Q.Why is it useful to define radius of gyration?

Radius of gyration lets any object's rotational inertia be written in the same simple point-mass-like form $I=MK^2$, and can be found experimentally even for irregular objects whose I cannot be calculated theoretically. …

Theoretical calculation of moment of inertia by integration ($I=\int r^2\,dm$) is only possible when an object's mass distribution is mathematically simple and integrable -- a real, irregular object generally has neither property known accurately enough. Radius of gyration solves this practically: by defining K through $I=MK^2$, ANY object's moment of inertia -- regardless of how complicated or irregular its shape -- can be expressed in the compact, point-mass-like form $MK^2$, as if its entire mass really were concentrated at the single effective distance K from the axis. This is useful because K can be measured EXPERIMENTALLY (e.g. via a torsion pendulum or a compound pendulum) purely from how the object actually behaves when set into rotation, without ever needing to know its exact internal mass distribution or carry out any integration -- which is essential for real engineering components (flywheels, rotors, vehicle wheels) whose shapes are far too irregular for …

Contrast the limitations of calculating I by integration (needs an integrable, known mass distribution) with what defining K via $I=MK^2$ achieves (a …

Describing radius of gyration merely as 'a formula rearrangement' without explaining WHY it is practically useful -- specifically, that it enables exper …