What Is Moment of Inertia and How Is It Calculated?
Moment of inertia, also called rotational inertia, is a measure of how strongly an object resists changes to its rotational motion about a given axis. It is the sum of each mass element multiplied by the square of its distance from the axis, expressed in kilogram-square-meters (kg·m²) in the SI system.
The general definition is I = Σ mᵢ rᵢ² (or ∫ r² dm for continuous bodies), where I is the moment of inertia, mᵢ is each mass element, and rᵢ is its perpendicular distance from the axis. For shapes with a different axis, the parallel axis theorem states I = I_cm + m d², where I_cm is the value about a parallel axis through the center of mass and d is the distance between the two axes.
According to the International System of Units (SI) as maintained by the BIPM, rotational inertia has the dimensions of mass times length squared. Newton's second law for rotation, τ = I α, links net torque τ (N·m) to angular acceleration α (rad/s²), so doubling the moment of inertia halves the angular acceleration for the same torque.
How to Use This Moment of Inertia Calculator
Pick the shape and axis, then enter the mass and dimensions. Results update instantly. Here's what each field means:
- Shape and axis: The geometry and where the rotation axis passes. The same rod has I = (1/12) m L² about its center but ⅓ m L² about its end — four times larger.
- Mass m: Total mass in kilograms, assuming uniform density.
- Dimensions: Radius, length, width or height in meters. Convert centimeters by dividing by 100 (50 cm = 0.5 m).
- Parallel axis offset d: Optional distance in meters from the axis through the center of mass to your actual axis. Leave it at 0 if the axis passes through the center.
Inputs are SI (kg and m) worldwide; results are also shown in imperial lb·ft² (1 kg·m² ≈ 23.73 lb·ft²) and the CGS unit g·cm² (1 kg·m² = 10,000,000 g·cm²), which many US and older textbooks use.
Moment of Inertia Formulas for Common Shapes
These are the standard formulas for uniform-density bodies, as listed in university physics and engineering references such as Halliday & Resnick's Fundamentals of Physics.
| Shape and axis | Formula | Example (m = 10 kg) |
|---|---|---|
| Solid cylinder / disk | ½ m r² | r = 0.5 m → 1.25 kg·m² |
| Hollow cylinder / ring | ½ m (r₁² + r₂²) | r₁ = 0.4, r₂ = 0.5 → 1.025 kg·m² |
| Solid sphere | ⅖ m r² | r = 0.5 m → 1.0 kg·m² |
| Thin hollow sphere | ⅔ m r² | r = 0.5 m → 1.667 kg·m² |
| Thin rod, center axis | (1/12) m L² | L = 1 m → 0.833 kg·m² |
| Thin rod, end axis | ⅓ m L² | L = 1 m → 3.333 kg·m² |
| Rectangular plate (⟂ axis) | (1/12) m (a² + b²) | a = 0.6, b = 0.8 → 0.833 kg·m² |
| Point mass | m r² | r = 0.5 m → 2.5 kg·m² |
Frequently Asked Questions
What is the moment of inertia of a solid cylinder?
The moment of inertia of a solid cylinder or disk about its central axis is I = ½ m r², where m is mass in kilograms and r is radius in meters. A 10 kg disk with a 0.5 m radius has I = 1.25 kg·m².
How accurate is this moment of inertia calculator?
The calculator uses exact closed-form textbook formulas, so results are exact for uniform-density, ideal shapes. Real objects with hollow sections, uneven density, or attachments will differ, and complex parts should be measured experimentally or modeled in CAD.
What is the difference between moment of inertia and mass?
Mass measures resistance to linear acceleration, while moment of inertia measures resistance to angular acceleration. Moment of inertia depends on both mass and how far that mass sits from the rotation axis, so the same object has different values for different axes.
How do I reduce the moment of inertia of a rotating part?
Reduce mass, or move mass closer to the axis, because inertia scales with distance squared. Halving a radius cuts a disk's inertia by a factor of four, which is why lightweight flywheels for fast response are small and why figure skaters pull their arms in to spin faster.