Simple Pendulum

The setup

A point mass m hangs from a massless string of length L, free to swing. The string makes angle θ with vertical at any moment.

Forces — gravity (mg, downward) and tension (T, along string). Decompose gravity:

• Component along string — mg·cos θ (cancelled by tension).
• Component perpendicular to string (tangent to arc) — -mg·sin θ.

This perpendicular component is the restoring force.

Equation of motion

Tangential acceleration along arc = L·θ̈. Tangential force = -mg·sin θ.

From F = ma:

m·L·θ̈ = -m·g·sin θ
θ̈ = -(g/L)·sin θ

Mass cancels — pendulum motion doesn't depend on mass.

For small angles (θ < ~15°), sin θ ≈ θ:

θ̈ ≈ -(g/L)·θ = -ω²·θ

This is simple harmonic motion with ω = √(g/L). Period:

T = 2π/ω = 2π·√(L/g)

Period vs length

Length L	Period T	Frequency f
0.25 m	1.00 s	1.00 Hz
1.00 m	2.01 s	0.50 Hz
2.00 m	2.84 s	0.35 Hz
5.00 m	4.49 s	0.22 Hz
10.00 m	6.34 s	0.16 Hz
1.00 m on Moon (g=1.62)	4.94 s	0.20 Hz

A "second pendulum" (T = 2 s) requires L ≈ 99.4 cm on Earth. Used in early metric system definition.

Period vs amplitude (large angles)

Exact formula (uses elliptic integrals):

T = T₀ · (4/π) · K(sin(θ₀/2))

where K is the complete elliptic integral of the first kind. Series expansion:

T = T₀ · (1 + (1/16)θ₀² + (11/3072)θ₀⁴ + ...)

where θ₀ is amplitude in radians.

Amplitude θ₀	Period correction	Period (relative to T₀)
5°	+0.05%	1.0005
15°	+0.4%	1.004
30°	+1.7%	1.017
45°	+4.0%	1.040
60°	+7.3%	1.073
90°	+18.0%	1.180

Foucault pendulum

A long, free pendulum's swing-plane appears to rotate over hours due to Earth's rotation. At the poles, full 360° rotation per sidereal day. At latitude φ, rotation per day = 360°·sin(φ).

Location	Latitude	Rotation per hour
North Pole	90°	15°/hr (full rotation in 24 hr)
Stockholm	59.3°	12.9°/hr
Paris (original Foucault)	48.9°	11.3°/hr
San Francisco	37.8°	9.2°/hr
Equator	0°	0 (no rotation visible)

The pendulum's swing-plane stays fixed in the inertial frame; Earth rotates underneath. Direct evidence of Earth's rotation, dramatic when seen for the first time.

JavaScript — pendulum simulation

// Small-angle period
function pendulumPeriod(L, g = 9.81) {
  return 2 * Math.PI * Math.sqrt(L / g);
}

console.log(`1m on Earth: T = ${pendulumPeriod(1).toFixed(3)} s`); // 2.006
console.log(`1m on Moon: T = ${pendulumPeriod(1, 1.62).toFixed(3)} s`); // 4.939

// Compute g from period and length
function gFromPendulum(period, length) {
  return 4 * Math.PI * Math.PI * length / (period * period);
}

console.log(`From T=2.006s, L=1m: g = ${gFromPendulum(2.006, 1).toFixed(4)} m/s²`); // 9.808

// Numerical simulation (large amplitude)
function pendulumSimulate(theta0, L = 1, g = 9.81, dt = 0.0001, t_max = 10) {
  let theta = theta0;
  let omega = 0;
  const states = [];
  for (let t = 0; t < t_max; t += dt) {
    const alpha = -(g / L) * Math.sin(theta);
    omega += alpha * dt;
    theta += omega * dt;
    if (Math.abs(t * 100 - Math.round(t * 100)) < dt) {
      states.push({ t, theta, omega });
    }
  }
  return states;
}

// Find period from simulation
function measurePeriod(states) {
  let lastSign = Math.sign(states[0].theta);
  let crossings = [];
  for (const s of states) {
    if (Math.sign(s.theta) !== lastSign && lastSign !== 0) {
      crossings.push(s.t);
      lastSign = Math.sign(s.theta);
    }
  }
  if (crossings.length < 2) return null;
  // Half period from one zero crossing to next; full period = 2× this
  return 2 * (crossings[1] - crossings[0]);
}

// Compare small vs large amplitude
const small = pendulumSimulate(0.1); // 5.7°
console.log(`Small amplitude (5.7°): T = ${measurePeriod(small)?.toFixed(3)} s (theory: 2.006)`);

const large = pendulumSimulate(Math.PI / 2);  // 90°
console.log(`Large amplitude (90°): T = ${measurePeriod(large)?.toFixed(3)} s (~18% longer)`);

// Foucault rotation rate
function foucaultRate(latitudeDeg) {
  return 360 * Math.sin(latitudeDeg * Math.PI / 180);  // degrees per day
}

console.log(`North Pole: ${foucaultRate(90).toFixed(1)}°/day`);
console.log(`Paris (48.9°N): ${foucaultRate(48.9).toFixed(1)}°/day`);
console.log(`Equator: ${foucaultRate(0).toFixed(1)}°/day`);

Where pendulums show up

• Timekeeping. Pendulum clocks (Huygens 1656); precision pendulum clocks (Shortt-Synchronome ~1920s) reached 1 sec/year accuracy.
• Gravity measurements. Historical g measurements via period; gravimeter pendulums sensitive to local gravity anomalies.
• Earth rotation demos. Foucault pendulums in museums and observatories.
• Engineering. Tuned mass dampers (large pendulum-like structures) in skyscrapers (Taipei 101) damp wind-induced oscillations.
• Physics education. Classic introduction to oscillations, SHM, energy conservation.
• Forensic investigations. Crash analysis using pendulum-impact tests; cadaver studies use pendulum dummies for impact reconstruction.
• Music. Metronomes use small pendulums for tempo. Bell ringing — large pendulums require careful timing.

Common mistakes

• Using small-angle formula at large amplitudes. Beyond ~15°, the simple T = 2π√(L/g) formula understates the period. Use the elliptic integral or numerical simulation.
• Confusing pendulum and spring periods. Pendulum: T = 2π√(L/g) — depends on length and g, not mass. Spring: T = 2π√(m/k) — depends on mass and spring stiffness, not g. Different systems.
• Ignoring the difference between physical and simple pendulum. Simple — point mass on string. Physical — extended body (e.g., a meter stick). Period formulas differ.
• Forgetting that the bob's path is an arc, not horizontal. The bob moves in a circular arc, so its motion isn't pure horizontal SHM. Small-angle approximation projects it as such.
• Mixing radians and degrees. sin θ ≈ θ approximation requires radians. Be careful if you compute angles in degrees.
• Treating Foucault rotation as a force. The pendulum swing-plane stays fixed in the inertial frame; Earth rotates underneath. From Earth's rotating frame, it appears the pendulum rotates due to Coriolis force.