How to Use Mass Moment of Inertia Calculator
The Mass Moment of Inertia Calculator computes rotational inertia for common shapes and applies the parallel axis theorem.
- Select the Shape — Choose from the dropdown: solid sphere, hollow sphere, solid cylinder, hollow cylinder, thin rod, rectangular plate, or point mass.
- Enter Mass and Dimensions — Fill in the mass and the required geometric parameters (radius, length, inner/outer radius, plate dimensions, or offset distance).
- Read the Result — The Mass Moment of Inertia Calculator displays I about the primary axis in your chosen unit system. If you entered a parallel-axis offset d, the shifted I is shown alongside.
- Switch Unit Systems — Use the unit toggle to compare results in SI (kg·m²) and Imperial (slug·ft² or lb·in²) simultaneously.
Formula & Theory — Mass Moment of Inertia Calculator
The Mass Moment of Inertia Calculator implements the standard closed-form formulas for each shape:
| Shape | Formula |
|---|---|
| Solid sphere | I = (2/5) m r² |
| Thin hollow sphere | I = (2/3) m r² |
| Solid cylinder / disk | I = (1/2) m r² |
| Hollow cylinder (annulus) | I = (1/2) m (r₁² + r₂²) |
| Thin rod (about centre) | I = (1/12) m L² |
| Thin rod (about end) | I = (1/3) m L² |
| Rectangular plate (cm) | I = (1/12) m (a² + b²) |
| Point mass | I = m r² |
Parallel Axis Theorem
I = I_cm + m × d²where I_cm is the moment about the centre-of-mass axis and d is the perpendicular distance to the new (parallel) axis.
Physical Intuition
Objects with mass concentrated near the rotation axis (small r) have low I and spin up quickly with little torque. Objects with mass far from the axis (large r) have high I — this is the principle behind flywheels (high I for energy storage) and figure skaters tucking their arms to spin faster (reducing I).
Unit Conversion Reference
| From | To | Factor |
|---|---|---|
| 1 kg·m² | g·cm² | × 10,000,000 |
| 1 kg·m² | slug·ft² | × 0.7376 |
| 1 kg·m² | lb·in² | × 3,417.2 |
Use Cases for Mass Moment of Inertia Calculator
The Mass Moment of Inertia Calculator supports engineers, physicists, and students across many rotating-system disciplines:
- Mechanical engineering design — Size motors, gearboxes, and brake systems for rotating machinery by calculating the total system inertia including shafts, disks, and attached loads.
- Robotics — Compute joint torque requirements for robot arms and legs; accurate inertia values are essential for stable and efficient trajectory control.
- Aerospace — Calculate satellite moment of inertia for spin stabilisation, reaction-wheel sizing, and attitude control system design.
- Automotive engineering — Analyse flywheel, crankshaft, and wheel inertia to optimise engine response and drivetrain efficiency.
- Physics education — Rotational mechanics coursework requires moment-of-inertia calculations for lab reports, homework, and exam preparation.
- Sports science — Model bat swing inertia, discus throw, or bicycle wheel dynamics to understand how mass distribution affects athletic performance.