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Re: scaling laws



At 14:26 -0400 10/10/03, Robert Cohen wrote:

Thanks Hugh but I think this demonstrates my point. Besides
knowing that there is a greater surface area to volume ratio
for a mouse, one also needs to know that (a) rate of metabolism
is proportional to total blood volume, (b) heat generated is
proportional to metabolism, (c) heat loss is proportional to
surface area and (d) volume pumped by one beat of a heart
is proportional to total blood volume.

I didn't think that I was implying that it was 100% scaling. I
suspect that nothing is that, except maybe geometry by itself. Yes,
there is a little something more, but the "something more" is pretty
basic and should be readily assimilated by beginning students.

All this comes back to the conceptual idea that triggered the thread.
One can do a rough calculation of surface area to volume for the
mouse and the elephant, and it turns out that the mouse has about 36
times as much surface area per unit volume as the elephant. At this
point is sufficient to know that the process of metabolism produces
heat within the animal's body, and because both are mammals, the
metabolic processes are at least very similar, and hence will produce
about the same amount of heat per unit mass. Clearly the mouse has a
whole lot more skin per unit mass than the elephant and so what heat
its metabolic processes produce, will be quickly radiated away, while
the elephant radiates is excess heat away much more slowly relative
to its mass, and hence doesn't have to produce heat at as great a
rate per unit mass as does the mouse.

This can also be phrased in terms of fraction of its body mass that
the animal has to east daily to stay alive. For mice, that fraction
is probably between 10 and 50% (I'm guessing here--I don't know the
actual figure), even though it doesn't eat that much in any absolute
terms, it is a significant fraction of its mass. On the other hand,
the elephant eats a lot every day, but it doesn't come close to the
fraction of its body mass that the mouse does.

It is sufficient that they understand that the excess heat leaves
through the surface of the body. They are familiar with this because
that happens to them (why do they sweat when the exercise?). More
surface are per unit mass, means more heat loss per unit mass. The
metabolic proportionalities are not needed to appreciate what is
going on, if all that is desired from this exercise is to appreciate
that the mouses metabolism must be faster than that of the elephant
if it wants to stay alive.

Hugh
--

Hugh Haskell
<mailto:haskell@ncssm.edu>
<mailto:hhaskell@mindspring.com>

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