Rotational mechanics tool

Moment of Inertia Calculator

Calculate moment of inertia for common shapes using mass and radius or length.

Moment of inertia solver

Calculate moment of inertia

Select an object shape, enter its mass and radius or length, then calculate I.

Formula: I = ½mr²

kg

Enter a positive mass in kilograms.

m

Enter a positive radius in meters.

Try an example

Calculator guide

How this calculator works

Moment of Inertia Calculator determines the resistance of an object to rotational acceleration based on mass distribution, geometry, and rotational axis. It helps physics students, engineers, and laboratory users analyze rods, disks, spheres, flywheels, and rotating mechanical systems.

Formula explanation

Moment of inertia describes resistance to rotational acceleration and depends on mass distribution.

Formula

Moment of Inertia = Σ(mr²) for point masses; specific shapes use standard rotational inertia formulas

Worked example

Example: Objects with mass farther from the rotation axis have greater inertia.

Assumptions

  • Mass and geometry measurements are accurate.
  • The selected object model matches the physical system being analyzed.
  • Input values use compatible units.

Examples

  • Example: Calculate the moment of inertia of a rotating object when mass and radius are known.
  • Example: Engineers compare different shapes to optimize rotational performance in mechanical systems.

Common mistakes

  • Using incorrect geometry assumptions.
  • Ignoring rotation axis.
  • Confusing mass with inertia.

Variables

  • Mass of object
  • Distance from rotational axis
  • Shape and geometry
  • Moment of inertia
  • Rotational axis
  • Measurement units

Limitations

  • Results depend on accurate mass distribution and geometry assumptions.
  • Complex objects with irregular mass distributions may require advanced integration methods, CAD analysis, or experimental measurements.

Scientific references

  • OpenStax University Physics: Rotational Motion
  • NIST SI Units and Measurement References
  • Classical mechanics principles

Content review

Reviewed by: ScienceCalcHub Physics Review Team | Last reviewed: 2026-08-30

Applications

  • Physics education
  • Rotational mechanics analysis
  • Engineering design studies
  • Flywheel and rotating machinery design
  • Laboratory rotational motion experiments

Frequently asked questions

How is moment of inertia calculated?

Moment of inertia is calculated by considering how mass is distributed relative to the axis of rotation. For point masses, it is calculated using I = mr².

Why is moment of inertia important?

Moment of inertia determines how much resistance an object has against changes in rotational motion.

What factors affect moment of inertia?

Moment of inertia depends on the object's mass, shape, size, and location of the rotational axis.

How does shape affect moment of inertia?

Shape affects moment of inertia because objects with the same mass can have different resistance to rotation depending on how their mass is distributed around the axis.

Accuracy and transparency

Created and maintained by our editorial team

This physics calculator is maintained by the ScienceCalcHub Editorial Team. Its calculation logic is tested with representative inputs, while the supporting guidance is checked for formula clarity, units, assumptions, and common mistakes.

Written by
ScienceCalcHub Editorial Team
Reviewed by
ScienceCalcHub Scientific Review Team
Review standard
Formula accuracy, units, examples, and educational clarity

Learn more about our formula-review and correction process, explore our calculation methodology, or view our scientific references.

  • Calculation logic tested
  • Variables and units explained
  • Assumptions stated clearly
  • Corrections handled transparently

Physics overview

What is moment of inertia?

Moment of inertia is the rotational equivalent of mass. It describes how difficult it is to change an object's rotational motion around a selected axis.

Standard formulas

Moment of inertia formulas by shape

Point mass

I = mr²

Solid disk or cylinder

I = ½mr²

Thin hoop or ring

I = mr²

Solid sphere

I = ⅖mr²

Thin spherical shell

I = ⅔mr²

Rod through center

I = ¹⁄₁₂mL²

Rod through one end

I = ⅓mL²

Formula variables

What each symbol means

  • I is moment of inertia in kg·m².
  • m is mass in kilograms.
  • r is radius in meters.
  • L is rod length in meters.

Worked example

Solid disk with mass 4 kg and radius 3 m

  1. Use the solid disk formula: I = ½mr².
  2. Substitute the values: I = ½ × 4 × 3².
  3. Square the radius: 3² = 9.
  4. Calculate the result: I = 18 kg·m².

Rotation axis

Why the axis matters

The same object can have different moments of inertia depending on the selected axis. A rod rotating through its center has a lower moment of inertia than the same rod rotating through one end.

Mass distribution

Distance from the axis changes I

Mass farther from the axis contributes more strongly because radius or length is squared in the formula. This is why hoops usually have greater moment of inertia than solid disks of equal mass and radius.

Units

Standard SI units

Use kilograms for mass and meters for radius or length. The resulting moment of inertia is expressed in kilogram square meters, written as kg·m².

Real-world applications

Where moment of inertia is used

  • Flywheels and rotating machinery.
  • Wheels, gears, and pulleys.
  • Sports equipment and body rotation.
  • Turbines and electric motors.
  • Satellites and spacecraft attitude control.

Calculation limits

Important assumptions

  • The object is treated as an ideal geometric shape.
  • Mass is assumed to be distributed uniformly.
  • The specified axis matches the selected formula.
  • Mass and dimensions must be greater than zero.
  • Units are not converted automatically.

Related tools

Use the Rotational Kinetic Energy Calculator to calculate energy using moment of inertia and angular velocity.

Use the Torque Calculator to calculate torque, force, or lever arm distance.

Questions and answers

Calculator FAQ

How is moment of inertia calculated?

Moment of inertia is calculated by considering how mass is distributed relative to the axis of rotation. For point masses, it is calculated using I = mr².

Why is moment of inertia important?

Moment of inertia determines how much resistance an object has against changes in rotational motion.

What factors affect moment of inertia?

Moment of inertia depends on the object's mass, shape, size, and location of the rotational axis.

How does shape affect moment of inertia?

Shape affects moment of inertia because objects with the same mass can have different resistance to rotation depending on how their mass is distributed around the axis.