Mass measures resistance to translational acceleration. For rotation, resistance also depends on how far the mass lies from the rotation axis. This rotational inertia is measured by the moment of inertia.

For point masses rotating about a fixed axis,

$$I=\sum_i m_i r_i^2,$$

where $r_i$ is the perpendicular distance from mass $m_i$ to the axis.

Distribution matters

Two objects with the same total mass can have different moments of inertia. Moving mass farther from the axis increases $I$ because the distance appears squared.

For example, moving a point mass from radius $r$ to $2r$ multiplies its contribution by four:

$$m(2r)^2=4mr^2.$$

The axis matters

Moment of inertia is not a property of an object alone; it is defined relative to a chosen rotation axis.

The same rigid body can therefore have different moments of inertia about different axes.

Continuous bodies

For a continuous body, the same idea applies by adding the contributions of many very small pieces of mass. Integration provides the corresponding continuous calculation once that mathematical tool is available.

Moment of inertia plays the same structural role in fixed-axis rotational dynamics that mass plays in translational dynamics.