Question

what is the moment of inertia of a segment scale in proportion to length in terms if scaling principles?

Answer 1

When scaled against body mass, forelimb inertial properties, apart from mass, scale with positive allometry. Fore- and hindlimb mass scale according to geometric similarity (limb mass is propotional to body mass1.0), as do the remaining hindlimb inertial properties.

Across terrestrial mammals lies an astounding diversity of limbs, in terms of the relative proportions of segments, the position of muscle origins and insertions, posture, and function.

Mass, one measure of limb size, is the limb's resistance to linear acceleration. The limb's center of mass is a measure of its mass distribution, a measure of shape, along its proximo-distal length. Regarding rotational or swinging movements, a limb's resistance to angular acceleration is its moment of inertia (MOI). With respect to MOI, the radius of gyration is an alternative measure of the limb's proximo-distal mass distribution. Notably, the radius of gyration (r) is a function of the ratio of limb MOI to mass (m):

or

Assume the pendulum scales with geometric similarity, its length would change with M1/3, mass with M1 and moment of inertia (ML2) with M5/3. We consider two scenarios for the scaling of maximum muscle force. In the first scenario, we assume that muscle force maintains dynamic similarity between animals of different sizes by scaling force in direct proportion to animal mass: Fmusc is directly propotional to M1.

In the second scenario, we instead assume muscle force scales with cross-sectional area: Fmusc is directly propotional to M2/3. In this simple pendulum model, the applied muscle torque Tmusc is the product of the muscle force and a constant muscle moment arm Rmusc:

We assume that the muscle moment arm scales with geometric similarity (M1/3).