Radius of Gyration

For a body of regular shape with uniform mass distribution, the moment of inertia formula naturally involves the body's total mass together with geometric quantities such as its radius, length or breadth. What is needed is a single, more general way of expressing moment of inertia that captures not just the mass, shape and size of an object, but also (implicitly) its orientation relative to the chosen axis — general enough to apply even to bodies of irregular shape or non-uniform mass distribution. This general expression is

where $M$ is the total mass of the object and $K$ is called the radius of gyration.

The radius of gyration of an object (about a given axis) is the perpendicular distance from that axis to an equivalent single point mass which, if it carried the object's entire mass, would reproduce exactly the same moment of inertia as the real, spread-out body. Since it is a distance, its SI unit is m, and its dimension is $[L]$.

Derivation as a root-mean-square distance. Model a rotating rigid body as made up of point masses $m_1,m_2,m_3,\ldots,m_n$ at perpendicular distances $r_1,r_2,r_3,\ldots,r_n$ from the axis. Its moment of inertia is

If, for simplicity, all $n$ individual masses are taken to be equal ($m_1=m_2=\cdots=m_n=m$), this becomes

Since $nm=M$, the total mass, comparing with $I=MK^2$ gives

$K=\sqrt{\frac{r_1^2+r_2^2+\cdots+r_n^2}{n}}.$ …