Rotational Inertia

Definition

Rotational inertia (moment of inertia) is:

I = ∫ r²·dm

For discrete masses:

I = Σ m_i · r_i²

The r is the perpendicular distance from each mass element to the rotation axis. Mass farther from the axis contributes much more (proportional to r²).

Units: kg·m². It's a scalar for fixed-axis rotation; a tensor for general 3D rotation.

Standard moments of inertia

Object	Axis	I
Point mass	At distance r	m·r²
Hoop / thin ring (radius R)	Through center, perpendicular	m·R²
Solid disk / cylinder (radius R)	Through center, perpendicular	½·m·R²
Solid sphere (radius R)	Through center	(2/5)·m·R²
Hollow sphere (thin shell, radius R)	Through center	(2/3)·m·R²
Rod (length L)	Through center, perpendicular	(1/12)·m·L²
Rod (length L)	Through end	(1/3)·m·L²
Rectangular plate (a×b)	Through center, perpendicular	(1/12)·m·(a²+b²)
Solid cube (side L)	Through center, face-perpendicular	(1/6)·m·L²

Parallel-axis theorem

If you know I about an axis through the center of mass (I_cm), you can find I about any parallel axis at distance d:

I = I_cm + m·d²

Example — rod of length L. I about center = (1/12)mL². I about end (parallel to center, at distance d = L/2):

I_end = (1/12)mL² + m·(L/2)² = (1/12)mL² + (1/4)mL² = (1/3)mL² ✓

Rotational kinetic energy

For an object rotating about an axis:

KE_rot = ½ · I · ω²

This is the rotational analog of ½mv². Adds to translational KE for objects both moving and rotating (e.g., rolling balls).

Total KE for rolling without slipping: KE = ½·m·v² + ½·I·ω². For a solid sphere rolling, the rotational part adds (2/5)·½·m·v² = (1/5)·m·v². Total = (7/10)·m·v². So rolling spheres have more KE than sliding ones at the same v.

Rolling on inclines — order of arrival

Roll different shapes down the same frictionless ramp from height h. Energy conservation gives v at bottom:

m·g·h = ½·m·v² + ½·I·ω²
With ω = v/r and I = β·m·r² (where β is the geometric factor):
m·g·h = ½·m·v² + ½·β·m·v² = ½(1+β)·m·v²
v = √(2gh/(1+β))

Shape	β (I/(mr²))	v at bottom	Order
Sliding (no rotation)	0	√(2gh)	Fastest
Solid sphere	2/5	√(10gh/7) ≈ 0.845√(2gh)	Fast
Solid disk	1/2	√(4gh/3) ≈ 0.816√(2gh)	Medium
Hollow sphere	2/3	√(6gh/5) ≈ 0.775√(2gh)	Slower
Hoop / hollow cylinder	1	√(gh) ≈ 0.707√(2gh)	Slowest

Hoops have the most "outer" mass relative to size (β=1), so they have the highest fraction of energy going into rotation, lowest into translation. Slowest race down.

JavaScript — moment of inertia computations

// Moment of inertia for common shapes
const I = {
  pointMass: (m, r) => m * r * r,
  hoop: (m, R) => m * R * R,
  solidDisk: (m, R) => 0.5 * m * R * R,
  solidSphere: (m, R) => (2/5) * m * R * R,
  hollowSphere: (m, R) => (2/3) * m * R * R,
  rodCenter: (m, L) => (1/12) * m * L * L,
  rodEnd: (m, L) => (1/3) * m * L * L,
};

// Parallel-axis theorem
function parallelAxis(I_cm, m, d) {
  return I_cm + m * d * d;
}

// Race down incline: which gets to bottom first?
function timeDownIncline(beta, h, ang_deg = 30, g = 9.81) {
  // Acceleration along incline = g·sin(θ) / (1 + β)
  const ang = ang_deg * Math.PI / 180;
  const a = g * Math.sin(ang) / (1 + beta);
  const length = h / Math.sin(ang);  // length along incline for height h
  // x = ½at² → t = √(2x/a)
  return Math.sqrt(2 * length / a);
}

console.log(`Sliding: ${timeDownIncline(0, 1).toFixed(2)} s`);          // Fastest
console.log(`Solid sphere: ${timeDownIncline(2/5, 1).toFixed(2)} s`);
console.log(`Solid disk: ${timeDownIncline(1/2, 1).toFixed(2)} s`);
console.log(`Hollow sphere: ${timeDownIncline(2/3, 1).toFixed(2)} s`);
console.log(`Hoop: ${timeDownIncline(1, 1).toFixed(2)} s`);              // Slowest

// Rotational kinetic energy
function rotationalKE(I, omega) {
  return 0.5 * I * omega * omega;
}

// Skater spinning: I = 5 kg·m² → 1 kg·m²
function skaterSpinUp(I_initial, omega_initial, I_final) {
  const L = I_initial * omega_initial;  // conserved
  const omega_final = L / I_final;
  const KE_initial = rotationalKE(I_initial, omega_initial);
  const KE_final = rotationalKE(I_final, omega_final);
  return {
    omega_final,
    KE_initial,
    KE_final,
    KE_increase: KE_final - KE_initial  // skater does work pulling arms in
  };
}

console.log(skaterSpinUp(5, 2, 1));
// L = 10; omega_final = 10 rad/s; KE goes from 10 J to 50 J (5× increase!)
// Skater's muscles do this work pulling arms in

Where rotational inertia shows up

• Engineering — flywheels. Energy storage in mechanical systems, smoothing rotational motion (engine flywheels, KERS in F1).
• Sports. Skating spin control (pull arms in for fast spin), diving (tuck for somersaults), tennis racket sweet spot, golf club design.
• Astronomy. Galaxy rotation curves (mass distribution), pulsars (compact = small I = fast spin), planetary spin axes.
• Vehicle dynamics. Wheels with low rotational inertia (lightweight rims) accelerate quicker; trucks have heavy wheels for stability.
• Robotics. Joint inertia affects servo torque requirements, response time, control complexity.
• Spacecraft. Reaction wheels exploit conservation of angular momentum for attitude control. Tensor analysis for satellite stability.
• Music. Drum heads (modes depend on I), cymbals, gongs — all have characteristic frequencies determined by mass and rotational dynamics.

Common mistakes

• Treating I as a property like mass. Mass is intrinsic; I depends on the chosen rotation axis. Same object has different I about different axes.
• Forgetting r² dependence. Doubling distance from axis means 4× more contribution to I. Mass distribution matters MUCH more than total mass.
• Confusing solid and hollow. Solid sphere I = (2/5)mR². Hollow sphere I = (2/3)mR² (mass all at outer radius). Different by 5/3 ratio.
• Using parallel-axis theorem about non-parallel axes. Theorem only works for axes parallel to the central axis. For perpendicular axes, use perpendicular-axis theorem (planar bodies only).
• Ignoring rotational KE in energy budgets. A rolling ball has both translational and rotational KE. Energy conservation needs both.
• Confusing scalar I with full tensor. For symmetric rotation, scalar I is fine. For 3D unconstrained rotation, you need the moment of inertia tensor.