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Discussion questions
A. A toy fire engine is 1/30 the size of the real one, but is constructed from the
same metal with the same proportions. How many times smaller is its weight.
How many times less red paint would be needed to paint it.
B. Galileo spends a lot of time in his dialog discussing what really happens
when things break. He discusses everything in terms of Aristotle’s now-
discredited explanation that things are hard to break, because if something
breaks, there has to be a gap between the two halves with nothing in between,
at least initially. Nature, according to Aristotle, “abhors a vacuum,” i.e. nature
doesn’t “like” empty space to exist. Of course, air will rush into the gap
immediately, but at the very moment of breaking, Aristotle imagined a vacuum
in the gap. Is Aristotle’s explanation of why it is hard to break things an
experimentally testable statement. If so, how could it be tested
experimentally.
1.3Scaling Applied to Biology
Organisms of different sizes with the same shape
The first of the following graphs shows the approximate validity of the
proportionality m
.
L
3
for cockroaches (redrawn from McMahon and
Bonner). The scatter of the points around the curve indicates that some
cockroaches are proportioned slightly differently from others, but in general
the data seem well described by m
.
L
3
. That means that the largest cock-
roaches the experimenter could raise (is there a 4-H prize.) had roughly the
same shape as the smallest ones.
Another relationship that should exist for animals of different sizes
shaped in the same way is that between surface area and body mass. If all
the animals have the same average density, then body mass should be
proportional to the cube of the animal’s linear size, m
.
L
3
, while surface
area should vary proportionately to L
2
. Therefore, the animals’ surface areas
should be proportional to m
2/3
. As shown in the second graph, this relation-
ship appears to hold quite well for the dwarf siren, a type of salamander.
Notice how the curve bends over, meaning that the surface area does not
increase as quickly as body mass, e.g. a salamander with eight times more
body mass will have only four times more surface area.
This behavior of the ratio of surface area to mass (or, equivalently, the
ratio of surface area to volume) has important consequences for mammals,
which must maintain a constant body temperature. It would make sense for
the rate of heat loss through the animal’s skin to be proportional to its
surface area, so we should expect small animals, having large ratios of
surface area to volume, to need to produce a great deal of heat in compari-
son to their size to avoid dying from low body temperature. This expecta-
tion is borne out by the data of the third graph, showing the rate of oxygen
consumption of guinea pigs as a function of their body mass. Neither an
animal’s heat production nor its surface area is convenient to measure, but
in order to produce heat, the animal must metabolize oxygen, so oxygen
consumption is a good indicator of the rate of heat production. Since
surface area is proportional to m
2/3
, the proportionality of the rate of oxygen
consumption to m
2/3
is consistent with the idea that the animal needs to
produce heat at a rate in proportion to its surface area. Although the smaller
animals metabolize less oxygen and produce less heat in absolute terms, the
amount of food and oxygen they must consume is greater in proportion to
their own mass. The Etruscan pigmy shrew, weighing in at 2 grams as an
Chapter 1Scaling and Order-of-Magnitude Estimates