Definition

Moment of inertia

I = i \sum m_{i} r_{i}^{2} = \int r^{2} d m

Statement

The sum, over every mass element, of mass times the square of its distance from the rotation axis.

Symbols and units

$I$	moment of inertia	kg·m²
$m_{i}, r_{i}$	mass and distance from axis	kg, m

When it applies

Any rigid body, about a specified axis.

Derivation

Weight each mass element by the square of its distance from the axis, since elements farther out contribute disproportionately more to rotational inertia:

I = i \sum m_{i} r_{i}^{2}

For a continuous body, the sum becomes an integral over mass elements dm:

I = \int r^{2} d m

Worked examples

Moment of inertia of two point masses

Two 1.0 kg masses at r = 0.50 m from the axis, on opposite sides:

I = \sum m_{i} r_{i}^{2} = 2 (1.0) (0.50)^{2} = 0.50 kg\cdotm^{2}

Watch out for

Moment of inertia depends on the choice of axis — the same object has a different I for every axis it could rotate about.

Connects to

Parallel axis theorem