Müller’s Micrograph
Crystal Structure
Crystal Power
Density
Elasticity
Tension and Compression
Arches
Scaling
The treatment of the crystalline nature of solids and bonding are very brief in this chapter. More emphasis is on elasticity, tension and compression, and the application to arches. Students should find the section on “Scaling” of particular interest. A fascinating source of additional scaling examples is George Barnes’ fascinating article, “Physics and Size in Biological Systems”—April 1989 issue of The Physics Teacher.
Scaling is becoming enormously important as more devices are being miniaturized. Researchers are finding that when something shrinks enough, whether it is an electronic circuit, motor, film of lubricant, or an individual metal or ceramic crystal, it stops acting like a miniature version of its larger self and starts behaving in new and different ways. Palladium metal, for example, which is normally composed of grains about 1000 nanometers in size, is found to be five times as strong when formed from 5 nanometer grains.
There are 2 OHTs for this chapter: Figures 12.16 and 12.17.
In the Practicing Physics book:
• Scaling
• Scaling Circles
There are problems for this chapter in the Problem Solving in Conceptual Physics student ancillary.
Three activities, one on scaling, one on density, and one on elasticity, and one experiment on Hooke’s law are in the Laboratory Manual.
In the Next-Time Questions book:
• Infant Growth • Material Strength
• Wet Gravel
This chapter may be skipped with no particular consequence to following chapters. If this chapter is skipped and Chapter 13 is assigned, density should be introduced at this time.
SUGGESTED LECTURE PRESENTATION
Crystal Structure: Begin by calling attention to the micrograph held by John Hubiz in the chapter opener photo. You may want to explain how Dr. Müller made the photograph. The micrograph is evidence not only for the crystalline nature of the platinum needle, but also evidence for the wave nature of atoms is seen in the resulting diffraction pattern. It is easy to imagine the micrograph as a ripple tank photo made by grains of sand sprinkled in an orderly mosaic pattern upon the surface of water.
Density: Measure the dimensions of a large wooden cube in cm and find its mass with a pan balance. Define density = mass/volume. (Use the same cube when you discuss flotation in the next chapter.) Some of your students will unfortunately conceptualize density as massiveness or bulkiness rather than massiveness per bulkiness, even when they give a verbal definition properly. This can be helped with the following:
CHECK QUESTIONS: Which has the greater density, a cupful of water or a lakeful of water? A kilogram of lead or a kilogram of feathers? A single uranium atom or the world?
I jokingly relate breaking a candy bar in two and giving the smaller piece to my friend who looks disturbed. “I gave you the same density of candy bar as I have.”
Contrast the density of matter and density of atomic nuclei that comprise so tiny a fraction of space within matter. From about 2 gm/cm3 to 2 ´ 1014 gm/cm3. And in a further crushed state, the interior of neutron stars, about 1016 gm/cm3.
Elasticity:
DEMONSTRATION: Drop glass, steel, rubber, and spheres of various materials onto an anvil and compare the elasticities.
DEMONSTRATION: Hang weights from a spring and illustrate Hooke’s law. Set a pair of identical springs up as in Problem 4, and ask the class to predict the elongation before suspending the load.
Tension, Compression, and Arches: Bend
Compare a cantilever and a simple beam.
Discuss the shape of an I-beam and Exercise 21.
Discuss the strength of arches. Before the time of concrete, stone bridges and the like were self-supporting by virtue of the way they pressed against one another—in an arch shape. Wooden scaffolding allowed their construction, and when the keystone was inserted, the structures stood when the scaffolding was removed. The same practice is used today.
