# Biophysics Problem 36

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

#### First Step

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 $'L'.$

#### Calculations

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

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