Answer to the Question 12/99
CREATURES GREAT AND SMALL
The question was:
Why is it that creatures of such different sizes as a flea, a grasshopper,
a human, and a lion can jump (i.e., raise their center of gravity)
1 meter in the air at most (give or take a factor of two)?
(3/00) The problem has been solved (6/12/99) by
Ansgar Esztermann (e-mail
ansgar@thphy.uni-duesseldorf.de), who remarked that got his idea
about scaling laws from a book by Isaac Asimov. (Additional list of people who suggested
correct, but slightly different solution can be found below.)
The solution:
The height of a jump is proportional to the ratio between the energy which
can be supplied to the body E and the weight of the body W.
The weight W is proportional to L3, where L
is the linear dimension of the solid. The energy E is proportional
to the product of the force produced by the foot multiplied by the distance along
which this force acts. The latter distance is proportional to the foot size and therefore
proportional to L, while the force is proportional to the cross-section
of the muscles and therefore is proportional to L2. Thus
E is proportional to L3, and the ratio
E/W is independent of the size of the body L !
Mark de Graaf from Technische Universiteit Eindhoven in Netherlands
(e-mail M.J.d.Graaf@tue.nl),
Dan Peleg (e-mail
dan.peleg@analog.com), and
Louis Zammit Mangion from the Department of Physics of
University of Malta (e-mail
lzamm@phys.um.edu.mt)
also correctly solved the problem.
However, they directly claimed that the work performed by a muscle
is proportional to its volume (which is correct - since the force is
proportional to cross-section and distance is proportional to the length
of the muscle). However, it seems to us that the argument presented above
is "cleaner".
Comment: This and similar problems are discussed in the book of
J.M. Smith Mathematical Ideas in Biology (Cambridge University Press,
1968). See also the discussion
by F. Felber from Jaycor, San Diego,
California in a letter to Physics Today, p.11, March 1999.
See also the lecture:
"Scaling: why giants don't exist" by Michael Fowler from Physics Department of
University of Virginia. (We thank L.Z. Mangion for bringing this reference to our attention.)
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