(2006-01-03) Scaling according to Galileo
From ants to elephants, following the father of modern physics.
For a given material, the strength of a structure depends on its various
cross-sections and is thus proportional to the square of the overall size.
As weight is proportional to the cube of size, such a structure
would therefore collapse if scaled-up beyond a certain size.
This subject was first discussed by Galileo Galilei (1564-1642)
who put the following words in the mouth of Salviati,
on the "second day" of the
Dialogues Concerning Two New Sciences (1638).
You can plainly see the impossibility of
increasing the size of structures to vast dimensions [...]
If his height be increased inordinately,
he would be crushed under his own weight.
We may reasonably expect the dynamic forces which make a creature
jump to be proportional to the cross-section of its muscles,
just like static forces are expected to be proportional to
the cross-sections of its bones.
At the very least, this is a much better starting point than
the popular misguided assumption discussed in the
following article, which would
have us believe that humans built like scaled-up fleas
could jump over skycrapers!
(2006-01-03) On the jump of a flea
Are fleas really much better jumpers than people?
Les puces peuvent sauter 135 fois leur taille.
C'est comme si
un homme sautait aussi haut qu'un immeuble de
65 étages.
CaramBar
Info [inside candy wrapping]
French kiddy sensationalism notwithstanding,
the performance of creatures having
vastly different sizes should not be compared using
the linear scaling implied by the above
comparison between men and fleas...
Taking the above at face value, a jumping creature would be expected
to release an energy roughly proportional to its volume
(limbs apply a force proportional to the square of the
size, along a launching trajectory proportional to the size).
This mechanical energy is thus expected to be proportional to the creature's
volume or its mass [since all living tissues have roughly the same density].
Neglecting air resistance, this would mean that all jumping creatures are
expected to jump to about the same height, not
very different heights proportional to their sizes...
Fair comparisons of the jumping performances of various animals
are best based on the ratio of the aforementioned mechanical energy
to the mass of the creature
(this ratio is equal to half the square of the speed reached at liftoff).
People can jump up only slightly more than a foot in height.
Gifted athletes can do significantly better, but
an athlete who clears a bar several feet off the ground does so partly because
his center of gravity is already about 3 feet high to begin with,
and also because the center of gravity of his bent body
may stay under the bar...
The human flea (pulex irritans) is commonly
quoted as being able to jump about
a foot in length, or a few inches in height.
This is commensurate with human performance, as predicted.
The size of the flea is essentially irrelevant...
(2007-08-03) Drag coefficient & Reynolds number
On the resistive force exerted by a fluid on a sphere at constant velocity.
In 1883,
Osborne Reynolds
(1842-1912) introduced a dimensionless parameter
as he investigated the transition from laminar to turbulent flow
for fluids in pipes.
That parameter R was first called "Reynolds number"
by Arnold Sommerfeld as he used it in what's now known as the
Orr-Sommerfeld equation which he introduced in the paper
entitled "Ein Beitrag zur hydrodynamischen Erklärung der turbulenten
Flüssigkeitsbewegung" presented in Rome in 1908,
at the 4th International Congress of Mathematicians (3, 116-124).
The uniform motion of a sphere through a fluid involves
the following quantities:
- The mass of the sphere: m.
- The radius of the sphere: r.
- The dynamic viscosity
of the fluid: h.
- The density of the fluid: r.
- The speed of the sphere relative to the fluid: v.
- The resistive force: F.
Those form four relevant quantities:
r (in m),
v (in m/s),
F/m (acceleration, in m/s2 )
and h/r (kinematic viscosity,
in m2/s).
As two units are involved, there must be two dimensionless parameters which
are functions of each other. One is the drag coefficient
(C) the other is the aforementioned Reynolds number (R).
Other such pairs of parameters would be acceptable,
but this is the traditional choice which we do retain.
