Vogel_1998_chpt30

Vogel_1998_chpt30 - V034 5< 19 Cats" PAW Moi...

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Unformatted text preview: V034 5,. < 19%)) Cats" PAW: Moi Cfiapu/fi: WeeLtcxz/w‘eal a/a/ (c1; JAMWU a/WL (Deny/e , My“; “fa/(L: MM Moms“ Ma Q). C h a p t e r 3 THE MATTER ' or MAGNITUDE ize matters, and like evolution, it will pervade all that follows. For one thing, an effective design for large things often works poorly for small j span an enormous range, from a virtual macromolecule to the largest of _ " human structures. For yet another, nature’s products are generally smaller 7‘ ways, though, the influence of such physical factors on the two often ‘I proves profoundly different; practical reality depends very much on how big something is. ' ‘5. How much size matters has long been recognized. Galileo gave it his full attention, correctly Calculating that (in the absence of air resistance) an animal of any size ought to be able to jump as high as any other. This means that relative to body length, the small ones win hands down. (Even g in the real world of draggy air, fleas are truly impressive, clearing the bar .3? A532”: #7: ‘3?” mm -= m as .A '40 Val-J Lana asau uaLa‘JunLa at several hundred times their own length.)2 The great seventeenth—centu— ry French polymath Descartes put the matter this way: “The only differ— ence I can see between machines and natural objects is that the workings of machines are mostly carried out by apparatus large enough to be readi— ly perceptible by the senses (as is required to make their manufacture humanly possible), whereas natural processes almost always depend on parts so small that they utterly elude our senses.”3 What confuses our intuitions—but didn’t mislead Descartes—43 our own atypical size. The smallest fully competent organism (thus excluding viruses), a bacterium that gives you a mild pneumonia, is about 0.2 micrometers long, about a fifth of a thousandth of a millimeter and just visible as a dot in a good light microscope. The largest in volume is a large whale, a little more than 20 meters or 60 feet long. That’s roughly a . hundred million—fold range. On an appropriate geometric scale, as in Figure 3.1, we humans hug the upper end. At about 2 meters long, we’re about ten times shorter than the biggest whale but ten million times longer than the smallest bacterium. We use, for these comparisons, a geometric rather than an arithmetic scale, counting each additional zero, Organisms Commercial Human Devices 10 kilometers . very lorxg bridge 1 kilometer long freight train 100 meters . i tallest trees big ship 10 meters f largest fishes truck or bus 1 meter human child car engine 100 millimeters largest insect electric razor 10 millimeters ’ smallest fishes watch movement 1 millimeter i smallest insects smallest mechanisms 100 micrometers i plant cells 10 micrometers animal cells 1 micrometer 5- bacteria 100 nanometers virus FIGURE 3. 1 . The size ranges oforganisms and our mechanical devices. Some arbitrary judgments bcgfirgivcness. The longest (zfvery thin) organisms are probably some multinuclcate flngi, and the Great Wll of China is vastly longer than the longest bridge. Nor are subcellular items, such as microtabulcs and bacterial flagella, included 41 - The Matter of Magnitude or order of magnitude, as an equivalent increment. As a pioneer of bio— mathematics, D’Arcy Thompson, puts it, “It is a remarkable thing, worth pausing to reflect upon, that we can pass so easily and in a dozen lines from molecular magnitudes to the dimensions of a Sequoia or a whale. Addition and subtraction, the old arithmetic of the Egyptians, are not powerful enough for such an Operation.”4 (D’Arcy Thompson needs a few words. He’s almost exclusively known for 072 Growth and Form, a large bookwritten in 1917 and again in 1942, unquestionably the best— known work on mechanical aspects of biology. Part of the book’s contin- uing impact—it’s still in print—comes from its shear linguistic splendor; more, perhaps, reflects its accessibility, startling breadth, and creative insight. While certainly worth reading, .as biology it’s strange and anachronistic, a search for a kind of geometrical perfection in nature to whichevolution by natural selection is largely irrelevant. Nonbiologists such as architects often assume that On Growth and Form is in the main- stream of biology or biomechanics. So I hasten to explain that Thompson is a much—beloved godfather rather than someone whose intellectual genes we proudly carry.) Not only are most organisms smaller than we, but in most groups smallness is the ancestral condition and largeness the specizilization.5 Big fossils are impressive, but little ones are more likely to lead