Moment of Inertia

Moment of inertia is the rotational analogue of mass: just as a body's ordinary (inertial) mass

measures its reluctance to change its state of straight-line motion under a force, moment of inertia

measures a rigid body's reluctance to change its state of ROTATIONAL motion about a given axis, under a

torque.

For a single point particle of mass $m$, at a perpendicular distance $r$ from a chosen axis, its moment of

inertia about that axis is defined as

For a rigid body made up of many particles $m_1, m_2, \ldots$ at perpendicular distances $r_1, r_2, \ldots$ from the axis, the moment of inertia of the whole body is the sum of each particle's own

contribution:

and for a continuous body this sum becomes an integral, $I = \int r^2\,dm$, taken over every infinitesimal

mass element of the body.

Why moment of inertia is NOT determined by mass alone. Ordinary mass is a single, fixed number for a

given body, entirely independent of anything else. Moment of inertia is fundamentally different: the very

same body, with the very same total mass $M$, can have MANY different moments of inertia, depending

entirely on WHICH axis is chosen and on how the body's mass happens to be distributed relative to that

particular axis. Two bodies of exactly equal mass $M$ and equal outer radius $R$ -- a thin ring and a

uniform disc, say -- have different moments of inertia about the same central axis (Section 5.11), simply

because the ring's material is concentrated entirely at the largest possible distance $R$ from the axis,

while the disc's material is spread out at every distance from $0$ up to $R$: since the $r^2$ term in the

defining sum weights mass located farther from the axis much more heavily than mass located close to it,

the ring -- with all of its mass at the maximum radius -- ends up with the LARGER moment of inertia of the

two, even though both bodies have identical mass and identical outer size.

This dependence on both the amount AND the distribution of mass, relative to a specific chosen axis, is …