radius of gyration
The distance k = √(I/M) at which a point mass equal to the body's total mass would have the same moment of inertia.
Definition
The radius of gyration k is the single distance from a chosen axis at which, if all the body's mass were concentrated at that one radius, the moment of inertia would equal the body's actual I. It is defined by I = M·k², giving k = √(I/M). Units are metres.
For a solid sphere, k = R·√(2/5) ≈ 0.632·R. For a hollow spherical shell, k = R·√(2/3) ≈ 0.816·R. For a thin hoop, k = R exactly. The radius of gyration is a compact way to summarise how spread-out a body's mass is relative to an axis, without having to quote both the mass and moment of inertia separately.
In structural engineering, the radius of gyration is used to characterise beams under compressive load: the slenderness ratio L/k (length over radius of gyration) controls how a column buckles under axial force. Two beams with the same mass per unit length but different cross-sectional shapes can have very different buckling behaviour, captured entirely by their differing values of k. In flywheel design k determines how much rotational energy is stored at a given ω. The concept lets engineers compare bodies with different mass distributions using a single number of dimension length.