somewhere. Nature starts small. Organisms are basically built up from cells rather than divided into cells; the earliest fossils are microscopic. Human tech— nology goes the other way. Our ships, buildings, and bridges may be larg- er than ever, but the factor of increase has been small and the times involved have been long. More impressive is the way our systems (or their parts) have gotten smaller. The first steam engines were enormous, oper— ating slowly and at low pressures. Jet turbines are small, .fast, highvpres— sure devices. Most extreme of course are electronic devicestcompare today’s microscopic semiconductor junctions within large—scaleintegrated circuits with the huge vacuum tubes of the 19305. LENGTH, SURFACE AREA, AND VOLUME Length, surface, and volume aren’t at all the same kind of thing. Consider a pair of cubic boxes, as in Figure 3.2. If an edge of one is twice as long as that of the other, then the larger one will have not twice but four times the surface area of the smaller. At the same time the larger will have fully eight times the volume of the smaller. Similarly, if one of the cubes has 9 2X 2 Edge length: 1 Surface Area: 6 4 X ' 24 Volume: 1 8 X B Surface/Volume: 6 7/2 X 3 factor of increase FIGU RE 3 . 2. Two‘cubcr, one wit/7 side: twice a! long as those oft/7: other. 7776 bigger basfiur time: the area, erg/9t time: the volume, but only lad/ft/Je swfizce relative to volume of the smaller: edges ten times longer than the other, itlwill have a hundred times as much Surface area and no less than a thousand times as much volume, Put as a general rule, area increases as the square of length (22 = 4; 102 = 100), while volume increases as the cube of length (23 = 8; 103 = 1,000). The rule works for any set of similarly shaped objects, such as spheres or (at least roughly) salmon. When things grow big, volume increases more drastically than does surface area. Therefore, being big means having lots of inside relative to your outside; being small means having lots of outside relative to your inside. ' Biological objects, whether trees, people, or bacteria, don’t vary much 'in density—all are about as dense as water—so mass and weight follow the rules for volume. A fish twice as long as another of the'same shape will weigh about eight times as much; let cooks take notice. Thus variables that follow volume, such as weight, will increase faster than variables that follow surface or length. The consequences aren’t triv- ial. For instance, heat is generated throughout an animal’s insides but is lost across its surface. If two animals, a large one and a small one, pro— duced heat at the same rate (relative to their volumes), the larger one, vol— 43 - The Matterof Magnitude ume rich and surface poor, would be warmer. But body temperature varies little among mammals and birds of all sizes. We larger creatures simply produce less heat (relative to our volumes). We need proportion— ater less food. We can get by with a thinner shell of insulating fur, fat, or feathers. We can also walk or swim about in a cooler climate. Being warm-blooded would have been no enormous accomplishment for a large dinosaur but a lot more remarkable for the. small mammals contemporary with the dinosaurs. Indeed, warm»bloodedness occurs only in animals above a few grams in mass; as a fine convergence, the smallest birds (hummingbirds) are about the same size as the smallest mammals (shrews). Both hummingbirds and shrews are voracious eaters, and at night both let their temperatures drop, essentially hibernating, lest they starve before morning. For warm—blooded aquatic animals the minimum size is still greater. Warm—bloodedness- is no small trick for tiny animals; all that surface makes trouble. : Our technology makes elaborate use of heat~for instance, in fabri— cating materials. But we use large ovens and fabricate in large batches, so the actual energy requirements aren’t thatbad. For an organism only a millimeter or centimeter across, making a' hot spot either internally or externally would be far more costly, relative to its volume. Keeping a large building heated is cheaper, relative to its volume, than is heating a small house. Colonial bees can heat their nests communally; no solitary insect can do so. Nor is this business of heat exchange the only consequence of how the relationships among length, surface area, and volume depend on size. All processes that involve exchange of material with the surroundings are ruled by those relationships—for both better and worse. If you’re small, getting oxygen in and out is relatively easy, even without resort to lungs or gills, but at the same time you’re more vulnerable to chemical assault by predator or pollution since no part of your inside is very far from yoursurface. ' As noted, organisms have to grow in size without serious‘interruption of their functioning. That raises peculiar complications for size—depen- dent variables. An adult can’t be, and in fact isn’t, just an enlarged child, as you can see from Figure 3.3. Consider two people of ordinary corpu— lence. One is tall, and the other short. (It doesn’t matter if the short one is a small adult rather than a youngster.) The weight of each ought to follow the cube of height, at least roughly, and the soles of the feet of each will have to bear that weight. But soles—now we’re talking about an area. If 44 ' (Jars' Paws and Latapurts F I G U at 3 . 