Biomechanics BME 315
Rod Lakes
University of Wisconsin

Scaling concepts.

Consider animals or humans of different size L. Determine the ratio of strength to weight as a function of L. Assume that the strength (maximum force exerted by a limb) is limited by the maximum stress in muscle.

The body mass m goes as
m = K r L3
in which r is the density and K is a constant of proportionality. K is in fact a constant provided the shape does not depend on changes in size.

Observe that the maximum stress s in a bone or muscle does not depend significantly on organism size. Such maximum stress in fact is nearly constant for a wide range of species.
The force F in a muscle goes as
F = K1 sL2.
The constant of proportionality depends on the cross sectional shape of the muscle and the ratio of its size to the overall size of the organism. It is in fact constant provided the muscle content of the organism is in fact independent of the organism size L.

The ratio of strength to weight is, with s as stress,
F/W = F/mg = K1 sL2 / Kg r L3

 F/W = (K1s/Kgr) {1/L}.

So the ratio of strength to weight is inversely proportional to the organism size L provided the shape does not depend on changes in size. A massive dinosaur will not be able to get up off the ground as easily as a squirrel. A human athlete who weighs 150 pounds can do pull-up exercises more easily than an athlete of similar body composition who weighs 300 pounds.