 |
| Figure 1: A Bass |
The size of something is important since size affects what it
can do and how it reacts to its environment. For example, the
size of a musical instrument affects the sound it produces. A
tuba produces much lower pitches than could ever be produced by
a French horn. Size and sound are the same reason a bass is as
tall as an adult, a cello the height of a person sitting down
and a violin can fit neatly on your shoulder under your chin.
Each of these stringed instruments produces different pitches,
largely because of their sizes. The length, material, thickness
and tension of the string also determines the pitch of the sound.
 |
| Figure 2: Sequoia |
Size is one of the most important aspects of living organisms,
yet there is enormous size differences among living things. A
giant Sequoia tree is the largest known living organism. But it
doesn't move around, also, the wood of a tree is composed of cells
that are no longer living. So the bulk of a Sequoia is dead cells.
The blue whale is the largest animal, but zoologists, those people
who study animals, say because of their large size, whales would
not be able to live a terrestrial lifestyle. The 100-ton (108g)
landlubber would collapse under it's own weight. The smallest
known organism is the mycoplasma, which weighs in at about 0.1
pg, is 21 orders of magnitude smaller than the great blue whale.
Mycoplasma are somewhat like tiny bacteria.
 |
Figure 3: An elephant,
a mouse, and
a water strider represent the diverse
size of living creatures and how their
size affects the way they live. |
A five-ton elephant has a slower heart rate, respiration and metabolism
than a smaller animal, such as a mouse. A water strider insect
would not be able to "walk on water," if it were the size of the
mouse, and you would never have to worry about a nosy fly on the
wall, if flies were the size of that same mouse.
Although elephants, mice and water striders have diverse sizes,
one thing they share in common is that they are all composed
of cells. But the cells in each of these animals are all about
the same size, about 10 µm in diameter. There are just more
cells in larger organisms. However, there are exceptions to
the size limitation. Some fiber cells in the jute plant can
reach lengths of two meters. These long cells are the reason
why jute was a favored material for ropes before synthetic materials
were developed. Also, mycoplasma, are only 0.1 µm in diameter.
But generally, cells tend to stay in the 10 µm range.
 |
Figure 4: Transmission
electron
microscope (TEM) image of a
plant cell. All metabolic proc-
esses that keep an organism
living occur at the cellular level |
Cells carry out metabolic processes through their cell walls.
Metabolic process are all the chemical processes that occur
at the cellular level that keep a cell, and therefore the whole
organism alive. Protein synthesis, respiration, digestion and
waste removal are examples of metabolic processes. Although
a cell is small, it has a complex molecular composition. Molecules
can range in size from a few nanometers to a few micrometers.
Molecules inside and outside the cell, such as carbohydrates,
enzymes, hormones, nucleic acids must interact and react together.
If a cell a were not limited to a size of only microns in diameter,
then it would be more difficult for the various and sundry organelles,
molecules and even individual atoms to come in contact and do
their stuff. The amount of metabolism a cell does is directly
related to the volume of the cell, however, the ability to do
these things is limited by the surface area of the cell. Since
the surface area per volume decrease as objects get bigger,
there comes a time when a cell is unable to get any larger.
As things get bigger their volumes increase rapidly, but their
surface areas don't increase as quickly. Since a cell is mostly
composed of water, as a cell's volume increases, so does it's
mass. A simple example of the relationship of length, area and
volume is a cube.
 |
| Figure 5: Cubes of
varying sizes. The side of the first cube is 1 unit, the
second cube is 2 units, the third cube is 3 units and the
fourth cube is 4 units. |
| Parameters |
Case I |
Case II |
Case III |
Case IV |
| Length (L) |
1 |
2 |
3 |
4 |
| Face Area (L2) |
1 |
4 |
9 |
16 |
| Volume (L3) |
1 |
8 |
27 |
64 |
| Surface Area(L2 x 6 faces) |
6 |
24 |
54 |
96 |
| Area/Volume ratio |
6 |
3 |
2 |
1.5 |
|
| Table 1: Cubes with
increasing length (L). Comparing the first four cubes: as
each unit of length (L) increases from 1, to 2, 3, and 4
for each new set of cubes, their volumes (L3),
increase from 1 to 8, to 27 and then 64. Each time the length
doubles, the volume increases by eight-fold. Look at cubes
1 and 3. Each edge on the third cube is three times as long
as the edge on the first cube. The third cube has nine times
the surface area (L2 x 6 sides), but 27 times
the volume of the first cube. As the volume increases with
length (L), mass increases at the same rate.
