Moment of Inertia

Just as mass measures a body's inherent resistance to any change in its state of linear motion (its inertia), moment of inertia measures a body's resistance to a change in its state of rotational motion — it is the direct rotational analogue of mass, and it emerged naturally already in §5.2.3 and §5.2.5, as the quantity $\sum_i m_ir_i^2$ appearing in both the torque-angular-acceleration relation and the angular-momentum relation for a rigid body.

For a single point mass $m_i$ at perpendicular distance $r_i$ from a fixed axis:

For a rigid bulk object, built from many such point masses, the total moment of inertia about the axis is the sum over all of them:

Its SI unit is kg m$^2$, and its dimension is $[ML^2]$.

A crucial contrast with ordinary mass. Mass is, for all practical (non-relativistic) purposes, an invariable property of a given quantity of matter. Moment of inertia is emphatically not invariable in this way — it depends not merely on how much mass a body has, but critically on exactly how that mass is distributed relative to the particular axis chosen. The very same rigid body therefore has a different moment of inertia for every different choice of rotation axis, even for axes lying entirely outside the physical body itself.

Finding $I$ for a continuous, uniformly-distributed body. Treat an infinitesimally small mass element $dm$, at perpendicular distance $r$ from the axis, as a point mass, so its own contribution to the moment of inertia is

The moment of inertia of the entire bulk object is obtained by integrating this expression over the whole body: …