Angular Momentum

Definition

For a point particle:

L = r × p = r × (m·v)

Angular momentum is the cross product of position vector (from a reference point) and linear momentum. Magnitude: |L| = m·v·r·sin θ, where θ is the angle between r and v.

For a rigid body rotating about an axis:

L = I·ω

where I is the moment of inertia (about the axis) and ω is angular velocity. Both vectors along the axis (right-hand rule).

Units: kg·m²/s.

Conservation

Total angular momentum of an isolated system is constant in time.

"Isolated" means no external torque (or net external torque = 0). Internal forces (and torques) come in equal-opposite pairs by Newton's third law, so they cancel — total L is preserved.

By Noether's theorem (1918), this is the consequence of physics's rotational symmetry — physics looks the same in any orientation.

Conservation in action

System	Initial	Final	Mechanism
Figure skater pulls arms in	L = I_out · ω_slow	L = I_in · ω_fast (same L)	I decreases, ω increases proportionally
Earth-Moon tidal coupling	Earth rotation L_E + Moon orbital L_M	Slightly less L_E, slightly more L_M	Day lengthens; Moon recedes
Collapsing star	Slow rotation, large I	Fast rotation, tiny I (pulsar)	I drops by ~10⁵; ω rises by ~10⁵
Earth orbit (Kepler 2)	L = mvr at aphelion	Same L at perihelion	v faster when closer, slower when far
Diving (somersault)	Tucked: small I, fast spin	Extended: large I, slow spin	Tuck for fast rotations; extend to slow before water
Cat falling and landing on feet	Cat rotates upper/lower halves opposite	Total L = 0 (initial = 0)	Internal motion gives rotation without breaking conservation

Moment of inertia — the rotational mass

I is to ω as mass is to v. Larger I means more rotational inertia — harder to start or stop spinning.

Shape	Axis	Moment of inertia I
Point mass at radius r	Through point	m·r²
Hoop / thin ring	Through center, perpendicular	m·r²
Solid disk / cylinder	Through center, perpendicular	½m·r²
Solid sphere	Through center	(2/5)m·r²
Hollow sphere (thin shell)	Through center	(2/3)m·r²
Rod (length L)	Through center	(1/12)m·L²
Rod	Through end	(1/3)m·L²

For arbitrary shapes, I = ∫r²·dm (integral over the body). Mass farther from axis contributes more — moving mass outward increases I.

Precession — gyroscope physics

If you apply a torque to a spinning gyroscope (e.g., gravity on a tilted top), the result isn't just falling. The angular momentum vector L rotates around a vertical axis — precession.

Mathematically:

τ = dL/dt

If τ is perpendicular to L, dL is perpendicular to L — L's direction rotates. Magnitude unchanged. Precession rate:

Ω_precession = τ / L = m·g·d / (I·ω)

where d is the distance from pivot to center of mass. Faster spin (large L) = slower precession.

This is why bicycle wheels resist tipping — the rider's small "torque" of leaning shifts L's direction, but the wheel resists by precessing in the perpendicular direction (gyroscopic effect helps stability).

JavaScript — angular momentum problems

// Compute angular momentum for a point particle
function pointAngularMomentum(mass, position, velocity) {
  // L = r × p (in 3D, cross product)
  const [x, y, z] = position;
  const [vx, vy, vz] = velocity;
  return [
    mass * (y * vz - z * vy),
    mass * (z * vx - x * vz),
    mass * (x * vy - y * vx)
  ];
}

// 2D simplified: L = m * r * v_perp (perpendicular speed)
function angularMomentum2D(mass, r, v_perp) {
  return mass * r * v_perp;
}

// Figure skater: L conserved as I changes
function skaterFinalOmega(I_initial, omega_initial, I_final) {
  // L = I·ω = constant
  return (I_initial * omega_initial) / I_final;
}

// Skater with arms extended: I = 5 kg·m²; tucked: I = 1 kg·m²
console.log(`Tucked spin rate: ${skaterFinalOmega(5, 2, 1).toFixed(1)} rad/s`); // 10 rad/s = 1.6 rev/s

// Pulsar: collapsed star spin-up
function neutronStarOmega(M_solar, R_initial_km, R_final_km, omega_initial) {
  // For solid sphere, I = (2/5) M R². L = I·ω constant.
  const I_init = R_initial_km * R_initial_km;  // proportional
  const I_final = R_final_km * R_final_km;
  return omega_initial * I_init / I_final;
}

// Sun: ω ~ 2.7e-6 rad/s, R ~ 7e5 km
// Pulsar: R ~ 10 km
const pulsarOmega = neutronStarOmega(1, 7e5, 10, 2.7e-6);
console.log(`Pulsar ω: ${pulsarOmega.toExponential(2)} rad/s`); // ~13,000 rad/s = ~2000 rev/s ✓

// Kepler's 2nd law: equal areas in equal times → L conserved
function keplerCheck(r_aphelion, v_aphelion, r_perihelion, v_perihelion) {
  // L = m * r * v at the extremes (where v ⊥ r)
  const L_aph = r_aphelion * v_aphelion;
  const L_peri = r_perihelion * v_perihelion;
  return { aphelion: L_aph, perihelion: L_peri, conserved: Math.abs(L_aph - L_peri) < 0.001 * L_aph };
}

// Earth: r_peri = 1.471e11 m, v_peri = 30,290 m/s; r_aph = 1.521e11 m, v_aph = 29,290 m/s
console.log(keplerCheck(1.521e11, 29290, 1.471e11, 30290));
// L_aph = 1.521e11 × 29290 = 4.456e15
// L_peri = 1.471e11 × 30290 = 4.456e15 ✓

Where angular momentum shows up

• Astrodynamics. Planet/satellite orbits (Kepler's 2nd law), galaxy rotation, spin of pulsars, accretion disks.
• Sports. Figure skating, gymnastics (somersaults, twists), diving (tuck position), pole vault, baseball pitch spin.
• Aviation and aerospace. Helicopter rotor balance, gyroscopes for navigation, spacecraft attitude control via reaction wheels.
• Quantum mechanics. Quantized angular momentum (orbital ℓ, spin); selection rules in atomic transitions; magnetic resonance imaging.
• Condensed matter. Superconductors, magnons, spin coupling — all involve angular momentum.
• Robotics. Walking robots, drones, balance algorithms — angular momentum analysis is core.
• Astronomy — formation of solar systems. Collapse of gas clouds preserves angular momentum, leading to disk formation; spinning is why solar systems are flat.

Common mistakes

• Forgetting it's a vector. Angular momentum has direction (along rotation axis). For multi-body systems, vector-sum.
• Confusing L with momentum. Linear momentum p = mv. Angular L = r × p (or Iω). Different units, different conservation laws.
• Thinking I is fixed. Moment of inertia depends on mass distribution. Skaters, divers, planets — I can change as mass redistributes; this is the basis of L-conservation tricks.
• Ignoring direction in conservation. If torque flips direction, angular momentum vector flips — full conservation is in 3D vector sense.
• Confusing precession with rotation. Spin (rotation about an axis) and precession (the axis itself rotating) are different. Gyroscopes do both.
• Applying conservation when external torque exists. Friction on a spinning top eventually slows it (external torque from contact). Earth's rotation slows due to lunar tidal torque.