|
| Parameters |
Case I
(64 cubes) |
Case II
(1 cube) |
| Length (L) |
1 |
4 |
| Number of cubes |
64 |
1 |
| Volume (L3) |
64 |
64 |
Surface Area(L2 * 6 faces
* number of cubes) |
6144 |
96 |
| Surface Area/Volume ratio |
96 |
1.5 |
|
| Table 2: Comparision,
between 64 cubes, with L=1, and a cube with L=4. Compare
the set of cubes on the left to the single cube on the right.
Both have a volume (L3) of 64. However, the set with 64
cubes, has 64 times the surface area than the single cube.
The surface area increases while total volume remains constant.
|
The size of the basketball player can affect how many rebounds
and slam dunks he or she makes during a game. If Shaq, the 7-foot
tall, 300-pound, basketball player for the Lakers were twice as
tall, would he be twice as good a ball player?
 |
Figure 6: What if
a Shaq-sized
person was twice as tall? Would
he make twice as many rebounds? |
Probably not. In fact, he would not be able to play at all.
The same surface/volume ratio principle illustrated with the
cubes applies to Shaq. In general, if his height was doubled
and his proportions remained geometrically similar, then his
surface area would quadruple, but his volume and mass would
octuple! He would weigh roughly 2 400 pounds! Not only would
Shaq no longer be able to rebound, but, like the landlubber
blue whale he would be crushed under his own weight. His bones
would no longer be able to support him. Even if crutches and
other devices could support him, his heart would probably not
be able to pump his blood throughout his giant body.
Shaq would be crushed because gravity has a more powerful effect
on bigger things, than on smaller things. In fact, graviational
forces acting upon our mass rule our world. But gravity is negligible
to very small animals and things with high surface to volume ratios.
The dominant force in their worlds is surface force. This can
be illustrated by comparing sugar cubes to powdered sugar.
Sweet success: Why doesn't a sugar cube stick to me as well
as powdered sugar?
It is a matter of size. It is also a balance of forces. The
relative strength of forces acting on an object can depend on
the size of the object.
Semi-quantitative relationships: The relationships
among the sizes of the sugar and the strength of the forces
acting upon them can be shown as:
 |
| Figure 7: The ability
of a sugar cube to stick, depends on a balance of forces. |
Length of side for sugar cube = S
Area of sugar cube in contact with a surface = S2
Volume of sugar cube = S3
Vertical component of Adhesive Force = Fa = 
A = S2
(where ''
is a proportionality constant).
Gravitational Force (weight) = Fg = m * g =  Vg
= gS3
(where ''
is density of the substance and 'g' is the acceleration
due to gravity).
Now, the ratio of stickiness to weight = Fa / Fg
| Fa |
|
S2 |
|
1 |
| ---- |
= |
----- |
= |
--- . -- |
| Fg |
|
gS3
|
|
g S |
This formula assumes a continuum model and smooth surfaces.
Since, ,
and 'g'
are all constants, the above ratio depends only on 'S'. Observe
that the ratio is inversely proportional to the side 'S'. Therefore,
smaller the side, bigger the ratio, and bigger the side, smaller
the ratio.
If the ratio, Fa/Fg is greater than
1, it means that the numerator dominates over the denominator,
that is, Fa (Adhesive force) is greater than Fg
(Gravity force), therefore the cube sticks. If the ratio, Fa/Fg
is less than 1, it means that the numerator is dominated by
the denominato, that is, Fa (Adhesive force) is less
than Fg (Gravity force), therefore the cube falls.
As the cube gets smaller, adhesive forces increasingly dominate
the gravity forces, and thus the smaller sugar cubes stick to
us better than the large ones.
Interactions between objects, in the universe as we understand
it, are governed by four major forces. They are:
- Gravitational Force: Gravity acts between all bodies
with mass. It is probably the most familiar force at the macroscale
and cosmological scale. Yet, we know so little about it, especially
at the smallest scale, where it is the weakest of all of the
forces. It is always attractive. It is a long-ranged force
that is especially dominant at the cosmological and macroscopic
world.