5 . An adult and an infizn't of about five mohth: drawn to the same apparent height, each with head, limbs, and torso in correct propor- tion. To emphasize the change in shape. the hahy ha: been given adult posture. the tall person were just an enlarged version of the short one, then the tall one would impose more pounds (weight) on each square inch (area) of those soles. As it happens, though, we’re more subtly made. As investiga— tors at the Nike Research Center found out, tall people have dispropor— tionately large feet, just disproportionate enough to fix the weight per area of sole.6 Of course weight here is a kind of anticipated weight based on the cube of height; if you get Eat, your feet don’t enlarge in compensa— tion, even if they flatten a bit in response. Another example: A falling body falls faster and faster, until its increasing drag just equals its weight and it gets to a final, steady velocity. The surface of a falling object determines its drag while the volume of the object sets the weight that draws it earthward. Since a larger body has more volume for its surface and since weight equals drag at its final veloci— ty, the larger body will fall faster, as you can see from Table 3.1. Falling is mechanically hazardous for a human; it’s a serious danger for a nestling bird only because predators may lurk below. As a great biologist, B. S. 45 - The Matter of Magnitude TABLE 3. 1 . Finalfilling reedrfirrbmofwmr’s demiiZEZglfzéfjfl 1 meter 330 m/s or 738 mph ne , can’t simply assume that 10 centimeter 104 233 drag varies wit/2 the square ‘ 1 centimeter 15 33.6 of speed. I ’1): ignored 1 millimeter 3.6 8.1 tramonic phenomena in the 0.1 millimeter 0.27 0.6 calcubtiam.) Haldane, put it in an essay entitled “On Being the Right Size,” “a mouse is uninjured, a man is broken, a horse splashes.”7 (In a bleak book about coal mining in England, The Road to Wigzm Pier, George Orwell—who later wrote Animal Farm and 1984—wondered about how mice get into the mines: “. . . possibly by falling down the shaft—for they say that a mouse can fall any distance uninjured, owing to its surface area being so large rel- ; ative to its weight.”8 Common knowledge to Orwell and other British ' Socialists at least—Haldane’s essay first appeared in their newspaper.) In short, we big, terrestrial animals live in a gravity—dominated world. For an animal of more ordinary size, gravity matters a lot less. For aquatic animals, it’s of no great gravity at all. _SIZE AND FLIGHT No mechanical feat is more impressive than flight. Nature showed us that flight was possible, something that could not have been self—evident. But while flying animals pointed the way to airplanes, they misled us badly on E the particulars——mainly because the practical problems of flight are strongly size—dependent. Put another way, a flying machine’s size controls both its design and performance.9 First, a flying machine has two separate size—dependent missions, staying aloft and making headway. For a tiny flying insect, staying aloft against gravity’s pull is easy. But making headway against the drag of the air is tougher than for a bird or airplane. At issue, again, is how drag ‘ and weight scale with size. To stay aloft takes an upward force that just counterbalances weight; going forward requires a force equal to drag at that flying speed. Weight depends on volume, while drag depends on surface area. Halving body length reduces weight fully eightfold while reducing drag about fourfold. Thus the smaller creature finds weight less troublesome but drag more so. Thus it flies more slowly and finds W's»... 