Moment of Inertia Converter

Moment of inertia — also called rotational inertia or second moment of mass — is the rotational analogue of mass in linear dynamics. Just as mass resists changes in linear velocity (Newton's First Law), moment of inertia resists changes in angular velocity. It quantifies how difficult it is to accelerate or decelerate a rotating object, and it depends not only on the total mass but on how that mass is distributed relative to the axis of rotation. Mass close to the axis contributes little; mass far from the axis contributes greatly.

For a point mass m at distance r from the axis, moment of inertia I = m × r². For extended objects, it is the integral I = ∫r² dm over the entire body. This means that a hollow cylinder of the same mass as a solid cylinder has a higher moment of inertia — its mass is concentrated at a larger radius. This principle explains why figure skaters spin faster when they pull in their arms (reducing r) and slow down when they extend them (increasing r).

The SI unit of moment of inertia is the kilogram square meter (kg·m²). This is the standard in physics, mechanical engineering, aerospace engineering, and international scientific literature. Newton's second law for rotation — Torque = Moment of Inertia × Angular Acceleration (τ = I × α) — requires consistent units: when I is in kg·m² and α is in rad/s², torque τ comes out in N·m. Converting moment of inertia to consistent units before applying this equation is therefore essential for correct results.

In the United States and aerospace industries, the traditional unit is slug square foot (slug·ft²), part of the British Gravitational (BG) system. 1 slug·ft² = 1.35582 kg·m². Aircraft, spacecraft, and rocket moment of inertia values in US engineering documents are almost always given in slug·ft². Converting to kg·m² for simulation software (MATLAB, Simulink, ANSYS) or international specifications is a routine task for aerospace engineers.

The pound square foot (lb·ft²) uses the pound as a unit of mass rather than force. 1 lb·ft² = 0.042140 kg·m². This unit appears in some dynamics textbooks and older US engineering documents. The related pound square inch (lb·in²) = 2.926 × 10⁻⁴ kg·m² is used for smaller rotating components such as motor armatures and compact flywheels.

Ounce square inch (oz·in²) is extremely common in servo motor and stepper motor datasheets. Motor manufacturers specify rotor inertia in oz·in² because it gives convenient numerical values for small motors — a typical servo motor rotor might have 0.5–5 oz·in² of inertia. 1 oz·in² = 1.829 × 10⁻⁵ kg·m². When sizing servo systems, engineers compare load inertia to motor rotor inertia — the load-to-rotor inertia ratio should generally be under 10:1 for optimal performance — and converting oz·in² to kg·m² is a necessary step.

The pound-force foot square second (lbf·ft·s²) and pound-force inch square second (lbf·in·s²) come from the weight-based US system where force (lbf) rather than mass (lb or slug) is the base unit. These are related to slug units: 1 lbf·ft·s² = 1 slug·ft² = 1.35582 kg·m². They appear in older machinery manuals and some testing standards.

The metric-gravitational system uses kilogram-force meter square second (kgf·m·s²) and kilogram-force centimeter square second (kgf·cm·s²). 1 kgf·m·s² = 9.80665 kg·m². These units appear in older European machinery documentation and some industrial motor datasheets from manufacturers that used the gravitational metric system before full SI adoption.

Gram square centimeter (g·cm²) and gram square millimeter (g·mm²) belong to the CGS system and appear in precision instrument design, watch movement engineering, optical mirror actuators, and micro-robotics where masses and dimensions are in grams and centimeters or millimeters. This moment of inertia converter supports all 14 units — kilogram square meter, kilogram square centimeter, kilogram square millimeter, gram square centimeter, gram square millimeter, kgf·m·s², kgf·cm·s², oz·in², ozf·in·s², lb·ft², lbf·ft·s², lb·in², lbf·in·s², and slug·ft² — instantly, precisely to 12 significant digits, completely free.