Discuss the catenary, as shown by my grandson Manuel in Figure 12.14. From my understanding, the catenary idea likely originated with Robert Hooke, who discussed it with the famed architect, Christopher Wren. Wren wisely used this idea when he designed the dome to St. Paul ’s Cathedral in London
Area-Volume: Introduce the relationship between area and volume as Chelcie Liu does by showing the following: Have a 500-ml spherical flask filled with colored water sitting on your lecture table. Produce a tall cylindrical flask, also of 500 ml (unknown to your students), and ask for speculations as to how high the water level will be when water is poured into it from the spherical flask. You can ask for a show of hands for those who think that the water will reach more than half the height, and those who think it will fill to less than half the height, and for those who guess it will fill to exactly half the height. Your students will be amazed when they see that the seemingly smaller spherical flask has the same volume as the tall cylinder. To explain, call attention to the fact that the area of the spherical flask is considerably smaller than the surface area of the cylinder. We see a greater area and we unconsciously think that the volume should be greater as well. Be sure to do this. It is more impressive than it may first seem.
Scaling: Now for the most interesting part of your lecture. Have at least 8 large cubes on your lecture table as you explain Figures 12.16 and 12.17. Support figures with further examples as found in the Haldane and Thompson essays (Suggested Reading).
CHECK QUESTIONS: Which has more surface area, an elephant or a mouse? 2000 kilograms of elephant or 2000 kilograms of mice? (Distinguish carefully between these different questions.)
CHECK QUESTION: Cite two reasons why small cars are more affected by wind.
CHECK QUESTION: Why do cooks preparing Chinese food chop food in such small pieces to stir-fry quickly in a wok?
CHECK QUESTION: In terms of surface area to volume, why should parents take extra care that a baby is warm enough in a cold environment.
Your lecture can continue by posing exercises from the chapter end material and having your class volunteer answers. The examples posed in the exercises will perk class interest. (The answer to Problem 8 may need more explanation. How much more surface area is there for a body with twice the volume? Consider a cube; twice the volume means each side is the cube root of two, 1.26 times the side of the smaller cube. Its area is then 1.26 ´ 1.26 = 1.587 times greater than the smaller cube. So the twice as heavy person at the beach would use about 1.6 times as much suntan lotion.)
Regarding Figure 12.19, note that the span from eartip to eartip is almost the height of the elephant. The dense packing of veins and arteries in the elephant’s ears finds a difference in five degrees in blood entering and leaving the ears. A second type of African elephant that resides in cooler forested regions has smaller ears. Perhaps Indian elephants evolved in cooler climates.
Solutions to Chapter 12 Exercises
1. Both the same, for 1000 mg = 1 g.
2. Disagree, for it is the arrangement of atoms and molecules that distinguishes a solid and from a liquid.
3. The carbon that comprises most of the mass of a tree originates from CO2 extracted from the air.
4. Physical properties involve the order, bonding, and structure of atoms that make up a material, and with the presence of other atoms and their interactions in the material. The silicon in glass is amorphous, whereas in semiconductors it is crystalline. Silicon in sand, from which glass is made, is bound to oxygen as silicon dioxide, while that in semiconductor devices is elemental and extremely pure. Hence their physical properties differ.
5. Evidence for crystalline structure include the symmetric diffraction patterns given off by various materials, micrographs such as the one shown by Professor Hubisz in the chapter-opener photo, and even brass doorknobs that have been etched by the perspiration of hands.
6. Density decreases as the volume of the balloon increases.
7. Iron is denser than cork, but not necessarily heavier. A common cork from a wine bottle, for example, is heavier than an iron thumbtack—but it wouldn’t be heavier if the volumes of each were the same.
8. The densities are the same, for they are both samples of iron.
9. Ice is less dense than water.
10. Its density increases.
11. Density has not only to do with the mass of the atoms that make up a material, but with the spacing between the atoms as well. The atoms of the metal osmium, for example, are not as massive as uranium atoms, but due to their close spacing they make up the densest of the metals. Uranium atoms are not as closely spaced as osmium atoms.
12. Aluminum has more volume because it is less dense.
13. Water is denser, so a liter of water weighs more than a liter of ice. (Once a liter of water freezes, its volume is greater than 1 liter.)
14. For one thing, drop both on a steel anvil. The steel ball will bounce higher.
15. The top part of the spring supports the entire weight of the spring and stretches more than, say the middle, which only supports half the weight and stretches half as far. Parts of the spring toward the bottom support very little of the spring’s weight and hardly stretch at all.