46 ’ \JaLa Lana u‘nu Vu§urun~u itself more severely affected by any wind—for better (taking advantage of air currents) or'worse (dealing with headwinds and navigational com- plications). ‘ g I Second, a Wing’s lift varies with its area, just as its drag does. Therefore, doubling length (while keeping shape unaltered) gives a~ craft four times the lift but eight times the weight, which doesn’t sound atispi— cious. One solution is having really large wings on the larger craft; anoth— er is to fly somewhat faster: Like drag, lift goes up with speed through the air. So here again, larger ought to mean faster, roughly what we see. among flying animals, from tiny insects to large birds and larger planes. It needn’t (and doesn’t) mean very much faster, .though, since a doubling of ‘speed increases both lift and drag, not twice but-about four times. A fruit fly might hit three miles per hour, a bumblebee can do twelVe or so, only large birds can exceed forty or fifty miles per hour, while for airplanes that’s about the slowest they can go. V A third size-dependent factor complicates design and performance. The best wings produce a lot of lift while suffering little drag. This rela- tionship between lift and drag depends slightly but significantly on size and speed, especially for small wings traveling slowly. Lift relative to drag gets worse as the craft gets smaller. The culprit is a property of fluids, whether liquids or gases, known as viscosity. Put simply, viscosity is a flu— id’s resistance to flowing, its internal stickiness. Its influence gets steadily more pernicious as systems get smaller and slower. The wing of a tiny insect moves through airia bit likea rod pushed through thick syrup; its shape, upon which its lift depends, is considerably obscured by the air carried with it. Very small and slow wings have more drag relative to their lift; here nature is the designer with the harder assignment. Consider gliders, whether living or not. The angle at which a glider descends in still air depends almost entirely on that ratio of lift to drag; maximization of the ratio gives the flattest, most nearly horizontal glide. Birds can’t make glides as flat as sailplanes, simply because they’re smaller, as in Table 3.2. In practical terms, a bird can’t glide as far from a given height. Is human technology better since our wings have less drag for their lift than those of birds? The comparison is so clouded by size that no simple judgment is fair or useful. As gliders, insects are still worse off than birds, and only large ones, such as locusts and butterflies, do much glid- ing at all.10 Gliding through moving air makes matters still murkier. Both human gliders and gliding animals ride air currents to prolong and direct their 47 - The Matter of Magnitude TABLE 3.2. Lam (and that best) angle: of descent fir a variety of gliders. . MINIMUM FLIER GLIDE ANGLE Sailplane l.5° Small airplane, engine off 3.0" Albatross 3.0° Falcon ' 55° Pigeon 95° Monarch butterfly 12 ° Flies, etc. (calculated) - 30 ° flights; it’s called soaring. 50 time aloft may be quite as important as the distance that might be covered in a simple still-air glide. Time aloft depends equally on descent angle and descent speed. Smaller generally means bOth steeper and slower, so while the gliding bird may descend more steeply than the sailplane, it descends more slowly. Thus time aloft is about the same for an eagle and a sailplane. Gliding insects descend still more steeply but more slowly yet, and the monarch butterfly is little worse than bird or plane. , ‘ Size affects the design of flying machines in a fourth way; a subtle one that misled most of our early attempts to fly. Flying animals use beating wings to produce both lift and thrust. Efficient human aircraft (which helicopters and harrier jets are not) divide lift and thrust production between fixed wings and propellers. Should birds emulate our separation of the two functions? Or why can we achieve decent efficiency only by disentangling the two while birds needn’t bother? An aircraft must push air rearward faster than its flying speed in order to keep going forward. At the same time it has to push air downward faster than its ascending to keep up its ascent. But ascent speeds are tiny ' compared with forward speeds; indeed, for most of any flight of bird or airplane. the ascent speed is zero. For level flight, any downward push will make lift, and the greatest efficiency (for a reason besr avoided at this point) is realized when the largest amount of air is given the least down— ward speed. If the craft is going forward, though, it must push air that’s already moving fast. Long, fixed wings deflect a lot of air downward; short propellers spin rapidly, so pushing less air but giving it that neces- sarily greater speed.” The separation of functions buys efficiency—small- er engines and less fuel—for airplanes. But flying animals, being smaller, proceed more slowly and don’t encounter such rapid, oncoming wind. I 48 ' lvats i’aws anti uarapuits M— Thus separating propeller and wings buys little advantage, and flapping wings make both lift and thrust quite nicely. We’ll return to this compari- son in Chapter 10. I SURFACE TENSION AND DIFFUSION If we humans, so large and lumbering, care little about viscosity, we care even less about surface tension and still less about diffusion.