Center of Mass

Definition

The center of mass is the mass-weighted average of position vectors:

r_cm = (m₁·r₁ + m₂·r₂ + ... + m_n·r_n) / (m₁ + m₂ + ... + m_n) = Σ(m_i·r_i) / M

For a continuous body:

r_cm = (1/M) · ∫r·dm

where M is total mass. Each component (x, y, z) is computed separately.

Key properties

Property	Statement
External force	F_ext = M·a_cm (system behaves as if all mass at COM)
Internal forces	Don't affect COM motion (3rd law cancellation)
Free flight	COM follows clean parabolic trajectory under gravity
Collisions	COM velocity unchanged by internal forces
Stability	Object tips when vertical line through COM exits base of support
Symmetry	For symmetric uniform body, COM is at the symmetric center

Common COM locations

Object	COM location
Uniform rod (length L)	Midpoint, L/2
Uniform sphere	Geometric center
Uniform triangle	Centroid (1/3 from each side toward opposite vertex)
Uniform L-shape	Inside the L, computed from rectangles
Hollow ring	Center (in empty middle)
Boomerang	In the bend (outside material)
Standing person	Roughly at navel, centered
Earth-Moon system	4670 km from Earth's center, inside Earth
Solar system	Just outside Sun's surface, mostly determined by Jupiter

COM motion in projectiles

Throw a hammer (or any irregular object) into the air spinning. The hammer rotates wildly, but its COM follows a clean parabolic trajectory.

Why — only gravity (external force) acts. F_ext = mg, downward. So COM acceleration = g, downward. Every point of the hammer moves through complex curves, but the COM is just doing free-fall.

This works for ANY rigid body, ANY motion (rotation, deformation, breaking apart). External forces dictate COM motion; internal forces dictate everything else.

Balance and stability

An object is in stable equilibrium if the vertical line through its center of mass falls within its base of support.

If COM is directly over the support, slight perturbations create torques that restore the object. If COM is at the edge, any tilt causes it to fall. If beyond the edge, it definitely tips.

Situation	Stability
Person standing upright	Stable — COM over feet
Person leaning past their toes	Unstable — COM ahead of base
Wide-base statue (Buddha)	Very stable — large base
Pencil balanced on tip	Unstable — tiny base, COM far above
Race car (low and wide)	Very stable — low COM, wide track
SUV (high COM)	Less stable — easier to tip when cornering

JavaScript — center of mass calculations

// COM of n point masses
function centerOfMass(masses, positions) {
  // positions is array of arrays (e.g., [[x1, y1], [x2, y2], ...])
  const totalMass = masses.reduce((s, m) => s + m, 0);
  const dim = positions[0].length;
  const com = new Array(dim).fill(0);
  for (let i = 0; i < masses.length; i++) {
    for (let d = 0; d < dim; d++) {
      com[d] += masses[i] * positions[i][d];
    }
  }
  return com.map(v => v / totalMass);
}

// 3-body system
const com1 = centerOfMass([1, 2, 3], [[0, 0], [1, 0], [0, 1]]);
console.log(com1); // [(0+2+0)/6, (0+0+3)/6] = [0.333, 0.5]

// Earth-Moon system
const M_EARTH = 5.972e24;
const M_MOON = 7.342e22;
const D_EARTH_MOON = 384400e3;
const com_em = centerOfMass([M_EARTH, M_MOON], [[0, 0], [D_EARTH_MOON, 0]]);
console.log(`Earth-Moon barycenter: ${com_em[0].toExponential(3)} m from Earth's center`);
console.log(`Earth's radius: 6.371e6 m → barycenter is INSIDE Earth: ${com_em[0] < 6.371e6}`);

// Project a thrown object: COM follows g
function projectileCom(initial_x, initial_y, vx0, vy0, g = 9.81) {
  return (t) => ({
    x: initial_x + vx0 * t,
    y: initial_y + vy0 * t - 0.5 * g * t * t
  });
}

// Hammer thrown at 20 m/s, 30° upward, from 1.5 m height
const com_traj = projectileCom(0, 1.5, 20 * Math.cos(Math.PI/6), 20 * Math.sin(Math.PI/6));
console.log(com_traj(1)); // {x: 17.32, y: 6.6} — COM trajectory
// (regardless of how the hammer rotates!)

// COM of an L-shape: decompose into rectangles
function lShapeCom(width1, height1, width2, height2) {
  // Rectangle 1: (0..width1, 0..height1), area = width1·height1
  // Rectangle 2: (0..width2, height1..height1+height2)
  const m1 = width1 * height1;
  const m2 = width2 * height2;
  const cm1 = [width1/2, height1/2];
  const cm2 = [width2/2, height1 + height2/2];
  return centerOfMass([m1, m2], [cm1, cm2]);
}

console.log(lShapeCom(4, 1, 1, 3)); // L-shape: 4×1 base + 1×3 vertical
// Total mass = 4 + 3 = 7. COM = ((4·2 + 3·0.5)/7, (4·0.5 + 3·2.5)/7) = [1.36, 1.36]

Where center of mass shows up

• Mechanics — collision analysis. COM moves at constant velocity through collisions; useful for momentum-conservation arguments.
• Sports. Diving, gymnastics, dance — performers manipulate posture to control COM motion. Long jump — extend forward, then tuck for max distance.
• Vehicle design. Low-and-wide cars are stable; SUVs with high COM tip in turns. F1 cars have COM as low as possible (~30 cm above ground).
• Architecture and engineering. Stable structures need COM above support. Cranes use counterweights to balance loads.
• Astronomy. Earth-Moon barycenter, planet-star wobble (exoplanet detection), galactic dynamics, solar system center.
• Robotics. Walking robots constantly track COM relative to support polygon; bipedal balance.
• Aerospace. Rocket COM shifts as fuel burns; thrust must be aligned with COM to avoid spinning. Spacecraft COM determines orbit dynamics.

Common mistakes

• Confusing COM with geometric center. For uniform symmetric bodies, they coincide. For asymmetric or non-uniform mass distributions, they differ.
• Thinking COM must be inside the body. Hollow rings, donuts, boomerangs all have COMs in empty space. Geometry doesn't constrain mass-weighted average.
• Ignoring internal forces (correctly!) — but applying it wrong. Internal forces don't affect COM motion. But they DO affect rotation, deformation, stress.
• Treating COM motion as the whole story. COM motion is just the translation of the system. Rotation, deformation, etc. are separate degrees of freedom.
• Confusing center of mass with center of gravity. Same in uniform g; differ slightly otherwise. Often used interchangeably in casual physics.
• Forgetting that stability depends on base shape, not size. A wide base is stable. But COM well above small narrow base is unstable. Both shape and size matter.