16. All parts of the spring would stretch more nearly the same because the lower part of the spring would be supporting nearly as much weight as the upper part is supporting.
17. A twice-as-thick rope has four times the cross-section and is therefore four times as strong. The length of the rope does not contribute to its strength. (Remember the old adage, a chain is only as strong as its weakest link—the strength of the chain has to do with the thickness of the links, not the length of the chain.)
18. The concave side is under compression; the convex side is under tension.
19. Case 1: Tension at the top Case 2: Compression at the top
and compression at the bottom. and tension at the bottom.
20. Concrete undergoes compression well, but not tension. So the steel rods should be in the part of the slab that is under tension, the top part.
21. A horizontal I-beam is stronger when the web is vertical because most of the material is where it is needed for the most strength, in the top and bottom flanges. When supporting a load, one flange will be under tension and the other flange under compression. But when the web is horizontal, only the edges of the flanges, much smaller than the flanges themselves, play these important roles.
22. The design to the left is better because the weight of water against the dam puts compression on the dam. Compression tends to jam the parts of the dam together, with added strength like the compression on an arch. The weight of water puts tension on the dam at the right, which tends to separate the parts of the dam.
23. Like the dams in the preceding exercise, the ends should be concave as on the left. Then the pressure due to the wine inside produces compression on the ends that strengthens rather than weakens the barrel. If the ends are convex as on the right, the pressure due to the wine inside produces tension, which tends to separate the boards that make up the ends.
24. A triangle is the most rigid of geometrical structures. Consider nailing four sticks together to form a rectangle, for example. It doesn’t take much effort to distort the rectangle so that it collapses to form a parallelogram. But a triangle made by nailing three sticks together cannot collapse to form a tighter shape. When strength is important, triangles are used. That’s why you see them in the construction of so many things.
25. Scale a beam up to twice its linear dimensions, I-beam or otherwise, and it will be four times as thick. Along its cross-section then, it will be four times as strong. But it will be eight times as heavy. Four times the strength supporting eight times the weight results in a beam only half as strong as the original beam. The same holds true for a bridge that is scaled up by two. The larger bridge will be only half as strong as the smaller one. (Larger bridges have different designs than smaller bridges. How they differ is what architects and engineers get paid for!) Interestingly, how strength depends on size was one of Galileo’s “two new sciences,” published in 1683.
26. Catenaries make up the arches of the ends of an egg. Pressing them together strengthens the egg. Not so when pressing the sides, which do not constitute catenary shapes, and easily splay outward under pressure.
27. Since each link in a chain is pulled by its neighboring links, tension in the hanging chain is exactly along the chain—parallel to the chain at every point. If the arch takes the same shape, then compression all along the arch will similarly be exactly along the arch—parallel to the arch at every point. There will be no internal forces tending to bend the arch. This shape is a catenary, and is the shape of modern-day arches such as the one that graces the city of St. Louis
28. No, the rods would not be necessary if the shape of the arch were an upside down version of a hanging chain. Why? Because compression of the stones in the semi-circular design press outward. Compression in the hanging chain design (catenary) is everywhere parallel to the arch, with no net sideways components.
29. The candymaker needs less taffy for the larger apples because the surface area is less per kilogram. (This is easily noticed by comparing the peelings of the same number of kilograms of small and large apples.)
30. Kindling will heat to a higher temperature in a shorter time than large sticks and logs. Its greater surface area per mass results in most of its mass being very near the surface, which quickly heats from all sides to its ignition temperature. The heat given to a log, on the other hand, is not so concentrated as it conducts into the greater mass. Large sticks and logs are slower to reach the ignition temperature.
31. The answer to this question uses the same principle as the answer to the previous exercise. The greater surface area of the coal in the form of dust insures an enormously greater proportion of carbon atoms in the coal having exposure to the oxygen in the air. The result is very rapid combustion.
