Biophysics Problem 36

How does the maximum work done by animals of similar shape scale in terms of size L?

Recall that work proportional both to the force applied, and to the distance moved (in the direction of the force); i.e.

\(W = F \;s\)

where \(F\) is the force applied, and s is the distance moved in the direction of the force.

You could find a way to relate force \('F'\) to the scaling factor \('L',\) and then relate distance moved \('s'\) to \('L'.\) Then, you can multiply these two relations together to find out how work \('W'\) depends on \('L'.\)

First, figure out how force \('F'\) exerted by a muscle is related to \('L'.\)

Think of your own arm. If you exercise, you will become stronger, and the muscles will have increased in cross-sectional area. Thus the force exerted by a muscle will depend on area:

\(F\) is proportional to \(L^2\)

You should realize that the distance moved will depend on the length of the muscle, so

\(s\) is proportional to \(L\)

We indicated previously that \(W = F \times s,\) so that means

\(W \propto (L^2 \times L)\\ W \propto L^3\)