About

In rotational dynamics the Moment of Inertia (I) plays the same role that mass plays in linear motion. It measures an object's resistance to angular acceleration. Engineers designing flywheels, shafts, or structural beams must calculate this value precisely to predict stress and motion. This tool provides a library of common geometric shapes and their inertia formulas.

A critical feature for practical engineering is the Parallel Axis Theorem. Often an object does not rotate around its center of mass. This calculator allows users to define an offset distance d automatically adjusting the result using I = I_cm + Md². It also outputs the Radius of Gyration (k) which normalizes the mass distribution into a single length parameter.

Formulas

The calculation is based on the distribution of mass relative to the axis of rotation.

Parallel Axis Theorem:

I_new = I_cm + Md²

Radius of Gyration:

k = √IM

Users must ensure units are consistent. If Mass is in kg and Distance in meters the result is kg⋅m².

Reference Data

Shape	Axis	Formula (I)
Solid Sphere	Diameter	25MR²
Hollow Sphere	Diameter	23MR²
Solid Cylinder	Central Axis	12MR²
Rect Plate	Center (perp)	112M(h² + w²)
Slender Rod	Center	112ML²
Slender Rod	End	13ML²
Hoop / Ring	Central Axis	MR²

Frequently Asked Questions

It is a rule used to determine the moment of inertia of a rigid body about any axis, given the moment of inertia about a parallel axis through the object's center of mass and the perpendicular distance between the axes.

Because all the mass in a hollow cylinder is concentrated as far as possible from the axis (at radius R). In a solid cylinder the mass is distributed from 0 to R. Mass further from the axis creates more resistance to rotation.

It is the distance from the axis of rotation to a point where, if the entire mass of the body were concentrated there, its moment of inertia would remain the same.