32. More heat is lost from the rambling house due to its greater surface area.
33. An apartment building has less area per dwelling unit exposed to the weather than a single-family unit of the same volume. The smaller area means less heat loss per unit. (It is interesting to see the nearly cubical shapes of apartment buildings in northern climates—a cube has the least surface area for a solid with rectangular sides.)
34. For a given volume, a sphere has less surface area than any other geometrical figure. A dome-shaped structure similarly has less surface area per volume than conventional block designs. Less surface exposed to the climate = less heat loss.
35. The surface area of crushed ice is greater which provides more melting surface to the surroundings.
36. Rolling up presents less surface area to the surroundings.
37. Rusting is a surface phenomenon. For a given mass, iron rods present more surface area to the air than thicker piles.
38. More potato is exposed to the cooking oil when sliced thinly than in larger pieces. Thin fries will therefore cook faster than larger fries.
39. The wider, thinner burger has more surface area for the same volume. The greater the surface area, the greater will be the heat transfer from the stove to the meat.
40. Cupcakes have more surface area per amount of material than a cake, which means there is more area exposed to the heat that the oven will provide, which means cooking will be facilitated. This also means the cupcakes will be overcooked if they are cooked for the time specified for a cake. (Now you see why recipes call for a “shallow pan” or a “deep dish” when baking times are given.)
41. Mittens have less surface than gloves. Anyone who has made mittens and gloves will tell you that much more material is required to make gloves. Hands in gloves will cool faster than hands in mittens. Fingers, toes, and ears have a disproportionately large surface area relative to other parts of the body and are therefore more prone to frostbite.
42. Strength varies in approximate proportion to the area of arms and legs (proportional to the square of the linear dimensions). Weight varies in proportion to the volume of the body (proportional to the cube of the linear dimension). So—other things being equal—the ratio of strength to weight is greater for smaller persons.
43. Small animals radiate more energy per bodyweight, so the flow of blood is correspondingly greater, and the heartbeat faster.
44. As an organism increases in size, surface area decreases relative to the increasing size. Therefore, a large organism such as a human being must have a many-folded intestinal tract so that the area will be large enough to digest the needed food.
45. The inner surface of the lungs is not smooth, but is sponge-like. As a result, there is an enormous surface exposed to the air that is breathed. This is nature’s way of compensating for the proportional decrease in surface area for large bodies. In this way, the adequate amount of oxygen vital to life is taken in.
46. Cells of all creatures have essentially the same upper limit in size dictated by the surface area per volume relationship. The nourishment of all cells takes place through the surface by the process called osmosis. As cells grow they require more nourishment, but the proportional increase in surface area falls behind the increase in mass. The cell overcomes this liability by dividing into two cells. The process is repeated and there is life that takes the form of whales, mice, and us.
47. Large raindrops fall faster than smaller raindrops for the same reason that heavier parachutists fall faster than lighter parachutists. Both larger things have less surface area and therefore less air resistance relative to their weights.
48. A child, for a child has more surface area per volume, and therefore loses disproportionately more water to the air.
49. Scaling plays a significant role in the design of the hummingbird and the eagle. The wings of a hummingbird are smaller than those of the eagle relative to the size of the bird, but are larger relative to the mass of the bird. The hummingbird’s swift maneuvers are possible because the small rotational inertia of the short wings permits rapid flapping that would be impossible for wings as large as those of an eagle. If a hummingbird were scaled up to the size of an eagle, its wings would be much shorter than those of an eagle, so it couldn’t soar. Its customary rate of flapping would be insufficient to provide lift for its disproportionately greater weight. Such a giant hummingbird couldn’t fly, and unless its legs were disproportionately thicker, it would have great difficulty walking. The great difference in the design of hummingbirds and eagles is a natural consequence of the area to volume ratio of scaling. Interesting!
50. The idea of scaling, that one quantity, such as area, changes in a different way than another quantity, such as volume, goes beyond geometry. Rules that work well for a system of one size may be disastrous when applied to a system of a different size. The rules for running a small town well may not work at all for a large city. Other examples are left to you. This is an open-ended question that may provoke thought—or better, discussion.
