Why might a small increase in the dimensions of an object cause a large change in its volume?

Answer 1

Does size matter?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.

So why are cells so small?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.

Length, Area and VolumeAs 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.ParametersCase ICase IICase IIICase IVLength (L)1234Face Area (L2)14916Volume (L3)182764Surface Area(L2 x 6 faces)6245496Area/Volume ratio6321.5

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.

ParametersCase I

(64 cubes)Case II

(1 cube)Length (L)14Number of cubes641Volume (L3)6464Surface Area(L2 * 6 faces

* number of cubes)614496Surface Area/Volume ratio961.5

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.

Bigger than lifeThe 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.

The gravity of the situationShaq 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

FaS2 1----=-----=--- . --FggS3g 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.

Forces and ScaleInteractions 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. It's the little things that countSurface 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.

Don't sweat the small stuff: when size doesn't matterSize does not influence everything. There are some properties that respond the same way, no matter at which scale they occur. Packing of spheresHave 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."

DensityWater 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!

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