- Electromagnetic Force: Electromagnetic Force is responsible
for the forces that control atomic structure, chemical reactions
and all electromagnetic phenomena. It is a long-ranged force
operating from the macroscopic world all the way down to the
nanoworld . Electromagnetic Force is dominant at the molecular
and atomic scale.
- Weak Force: Weak Force is responsible for beta decay
of particles and nuclei. Beta decay occurs when an unstable
atomic nucleus changes into a nucleus of the same mass number
but different proton numbers. It is a short-ranged force operating
within the confines of the atomic nucleus.
- Strong Force: Strong Force gives the atomic nucleus
its great stability by holding the neutrons and protons tightly
together. It is a very powerful, but short-ranged force, operating
at a size scale similar to that of the atomic nucleus.
The repulsive and attractive nature of these forces, especially
the interplay of Gravitational force, and Electromagnetic force,
with other natural forces, explains, why some insects are able
to walk on water, but you cannot.
Surface area, gravitational forces and surface forces are all
in play when a water strider insect "walks on water". The perimeter
of a water strider's foot that is in contact with the water surface
is a function of the foot's surface area (L2). The
surface force of the water tension depends on the insect's foot
perimeter. But the weight to be supported by the surface force
depends on the volume of the insect. Weight is directly dependent
upon gravity, which in this case is not much, since the insect
is so small.
 |
Figure 8: Because
of a water
strider's small size, it would
not be able towel dry. Surface
adhesion forces would cause
the towel to adhere to the
water strider! |
If by chance the water strider did break the water tension and
take a plunge, it would not be able to dry off with a bug-size
towel. At this size, surface adhesion forces would keep the towel
stuck to it. Likewise, the water strider could put on a bathing
suit for it's dip and it would never have to worry about the suit
coming off when it hit the water during a high dive. First of
all, because of its small size, the insect would float gently
down as frictional forces acting upon its surface overcome the
weak influence of gravity. But also adhesion forces would keep
the suit on it for life. It would also be impossible for the bug
to read a book by the pool, since once the pages were scaled down
to bug-size, surface adhesion would keep the pages stuck together.
Small size and the small influence of gravity are also factors
for flies on the wall, not to mention the millions of other
insects, lizards, frogs, geckos and such that can climb walls
and windows.
Many walls have textures so they do not appear smooth, but
window glass appears smooth to us at our level of size and scale.
However, to small animals, such as geckos and flies, the window
is rough enough to provide "foot holds and toe holds" for climbing.
These wall and window climbers also have "feet" specially designed
for such feats.
The segments, or tarsi, at the end of insect legs possess
claw-like structures that help the insect hold on to apparently
smooth surfaces. These tarsal claws are used to grip the tiny
irregularities on surfaces that are actually rough at their
small scale. Geckos have a somewhat similar strategy. The surface
of the gecko's foot is divided up into miniscule "forests" of
"hooks" that are as small as 0.1 µm in diameter. The high-surface
area foot produces plenty of places to grab a wall surface.
Since these animals are small, they have relatively large
ratios of overall surface area to volume (S/V). One consequence
is that smaller animals are more influenced by surface forces
such as adhesion. When animals increase in size, the S/V ratio
also changes. This results in larger animals being more influenced
by gravity.
If a gecko on the wall were to drink Alice in Wonderland's
growth potion and isometrically grow in size by a factor of
10, its volume and its weight would increase by a factor of
about 125, whereas its surface in contact with the glass would
increase by a factor of about 100. The poor gecko would fall
from the wall because the somewhat increased adhesive forces
could no longer resist the greatly increased pull of gravity.
There are countless ways that size matters, from the sugar
sticking to your fingers, water striders walking on water and
geckos climbing walls. But there are many ways in which size
doesn't matter.
Size does not influence everything. There are some properties
that respond the same way, no matter at which scale they occur.
Have you ever noticed the pattern formed by the oranges so neatly
arranged for display at the grocery store? It is the same pattern
formed by marbles dumped in a container, as well as the individual
berries of a blackberry fruit, or the compound eyes of an insect,
or polystyrene spheres that are only a 200 nanometers in diameter,
and even individual silicon atoms.
All of these spheres, no matter at which scale they occur
all follow the same hexagonal close packing pattern. The reasons
for this pattern are explored in the IN-VSEE module "Music of
the spheres."
Water is denser than ice. That is why your ice cubes float in
your glass of water. Even when those ice cubes are the size of
Titanic-eating icebergs, they still float!
That's about the size of it.
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