Radius of Gyration

The radius of gyration, denoted $K$, of a rigid body about a given axis is defined as the

particular distance from that axis at which, IF the body's ENTIRE mass $M$ were imagined concentrated as a

single point, that point mass would have exactly the same moment of inertia about the axis as the real,

extended body actually has. In symbols, comparing the point-mass formula $I = MK^2$ (Section 5.10) with the

body's real, known moment of inertia $I$,

Radius of gyration is useful because it lets any body's moment of inertia be written in the single, simple

form $I = MK^2$ -- structurally identical to the point-mass formula -- REGARDLESS of how complicated the

body's own true mass distribution actually is; all of that geometric complexity is folded into the one

number $K$, which by definition already has the units of a length (SI unit: metre).

$K$ is not, in general, equal to the body's own geometric size, and it is not fixed for a body -- exactly

like moment of inertia itself, $K$ depends on which axis is chosen. For example, a thin ring of radius $R$

spinning about the central axis perpendicular to its plane has $I = MR^2$, so directly $K = R$ for that

particular axis (every particle of a ring is, of course, already at the same fixed distance $R$ from that

axis, so this result should be expected). A uniform disc of the very same radius $R$, about the very same

kind of central perpendicular axis, instead has $I = \tfrac12 MR^2$, giving $K = R/\sqrt2 \approx 0.707R$ --

noticeably smaller than the disc's own actual physical radius $R$, precisely because much of the disc's …