Radius of Gyration

k = √(I/M) is the distance from the axis at which all the mass, if concentrated as a thin ring, would give the same I.

Ring: k = R. Disc: k = R/√2. Solid sphere: k = R√(2/5). Hollow shell: k = R√(2/3). Rod about center: k = L/√12.

Useful in rolling motion: a = g sinθ / (1 + k²/R²).