- But for the tiny insect that accidentally touches a wing to a puddle of water, surface tension can be a matter of life or death. Nature gives it close attention, making surfaces that either wet easily or vigorously repel water, depend— ing on the roles they play. ' Surface tension comes from mutual attraction among the molecules of a liquid such as water. If molecules attract each other, they prefer to cluster—which is to say that they prefer to form the least amount of outer surface. Surface tension works much like a bunch of people trying to get close together, perhaps to lessen heat loss. A droplet of water tends to take the shape that gives it the least surface for its volume. If the droplet of water rests on a surface to which its molecules are less attracted than they are to each other, it will round up into a flattened sphere, as, for instance, a raindrop on a well-waxed car. In an orbiting spacecraft the droplet will be almost perfectly spherical. If, by contrast, the water molecules are more strongly attracted to the surface than to each other, the drOplet will spread as a thin film over the surface. We see that water rises in a thin glass tube since water and clean glass attract each other strongly; we note that mercury drops in the same tube because of its low attraction to glass. We commonly add a little detergent to water to reduce its surface tension and make it give a more cleansing wash. We notice (usually without knowing why) that an absorbent, say, a ‘ cotton ball or a cloth of natural fiber, compresses as it dries; “absorbent” means that water adheres to it, and the surface—minimizing tendency of water pulls the wet fibers inward as the water evaporates and loses vol— ume. One can use the phenomenon to make a self—starting, small—scale siphon; a glass of water will empty if you drape a piece of cotton fabric from the bottom of the glass up and over its rim and down into an adja— cent, lower sink. Still, such things are far from critical to our daily lives. But for a water Strider, as in- Figure 3.4, surface tension is life itself. Its legs have a waxy coating and don’t attract water; its weight depresses the ' Notice the dimple: in the water’s a i,i.it_444 a t 49 ' lnc matter or magnumut FIGURE 3.4. A water rtrider standing on #26 5103916: of a pond. surfizce under ear/7 leg. water's surface under each leg; the water pushes upward as it tries to flat- ten its surface and thus minimize its area; the water’s surface ends up depressed just enough so that its upward force offsets the water strider’s weight. If the insect lifts two legs off the water, the other four depress their surface'dimples slightly more. In its world surface tension is a major player"; some surface insects move forward by squirting detergent behind. What about us? Why can’t we walk on water? A downward force, weight, has to be balanced by an upward force, surface tension times the length of the fOOt-water-air contact line. The downward force is a matter of a vol- ume, the upward one of a length. We’re so big that we weigh too much for feet of any manageable edge length. My sixty-kilogram mass wOuld require feet with eight thousand meters (five miles) of edge, but. a ten- milligram mosquito—size insect needs a mere millimeter of total foot edge. The downside of being small enough to walk on water is being unable to get through the water’s surface. That surface has about the same . behavior for an insect about a millimeter long as the canvas wall of a tent has for us. We can dive in, we can tow a boat by repeatedly dipping the oars, and we can swim using crawl, butterfly, or backstroke; the tiny insect, thOugh, must remain above or below the surface. Nor is the surface of pond or puddle the only place surface tension matters. Water rises in the conduits of trees and evaporates from leaves. If you stop sucking on a straw, air enters the top, and the soda goes back down, so why doesn’t air go back into the leaves? Surface tension turns out to be criticalifor keeping the air out; the pores in a leaf ’3 cell walls. will let water evaporate out, but they’re just too small for air to get in. Around thirty atmospheres of pressure would be needed to pull air through the __._c_________EQ“__.«_W-______._._.....___,, .. . ...._~._._.