Biomechanics BME 315
Rod Lakes
University of Wisconsin

Scaling concepts.


Consider animals or humans of different size L. Determine the dependence on L of the running speed up a steep hill. The hill is steep enough that most of the effort goes into raising the body weight up the hill against gravity, but is not so steep to require climbing.

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.

The downward force is mg.
The power required to run at speed v up the hill is P_grav = mg v sin q with q as the slope of the hill. This neglects the power to accelerate the legs; recall we assumed the hill was sufficiently steep.

The metabolic power is assumed to be limited by transport of gases in the lungs, which have area proportional to L².
P_metabolic is proportional to L².

P_metabolic = P_grav so with A as a constant of proportionality, A g L³ v = L² so

v = 1/AgL

So, larger organisms have a disadvantage going up hill.

Actually P_metabolic is proportional to m^3/4 not m^2/3. The distinction is seen only over a wide range of body size. The difference is attributed to the fractal nature of the branching of tubules in the respiratory and circulatory system.

Question. A rhinoceros weighs four tons and can run 25 miles an hour on level ground. It does not appreciate your presence and decides to charge. Should you run on level ground or run up a nearby hill?