What Is the Moment of Inertia?
The mass moment of inertia (also called angular mass or rotational inertia) is a measure of how difficult it is to rotate an object around a specific axis. It plays the same role in rotational dynamics as mass does in linear dynamics — the higher the moment of inertia, the more torque is needed to achieve a given angular acceleration.
Moment of inertia depends both on the total mass of the object and on how that mass is distributed relative to the rotation axis. A concentration of mass far from the axis dramatically increases the moment of inertia.
Moment of Inertia Units
The physical dimension of moment of inertia is mass × length².
- SI unit: kilogram‑metre squared — kg·m²
- Imperial / US unit (slug system): pound-foot-second squared — lb·ft·s² (equivalent to slug·ft²)
- Imperial / US unit (pound-mass system): pound-foot squared — lb·ft²
Conversion: 1 kg·m² = 0.7376 lb·ft·s² = 23.73 lb·ft²
Moment of Inertia Equation
For a single point mass m located at a perpendicular distance r from the rotation axis:
I = m · r²
For a body composed of n point masses:
I = Σ mᵢ · rᵢ²
For continuous bodies with uniform mass distribution, the sum becomes an integral:
I = ∫ r² dm
Moment of Inertia Table
The table below lists the moment of inertia formulas for the most common rigid‑body shapes.
| # | Shape | Axis | Formula |
|---|---|---|---|
| 1 | Point mass | Any axis at distance r | I = m·r² |
| 2 | Solid cylinder / disk | Through central axis (symmetry axis) | I = ½·m·R² |
| 3 | Hollow cylinder — thin wall | Symmetry axis | I = m·R² |
| 4 | Hollow cylinder — thick wall | Symmetry axis | I = ½·m·(R₁² + R₂²) |
| 5 | Disk | Through diameter | I = ¼·m·R² |
| 6 | Solid sphere | Through center | I = 2/5·m·R² |
| 7 | Hollow sphere — thin shell | Through center | I = 2/3·m·R² |
| 8 | Hollow sphere — thick shell | Through center | I = 2/5·m·(R₂⁵−R₁⁵)/(R₂³−R₁³) |
| 9 | Thin rod | Through center (perpendicular) | I = 1/12·m·L² |
| 10 | Thin rod | Through one end (perpendicular) | I = 1/3·m·L² |
| 11 | Rectangular plate | Through center, perpendicular to plate | I = 1/12·m·(a² + b²) |
| 12 | Rectangular plate | Through center, parallel to side b | I = 1/12·m·a² |
| 13 | Rectangular cuboid | Through center, parallel to edge c | I = 1/12·m·(a² + b²) |
| 14 | Solid cone | Symmetry axis | I = 3/10·m·R² |
| 15 | Solid cone | Through apex, perpendicular to axis | I = 3/20·m·(R² + 4h²/3) |
| 16 | Hollow cone — thin shell | Symmetry axis | I = ½·m·R² |
| 17 | Torus | Symmetry axis (through center of tube) | I = m·(R² + 3/4·r²) |
| 18 | Torus | Axis through center of torus (diameter) | I = m·(5/8·r² + R²/2) |
| 19 | Thin hoop | Symmetry axis | I = m·R² |
| 20 | Thin hoop | Through diameter | I = ½·m·R² |
| 21 | Solid ellipsoid | Symmetry axis c | I = 1/5·m·(a² + b²) |
| 22 | Triangular plate | Through centroid | I = 1/18·m·(a² + b² + ab) |
| 23 | Regular polygon | Through center, perpendicular to plane | I = m·R²/2 · (1 + cos²(π/n)/3) |
FAQs
Does the axis of rotation affect the moment of inertia?
Yes, significantly. The same rigid body has different moments of inertia depending on the chosen axis. For example, a thin rod has I = 1/12·mL² about its center but I = 1/3·mL² about its end — a 4× difference. Use the parallel-axis theorem to shift between axes: I = I_cm + m·d², where d is the distance between the original (center-of-mass) axis and the new parallel axis.
What is the difference between lb·ft·s² and lb·ft²?
lb·ft·s² uses the slug as the unit of mass (1 slug = 14.594 kg). This is the fully coherent imperial unit: I = slug × ft². Its SI equivalent is 1 lb·ft·s² ≈ 1.3558 kg·m².
lb·ft² uses pound-mass directly (1 lbm = 0.4536 kg). It is common in engineering software but is not coherent with F = ma in imperial units. Its SI equivalent is 1 lb·ft² ≈ 0.04214 kg·m².
What is rotational kinetic energy?
The kinetic energy of a rotating body is:
E_rot = ½ · I · ω²
where ω is the angular velocity in radians per second. This is the rotational analogue of E_k = ½·m·v².
How is torque related to the moment of inertia?
Newton's second law for rotation states:
τ = I · α
where τ is torque (N·m or lb·ft) and α is angular acceleration (rad/s²). A larger moment of inertia means more torque is required to achieve the same angular acceleration.
Can I add moments of inertia for compound objects?
Yes. For a compound body consisting of several parts, the total moment of inertia about a common axis is simply the sum of the individual moments of inertia (applying the parallel-axis theorem when necessary to shift each part's I to the common axis).