Chapter 12 Problem Solutions
1. Density = = . Now the volume of a cylinder is its (round area) ´ (its height) (πr2h).
So density = = = 17.7 g/cm3.
2. A cubic meter of cork has a mass of 400 kg and a weight of about 4,000 N. Its weight in pounds is 400 kg ´ 2.2 lb/kg = 880 lb, much too heavy to lift.
3. 45 N is 2.25 times 20 N, so the spring will stretch 2.25 times as far, 9 cm. Or from Hooke’s law; F = kx, x = F/k = 45 N/(20 N/4 cm) = 9 cm. (The spring constant k = 5 N/cm.)
4. When the springs are arranged as in (a), each spring supports half the weight, stretches half as far (2 cm), and reads 5 N. In position (b) each spring supports the full weight, each stretches 4 cm, and each reads 10 N. Both springs stretch 4 cm so the weight pulls the combination down a total distance of 8 cm.
5. If the spring is cut in half, it will stretch as far as half the spring stretched before it was cut—half as much. This is because the tension in the uncut spring is the same everywhere, equal to the full load at the middle as well as the end. So the 10 N load will stretch it 2 cm. (Cutting the spring in half doubles the spring constant k. Initially k = 10 N/4 cm = 2.5 N/cm; when cut in half, k = 10 N/2 cm = 5 N/cm.)
6. If the linear dimensions of a storage tank are reduced to half, then the area is reduced to one-quarter and the volume is reduced to one-eighth. (Compare the cubes in Figure 12.15.) The same general rules apply for any shape, not just cubes.
7. (a) Eight smaller cubes (see Figure 12.15).
(b) Each face of the original cube has an area of 4 cm2 and there are 6 faces, so the total area is 24 cm2. Each of the smaller cubes has an area of 6 cm2 and there are eight of them, so their total surface area is 48 cm2, twice as great.
(c) The surface-to-volume ratio for the original cube is (24 cm2)/(8 cm3) = 3 cm-1. For the set of smaller cubes, it is (48 cm2)/(8 cm3) = 6 cm-1, twice as great. (Notice that the surface-to-volume ratio has the unit inverse cm.)
8. Both have the same density, so the person twice as heavy has twice the volume, but not twice the surface area. The twice-as-heavy person has more area than the smaller person, but less than twice as much. (How much more surface area is there for a body scaled in proportion to twice the volume? Consider the unit cube of the previous problem; twice the volume means each side is the cube root of two (1.26) times the side of the smaller cube. Its area is then 1.26 ´ 1.26 = 1.59 times greater than the smaller cube. So the twice as heavy person at the beach would use about 1.6 times as much suntan lotion.)
9. The big cube will have the same combined volume of the eight little cubes, but half their combined area. The area of each side of the little cubes is 1 cm2, and for its six sides the total area of each little cube is 6 cm2. So all eight individual cubes have a total surface area 48 cm2. The area of each side of the big cube, on the other hand, is 22 or 4 cm2; for all six sides its total surface area is 24 cm2, half as much as the separate small cubes.
10. The big sphere will have twice the diameter of the 1-mm spheres, and will have only one-half as much surface area as the total surface area of the eight little spheres. The scaling principles here are not restricted to spheres, and apply to any shapes that are similar to each other. The simplest shapes to consider are cubes, and with the help of Figure 12.15 we can better understand this exercise by considering the mercury to be in the form of little cubes. In the figure we can see that the combination of eight little cubes form a single cube of twice the linear dimensions. To simplify, suppose each little cube has a unit area for each face—then the total surface area for each cube is 6 units (because it has 6 sides). So all eight cubes have a total surface area of 8 ´ 6 = 48 units. But when they combine, note the big cube formed has 24 units total surface area. So their total surface area after combination is only half what it was before. Likewise if they were spheres. (Now do you see why mice and other creatures ball up in little clusters to reduce their total surface area in cold weather?)
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