Calculators
Moment of Inertia Calculator
Calculate rotational inertia for common uniform shapes and axes.
Estimate moment of inertia for common idealized shapes about the stated symmetry axis. Choose the shape carefully because mass distribution and axis location determine the result.
0.5 kg·m²
Formula for this idealized case: I = ½mr²
These standard formulas assume uniform mass distribution and the axis named in the shape option. Real parts with holes, offsets, mixed materials, or different axes need a composite-body or parallel-axis calculation.
About This Tool
Moment of inertia describes how strongly a body's mass distribution resists a change in rotational motion about a particular axis. Unlike mass alone, rotational inertia depends on where the mass lies relative to that axis. This calculator applies standard formulas for several uniform idealized shapes and converts mass, length, and inertia units automatically. It is useful for physics exercises, quick rotational estimates, and checking hand calculations when the object's geometry matches one of the listed cases.
How To Use It
- Select the idealized shape and axis that best match the rotational system.
- Enter the object's mass and choose its mass unit.
- Enter radius or distance from the axis for circular/spherical cases, or total rod length for a rod formula.
- Choose the desired moment-of-inertia unit and compare the displayed formula with your geometry before using the result.
Examples
Solid disk
A uniform 4 kg disk with radius 0.5 m has I = ½mr² = 0.5 kg·m² about its central symmetry axis.
Thin ring
A 4 kg thin ring with radius 0.5 m has I = mr² = 1 kg·m², twice the value of the equal-mass solid disk because more mass is concentrated farther from the axis.
Solid sphere
A uniform 5 kg sphere with radius 2 m has I = ⅖mr² = 8 kg·m² about a diameter through its center.
Rod about one end
A uniform 3 kg thin rod 2 m long has I = ⅓mL² = 4 kg·m² about an axis through one end perpendicular to the rod.
Useful Notes
Why the axis matters
Moment of inertia is always defined about an axis. The same object can have different inertia values about different axes because the distances of its mass elements from the axis change. Always match the selected formula's axis to the actual rotation axis.
Mass distribution matters
Moving mass farther from the axis increases rotational inertia because distance enters standard inertia expressions as a squared term. That is why a thin ring has more inertia than a solid disk of equal mass and outer radius about the same central axis.
Point mass and common shapes
A point mass at distance r uses I = mr². Uniform disks/cylinders, rings, spheres, spherical shells, and thin rods have geometry-specific coefficients obtained by integrating mass distribution over the body.
Moment of inertia and torque
For rotation about a fixed axis under suitable rigid-body assumptions, torque and angular acceleration are related by τ = Iα. A larger moment of inertia therefore requires more torque to produce the same angular acceleration.
Scope and composite objects
The calculator assumes uniform idealized geometry. An assembly can often be handled by summing component inertias about the same axis, while an axis displaced from a component's center of mass may require the parallel-axis theorem. Complex CAD parts generally need geometry-specific analysis.
FAQ
What is the SI unit of moment of inertia?
The SI unit for mass moment of inertia is kilogram metre squared (kg·m²).
Is moment of inertia the same as mass?
No. Mass measures translational inertia, while moment of inertia depends on both mass and how that mass is distributed relative to a chosen rotation axis.
Why does a ring have more inertia than a disk?
For equal mass and outer radius, a thin ring places all its mass at the outer radius, while a solid disk has mass distributed from the center outward. The ring therefore has a larger average squared distance from the axis.
Can I use radius for the rod formulas?
No. The rod formulas in this calculator use the rod's full length L. The interface changes the dimension label to rod length when a rod option is selected.
Can this calculate an irregular object's inertia?
Not directly. Irregular or composite objects require their actual mass distribution, often using component sums, the parallel-axis theorem, integration, CAD mass properties, or measurement.
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