Mass-spring: T = 2π√(m/k). Pendulum: T = 2π√(L/g). They look similar but have different physics. Pendulum period is independent of mass.
• Confusing damping regimes. Critically damped is the BORDERLINE case — fastest return without oscillation. Underdamped systems still oscillate (just decreasingly).
• Treating energy as oscillating differently. Total energy is constant (½kA²). KE and PE individually oscillate, but their sum is fixed.