-- air—water boundary in pores a ten—thousandth of a millimeter across. Again, the relevant size (here pore diameter) is small, so surface tension can be enlisted for a major role.12 Diffusion comes into play on an even smaller scale. It’s a conse- quence of the endless random and independent wandering of every mol- ecule of every gas or liquid; left alone, substances such as oxygen and nitrogen or fresh and salt water mix together. The usual demonstration of molecular diffusion involves opening a bottle of perfume in a class- room; after a short while everyone smells the aroma. The odorant is spread, it is claimed, by that random wandering of molecules—«to diffu— sion.13 Not so. Except for the tiniest bit next to one’s nasal epithelium, ‘ the perfume has been carried around by the irregular and turbulent motion of air in the room, something totally different from random molecular motion. 30 ubiquitous is such convective motion of air and water that diffusion is almost impossible“ to demonstrate on a perceptual— ly relevant scale. . v But as we go from small to smaller, to subcellular dimensions, diffusion becomes a potent agency of transport and mixing—within ourselves and all other organisms. Impulses most often go from one nerve cell to another by the diffusive spread of a transmitter substance. With a gap between cells of only about one fifty—thousandth of a’ millimeter, the diffusional'delay is about one ten—thousandth of a second. For subcellular distances, diffusion is certainly speedy. Indeed, almost all transport of material within animal- cells takes advantage of diffusion. But the usefulness of diffusion depends drastically on size; a tenfold increase in distance slows diffusive transport a full hundredfold. Animals made up of more than one or a few cells can’t ordinarily rely just on diffusion for moving material within themselves. They have to augment it with hearts, blood vessels, pumped lungs, diges— tive tubes, and other devices that force fluids to move.14 The machines of human technology, much larger than cells, only occasionally make use of diffusion. One blood—cleansing machine used for dialysis of people with kidney failure relies on diffusion in and out of very tiny pipes with a huge aggregate surface area. One famous (or infamous) process takes advantage of the different rates of diffusion of molecules of different sizes. The rare but fissionable uranium 235 moves faster than the common, bigger, and nonfissionable uranium 238 as they diffuse (as gases) through a porous barrier. Being big and impa- tient, we ordinarily resort to stirring and pumping to get around the slowness of diffusion over appreciable distances, doing just what ani— 51 -'l'he Matter of Magnitude mals do when they make circulatory systems. A cell-size creature might well wonder why we bother. ' GRAVITY AND INZERTIA We’ve now noted three phenomena—~viscosity, surface tension, and diffu— sion———especially important in small systems. Others—in particular, gravity 1 and inertia—~dominate large ones. Gravity hasalready raised problems here. 3 It causes large objects to fall faster than small ones. It doesn’t let large crea— tures support themselves with surface tension on water’s surface. And it makes it necessary for large aircraft to fly faster if they're to stay up with decent economy. j‘ Gravity’s size-related mischief takes more subtle forms as well. Consider the moving waves made by wind blowing across a body of water. What keeps them wavy is the water’s inertia. What makes the water flatten out are _ the water’s surface tension and weight. For ripples two—thirds of an inch or less between crests, surface tension is the more important thing flattening the water; its molecules pull together and minimize its surface area. For larg— er waves, weight—gravity—predominates, and water’s preference for flow— ing downward is what flattens the water. The shift makes big waves and small waves behave differently. In particular, the relationship between the size of waves and the speed at which they roll along depends on whether they’re big or small. For big waves, bigger is faster; increase their crest-to- crest distance, or wavelength, fourfold, and waves travel twice as fast. An ordinary boat can’t easily exceed the speed of waves as long as its hull, so a boat four times as long can go twice as fast before the cost of propulsion starts to rise disproportionately: Large ships go faster than small ones, and even small ones go faster than ducks and muskrats. But for tiny ripples, ones less than two—thirds of an inch apart, the rule is just the opposite: Smaller is faster. The world of a minute surface boat, such as a Whirligig beetle, must be something like a freeway with small, fast sports cars and large, slow vans. Inertia, a property of both solids and fluids, is the tendency for some— thing either to remain at rest or to keep moving unless persuaded otherwise by some external force. Put another way, to get an object going