Moment of Inertia

This calculator calculates the moment of inertia of a figure rotating around a given axis, also shows an animation of the rotating figure and the calculation formula.

Moment of Inertia

FigureHollow sphere Point Solid cylinder Solid sphere Thick walled cylindrical tube Thin circular loop Thin cylindrical shell with open ends Thin disk Thin disk with a concentric hole Thin rod Two points

Rotation aroundCenter Center of mass Cylinder axis Diameter Perpendicular to the center Perpendicular to the end

Mass

Second object mass

Radius

Distance

Length

Inner radius

Calculate

Calculation precision

Digits after the decimal point: 2

Moment of Inertia

 

Formula

 

Picture

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The moment of inertia is a measure of the inertial resistance of the object to changes in its rotational motion about the axis¹. When compared to linear motion, the moment of inertia is analogous to mass. Unlike linear motion, it is not only a property of the object as a mass, but depends on the position of the axis of rotation, as well as the distribution of mass within the object. The further the mass is concentrated from the axis of rotation, the greater the moment of inertia.

We can calculate the moment of inertia as sum of moments of individual particles that constitute the object by formula:
,
where
m_i - is a mass of a particle
r_i - is a shortest distance of a particle to the rotation axis.

For continuous objects this sum becomes the integral of mass elements dm located at a distance r from the axis of rotation:
.
This integral can be evaluated by expressing a mass element as a density times an element of length, area or volume.

It is also useful to apply Parallel Axis Theorem (Huygens–Steiner theorem) , which states moment of Inertia about an axis, which is parallel to the center of the mass axis:
,
where
I_cm - moment of inertia about a parallel axis through the center of the mass,
h - distance between the axes,
M - total mass of the object.

We use several pre-evaluated formulas for the moment of inertia of some figures in our calculator.
If the required formula is not there, write in the comments for which figures you need to calculate the moment of inertia and relative to which axis.