Moment of Inertia Converter

Convert moment of inertia between kg·m², kg·cm², lb·ft², lb·in², and other units with scientific precision.

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Moment of Inertia Units Explained

Kilogram Square Meter (kg·m²)

The SI unit of moment of inertia. It represents rotational resistance of a mass about an axis. I = Σ m × r².

Common uses: Rotational dynamics, mechanical engineering, international standards.

Kilogram Square Centimeter (kg·cm²)

Moment of inertia in metric units using centimeters. 1 kg·cm² = 0.0001 kg·m². Common in machinery and rotating equipment specifications.

Common uses: Motor specifications, machinery design, rotating equipment.

Gram Square Centimeter (g·cm²)

Moment of inertia using grams and centimeters. 1 g·cm² = 0.0001 kg·m². Used in older scientific literature and some specialized applications.

Note: Commonly used in optical and precision systems.

Pound Square Foot (lb·ft²)

Moment of inertia in US customary units. 1 lb·ft² ≈ 0.4214 kg·m². Common in US engineering.

Common uses: US mechanical design, automotive engineering, aerospace applications.

Pound Square Inch (lb·in²)

Moment of inertia for smaller mass distributions. 1 lb·in² ≈ 0.00291 kg·m². Used for compact equipment and precision systems.

Common uses: Motor design, precision machinery, aerospace components.

Slug Square Foot (slug·ft²)

Moment of inertia using slug mass units. 1 slug·ft² ≈ 1.3558 kg·m². Slug-based equivalent to lb·ft².

Note: Results in cleaner numerical relationships in US customary calculations.

Rotational Dynamics Equations

Moment of inertia is fundamental to rotational motion:

  • Torque: τ = I × α (analogous to F = m × a)
  • Rotational kinetic energy: KE = ½ × I × ω²
  • Angular momentum: L = I × ω

Moment of Inertia for Common Shapes

  • Solid cylinder (rotation about axis): I = ½MR²
  • Hollow cylinder (rotation about axis): I = ½M(R₁² + R₂²)
  • Solid sphere (rotation about axis through center): I = ⅖MR²
  • Thin rod (rotation about center): I = (1/12)ML²
  • Thin rod (rotation about end): I = (1/3)ML²
  • Rectangular plate (rotation about center): I = (1/12)M(a² + b²)

Parallel Axis Theorem

Moment of inertia about any axis parallel to an axis through the center of mass: I = I_cm + M × d²

  • I: Moment of inertia about the new axis
  • I_cm: Moment of inertia about center of mass
  • M: Mass of the object
  • d: Distance between axes

Common Applications

Moment of inertia is critical in:

  • Motor Design: Starting torque, acceleration capability
  • Flywheels: Energy storage, speed regulation
  • Vehicle Dynamics: Handling, stability, acceleration
  • Rotating Machinery: Vibration analysis, balancing
  • Gyroscopes: Stability and precession calculations
  • Structural Analysis: Beam deflection, buckling resistance
  • Precision Equipment: Speed and torque control

Polar Moment of Inertia vs. Second Moment of Area

Note: "Moment of inertia" has different meanings in different contexts:

  • Mass moment of inertia (I): Rotational property of mass, units: kg·m²
  • Second moment of area (J): Geometric property of cross-sections, units: m⁴
  • Polar moment of area (I_p): Second moment about central axis, units: m⁴

Relationship: I = ρ × J (where ρ is density)