takes force, and a moving object exerts a force when it stops. More specifically the force equals the mass of the object times the acceleration or deceleration that alters its motion. What matters here is that the force associated with inertia follows an object’s mass. A massive system can exert a lot of force by stop- ping suddenly. Conversely, bullets must have enormous speeds to offset 52- Votu ALAva uuu unnuruauu their limited masses, and the lower speeds of short—muzzle handguns are commonly offset by using heavier projectiles. Humans have long used stone—ended clubs and metal—headed hammers, sledges, mauls, picks, and axes—heavy things that stop abruptly. Large pieces of metal can be shaped by dropping even larger pieces on them, something of industrial impor- tance for well over a century. A large animal can inflict substantial damage on another by kicking; even a human can injure another by punching. But inertial aggression without weaponry has little value for creatures much smaller than we are; the most pugnacious ants don’t kick their antagonists. Even for us, the effectiveness of a punch depends on the inertia—the mass——of its mark. Kicking your cat isnasty; kicking a mouse is ineffectual. Put the other way around, small things, with less mass, are easier to start and stop—é—to accelerate and decelerate. As noted earlier, all of nature’s jumpers could, if air resistance didn’t matter, achieve about the same height. That implies that their takeoff speeds must be the same. But the short-legged flea achieves that speed in avastly Ishorter distance than does the long—legged kangaroo; the flea’s acceleration is much greater. Bigger may mean higher speed, but bigger also means loWer acceleration, a rule of thumb that works for both living and nonliving systems. Try catching a resting housefly in your hand! The jackrabbit starts faster than the best racehorse and the most violent drag racer. Once started, though, the large mover can coast better than the small one. Stopping a large ship takes miles; a ferry must reverse its engines or it will smash the dock; automo— biles must be equipped with brakes. BUt a swimming microorganism will halt almost instantly—typically in less than its body length. Its surface— dependent drag is relatively huge; its mass—dependent inertia is trivial. Inertia also affects how fluids flow. At small sizes, viscosity predomi— nates, and flows are orderly, laminar affairs. Each bit of fluid does nearly the ‘ same thing as its neighbors. At larger sizes, inertia increasingly offsets viscosi— ty, so the bits of fluid tend to keep doing whatever they have been, despite any different motion of their now more temporary associates; we call such flows, with their chaotic eddying,‘turbulent. (Figure 3.5 illustrates the differ— ence.) Flow immediately around an aircraft or ship is inevitably turbulent; flow around a microorganism is as assuredly laminar. (In between some tun— ing of transition points may be achieved by changes of shape or surface tex— ture.) Turbulent flow really stirs things up; laminar flow is surprisingly inef— fectual at mixing. A microperson’s spoon wouldn’t easily stir milk into coffee. Blood flow in all but the largest vessels of large mammals is laminar; almost all flows in industrial and household plumbing are turbulent. - .. .4- :v. nan-W, “mm—7:» .-.«.-. en-“g‘f«MW-awr‘wre-remwwmfigmwmamrmwwfignfimm‘wm“ i. i. 53 - The Matter of Magnitude \ "K ‘19, ’i ' mm, «7 ~ \.\u.. ~-~\ FIGURE 3.5. A: speed increases, theflow ofa liquid within ana’fiom a pipe thifi‘: fiom heing laminar (ohooe) to turbulent (he/ow). The bigger the pipe, the lower the speed at which the transition occurs. The two regimes don’t just differ in self-stirring; almost every rule for fluid flow comes in two versions. Viscosity is a kind of internal stickiness, and'it causes small objects moving slowly through fluids to carry along a lot of the fluid. Seeds of such plants as dandelions and milkweeds can thus descend slowly by using a bunch of fine hairs as an analog of a para- chute, as in Figure 3.6; the hairs carry along enough air so the bunch behaves like a balloon. But the device scales up'badly since for larger sizes and speeds, viscosity becomes less important than weight and inertia. So neither technology can use the fluffy seed solution for slowing the descent of larger items. For slowing these, rather curiously, the two have gone sep- arate ways. The parachutes used by humans or our bundles of baggage have only fairly crude analogs among terrestrial or arboreal organisms. Nature prefers another design, that of spinning, autogyrating seeds (fruits, strictly) of maples and other trees. While these passive autogyros scale up satisfactorily and have been considered for use by humans, para- chutes have consistently proved handler.15 Physical reality precludes using tufts of fluff if one is large, but it imposes no rigid choice between auto— gyro and parachute. COLUMN AND BEAM If you double the length of a column that supports a roof, how much fat— ter must you make it? Even a casual look at some of the rules used by mechanical engineers reveals still another role of size. These rules must Gen-w.