Moment of inertia

An object resists changes in its rotational motion. Its moment of inertia, $$$I$$$ , describes this resistance about a specified axis of rotation. It depends not only on the object's mass, but also on how that mass is distributed relative to the axis.

Moment of inertia of a point mass

For a point mass $$$m$$$ at perpendicular distance $$$r$$$ from the axis, the moment of inertia is:

I = mr^2

The distance is measured at right angles to the axis. Because it is squared, a mass twice as far from the axis contributes four times as much to the moment of inertia.

Moment of inertia of an extended object

An extended object has mass at different distances from the axis. Its moment of inertia is the sum of the contributions from its mass elements:

I = \sum_i m_i r_i^2

Here, $$$m_i$$$ is the mass of an element and $$$r_i$$$ is its perpendicular distance from the axis. The SI unit of moment of inertia is $\mathrm{kg,m^2}$ .

Mass farther from the axis contributes more strongly than the same mass closer to it. Two objects with the same total mass can therefore have different moments of inertia if their mass is distributed differently. Changing the position or orientation of the axis can also change the distances of the mass elements from it, and hence change the moment of inertia.