- " FIGURE 3 .6. Three ways to descend more slowly in air: the drag-incrtmingfibers ofa drin— delion seed, the lifl—producing autograting ramara afa maple, and a drag—increasing com/en— tiomzl parachute. apply equally to designs in nature. For a simple example, we’ll consider two circular cylinders (Figure 3.7) made of an ordinary material and car— rying loads that don’t vary over time. Look first at the upright column, a cylinder supporting its own weight and the weight of some load on top. As you might guess, failure by crushing will happen only in a short, fat column. We’ll worry about one long and thin enough so it fails by sudden buckling to one side or another, as when the ends of a piece of dry spaghetti are pushed together. What does it take to start such a collapse? The critical force varies with the fourth power of the column’s diameter divided by- the square of the column’s height. That’s a combination sufficiently hard on the unaided intuition to demand specific numbers. What, then, would happen if we make the column twice as big, doubling both diameter and height? The force that istarts buckling then goes up by 2“ divided by 22, or 16/4, or I fourfold. Swell, a twofold size increase gives a fourfold increase in resis? tance to buckling. ' But it’s really not at all good. If we’re being completely consistent about doubling size, we end up increasing fully eightfold the weight of both the column and whatever loads it. So scaling the entire system up by 55 - The Matter of Magnitude ‘4 FIGURE 3 .7. A cylindrical column wit/2 a load on top; a similar circular cylinder supported near it: ends and loaded in the middle. a factor of two gives a column that is four times stronger, to be sure, but one that must bear eight times the load! At best, the safety factor is halved; at worst, the column breaks. For the larger column to serve as well as the smaller one, it must be fatter. Worse, being fatter, it suffers still greater self-loading, 'which reguires it to be fatter still. At the least, the larger structure will need different proportions, and if the differences in size are very great, it may need a stiffer material or have to be designed differently. Large mammals have stiffer (and thus more fracture-prone) bones than small ones. The daddy longlegs (or harvestman) walks easily on spindly, multiply flexed legs, while the legs of the elephant are straight, substantial columns. The same rule applies to a cylinder serving as a horizontal beam between two supports. And it works whether the load acts at a single point in the middle or uniformly over its length. For the cylinder to bend downward in the same proportion after a doubling of size, something must be altered; its material must be stiffer or its thickness must be more than doubled. Put another way, if the distance between the heam’s sup— 56 ' \J'dto LdVVD auu Unlayultu ports, the beam’s diameter, and the length, width, and height of the load all are doubled, then the beam will sag downward not twice as far but four times as far. Once again, larger is weaker, relatively, whichever tech— nology is in charge. Thus, if the same design is used, the larger bridge Will incur a greater penalty from self—loading; scaled up sufficiently, it will col— lapse from self—loading alone. Elephants are bonier than cats but still must ‘ tread more carefully. These examples are just that—examples, picked from a rich diversity of size—related phenomena. But they Show how severely size affects design, both imposing constraints and affording opportunities. Gravity is impor- tant if you’re big, diffusion if you’re smalldAnd so on. Of particular rele— vance here is the fact that the technology of people must be different from that of nature simply because the two span different size scales. A rule for design may apply to both, but if the rule has in it a size—related factor, the particular way it applies will differ between the two. ...
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This note was uploaded on 08/28/2011 for the course ESE 585 taught by Professor Richardhoward during the Fall '11 term at Purdue.

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Vogel_1998_chpt30 - V034 5< 19 Cats" PAW Moi...

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